Finite Group Databasebeta

← #10814901

Subgroups — #10814901

Order 768.

710 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2925C211yesmaximal, characteristic, complemented
2#82506C211yesmaximal, characteristic
3#79000C211yesmaximal, characteristic, complemented
4#71646C211yesmaximal, characteristic, complemented
5#70529C211yesmaximal, characteristic, complemented
6#70519C211yesmaximal, characteristic, complemented
7#70467C211yesmaximal, characteristic, complemented
8#70000C211yesmaximal, characteristic
9#69999C211yesmaximal, characteristic
10#69996C211yesmaximal, characteristic, complemented
11#69996C211yesmaximal, characteristic, complemented
12#69900C211yesmaximal, characteristic, complemented
13#69900C211yesmaximal, characteristic, complemented
14#69890C211yesmaximal, characteristic, complemented
15#69889C211yesmaximal, characteristic, complemented
16#22804—31nomaximal, Sylow
17C3×C4×#41C2211yescharacteristic
18C2×#950C2211yescharacteristic
19C4×#235C2211yescharacteristic
20C3×#433—21no—
21C3×#433—21no—
22C3×#432—21no—
23C3×#431—21no—
24C3×#423C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C3×#408C2211yescharacteristic
27C3×#408C2211yescharacteristic
28C3×#405C2211yescharacteristic
29C3×#405C2211yescharacteristic
30C2×#837C2211yescharacteristic
31C2×#837C2211yescharacteristic
32#5233C2211yescharacteristic
33C2×#825C2211yescharacteristic
34C2×#825C2211yescharacteristic
35C2×#822C2211yescharacteristic
36C2×#822C2211yescharacteristic
37C2×#822C2211yescharacteristic
38C2×#822C2211yescharacteristic
39#5110C2211yescharacteristic
40#5110C2211yescharacteristic
41#5101C2211yescharacteristic
42#5101C2211yescharacteristic
43C2×#815C2211yescharacteristic
44C2×C8×Dic3C2211yescharacteristic
45#5030C411yescharacteristic, complemented
46#5030C411yescharacteristic, complemented
47#5026C411yescharacteristic, complemented
48#5026C411yescharacteristic, complemented
49#5018C411yescharacteristic, complemented
50#5018C411yescharacteristic, complemented
51#5014C411yescharacteristic, complemented
52#5014C411yescharacteristic, complemented
53#5002C2211yescharacteristic
54#4998C2211yescharacteristic
55C4×C4⋊Dic3C2211yescharacteristic
56#4924C2211yescharacteristic
57C2×#803C2211yescharacteristic
58#4840—21no—
59#4840—21no—
60#4835—21no—
61#4833—21no—
62#4555C2211yescharacteristic
63#4555C2211yescharacteristic
64#4550C2211yescharacteristic
65#4550C2211yescharacteristic
66#4532C2211yescharacteristic
67#4532C2211yescharacteristic
68#2943—31no—
69#2925—31no—
70#2925D311yescharacteristic, complemented
71#1975—31no—
72#1974—31no—
73#1929—31no—
74#1929—31no—
75#1929—31no—
76#1929—31no—
77#1849—31no—
78#1806—31no—
79#1806—31no—
80#1773—31no—
81#1772—31no—
82#1771—31no—
83C2×C3×#41C2311yescharacteristic
84C2×#235C2311yescharacteristic
85C2×C3×M16C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×C4×Q8C2×C411yescharacteristic
90C3×C4×Q8C2×C411yescharacteristic
91C3×C4×D4—21no—
92C3×C4×D4—21no—
93C3×C4×D4C2×C411yescharacteristic
94C3×C4×D4C2×C411yescharacteristic
95C3×#117—21no—
96C3×#117C2311yescharacteristic
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99#950—22no—
100#950—22no—
101C2×#216C2×C411yescharacteristic
102C2×#215C2×C411yescharacteristic
103C2×#214C2×C411yescharacteristic
104C2×#213C2×C411yescharacteristic
105C4×#67—21no—
106C4×#67—21no—
107#926C2311yescharacteristic
108C2×C4⋊Dic3—21no—
109C2×C4×Dic3C2311yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C2×#208C2311yescharacteristic
112C2×#208C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C4×D12C2×C411yescharacteristic
116C4×D12C2×C411yescharacteristic
117#872—21no—
118C42×D3—21no—
119C4×Dic6C2×C411yescharacteristic
120C4×Dic6C2×C411yescharacteristic
121C3×C4⋊C8—21no—
122C3×C4⋊C8—21no—
123C3×C4⋊C8D411yescharacteristic
124C3×C4⋊C8D411yescharacteristic
125C3×C4⋊C8D411yescharacteristic
126C3×C4⋊C8D411yescharacteristic
127C3×C22⋊C8—21no—
128C3×C22⋊C8—21no—
129C3×C22⋊C8—21no—
130C3×C22⋊C8—21no—
131C3×#97—21no—
132C3×#97—21no—
133C3×C4×C8—21no—
134C3×C4×C8—21no—
135#837—41no—
136#837—41no—
137#835C2×C411yescharacteristic
138#835C2×C411yescharacteristic
139#832C2×C411yescharacteristic
140#832C2×C411yescharacteristic
141C3⋊#95—21no—
142#825—41no—
143#825—41no—
144#822—22no—
145#822—22no—
146#822—41no—
147#822—41no—
148(C2×D3)⋊C8—21no—
149(C2×D3)⋊C8—21no—
150(C2×D3)⋊C8—21no—
151(C2×D3)⋊C8—21no—
152#815—21no—
153#815—21no—
154#815—22no—
155C8×Dic3—21no—
156C8×Dic3—21no—
157C8×Dic3—22no—
158#810C2×C411yescharacteristic
159#810C2×C411yescharacteristic
160#809C2×C411yescharacteristic
161#809C2×C411yescharacteristic
162#808C2×C411yescharacteristic
163#808C2×C411yescharacteristic
164#807C2×C411yescharacteristic
165#807C2×C411yescharacteristic
166C4⋊(C3⋊C8)—21no—
167C4⋊(C3⋊C8)—21no—
168#803C2×C411yescharacteristic
169#803C2×C411yescharacteristic
170#803C2×C411yescharacteristic
171#803C2×C411yescharacteristic
172C2×#135—31no—
173C4×#41—31no—
174C4×#41C2×D311yescharacteristic
175#442—31no—
176#442—31no—
177#441—31no—
178#441—31no—
179#440—31no—
180#440—31no—
181#440—31no—
182#440—31no—
183#433—21no—
184#433—21no—
185#433—61no—
186#433—61no—
187#432—21no—
188#432—61no—
189#431—21no—
190#431—61no—
191C2×#108—31no—
192C2×#108—31no—
193C2×#108—31no—
194C2×#108—31no—
195#427—31no—
196#423—31no—
197#423C2×D311yescharacteristic
198C2×C4⋊C8C2×D311yescharacteristic
199C2×#104—31no—
200C2×#104—31no—
201C2×#104—31no—
202C2×#104—31no—
203#416—31no—
204#416—31no—
205#408—31no—
206#408—31no—
207#408C2×D311yescharacteristic
208#408C2×D311yescharacteristic
209#405—31no—
210#405—31no—
211#405C2×D311yescharacteristic
212#405C2×D311yescharacteristic
213C2×#97—31no—
214C2×#97—31no—
215C2×C4×C8—31no—
216C4×C4⋊C4—31no—
217#339—31no—
218#339—31no—
219#339—31no—
220#339—31no—
221#336—31no—
222#336—31no—
223C3×#41—21no—
224C3×#41—21no—
225C3×#41—21no—
226C3×#41—21no—
227C2×C3×Q8C22×C411yescharacteristic
228C2×C3×D4—21no—
229C2×C3×D4C22×C411yescharacteristic
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4—21no—
233C22×C3×C4#4111yescharacteristic
234C22×C3×C4#4111yescharacteristic
235C22×C3×C4C2411yescharacteristic
236C2×#67—21no—
237C22×Dic3—21no—
238C22×Dic3—21no—
239#235—21no—
240#235—21no—
241#235—21no—
242#235—21no—
243C2×D12C22×C411yescharacteristic
244C2×C4×D3—21no—
245C2×C4×D3—21no—
246C2×C4×D3—21no—
247C2×Dic6C22×C411yescharacteristic
248C3×M16—21no—
249C3×M16—21no—
250C3×M16—21no—
251C3×M16—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C2×C3×C8—21no—
258C2×C3×C8C2×D411yescharacteristic
259C2×C3×C8C2×D411yescharacteristic
260C2×C3×C8C2×D411yescharacteristic
261C2×C3×C8C2×D411yescharacteristic
262C3×C4⋊C4—21no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4C22×C411yescharacteristic
265C3×C4⋊C4C22×C411yescharacteristic
266C3×C22⋊C4—21no—
267C3×C22⋊C4—21no—
268C3×C22⋊C4—21no—
269C3×C42—21no—
270C3×C42—21no—
271C3×C42—21no—
272C3×C42C22×C411yescharacteristic
273C3×C42C22×C411yescharacteristic
274C3×C42C2×D411yescharacteristic
275C3×C42C2×D411yescharacteristic
276#217—21no—
277#216—22no—
278#215—22no—
279#214—22no—
280#213—22no—
281(C2×D3)⋊C4—21no—
282(C2×D3)⋊C4—21no—
283C4⋊Dic3—42no—
284C4⋊Dic3C22×C411yescharacteristic
285C4⋊Dic3C22×C411yescharacteristic
286Dic3⋊C4—21no—
287Dic3⋊C4—21no—
288C4×Dic3—21no—
289C4×Dic3—21no—
290C4×Dic3—21no—
291C4×Dic3—21no—
292C4×Dic3—21no—
293C4×Dic3—21no—
294C4×Dic3—21no—
295#208—21no—
296#208—21no—
297#208—21no—
298#208—21no—
299C2×C3⋊C8—21no—
300C2×C3⋊C8—21no—
301C2×C3⋊C8—21no—
302C2×C3⋊C8—21no—
303C2×C3⋊C8C22×C411yescharacteristic
304C2×C3⋊C8C22×C411yescharacteristic
305C2×C3⋊C8C22×C411yescharacteristic
306C2×C3⋊C8C22×C411yescharacteristic
307C2×C3⋊C8#4112yes—
308C2×C3⋊C8#4112yes—
309C2×#41—31no—
310C2×#41C22×D311yescharacteristic
311#135—62no—
312#135—62no—
313C2×Q16—31no—
314C2×SD16—31no—
315C2×SD16—31no—
316C2×D8—31no—
317C2×M16—31no—
318C2×M16—31no—
319C2×M16C22×D311yescharacteristic
320C2×M16C22×D311yescharacteristic
321C22×C8—31no—
322C22×C8—31no—
323C22×C8C22×D311yescharacteristic
324C22×C8C22×D311yescharacteristic
325C4×Q8—31no—
326C4×Q8—31no—
327C4×Q8C4×D311yescharacteristic
328C4×Q8C4×D311yescharacteristic
329C4×D4—21no—
330C4×D4—21no—
331C4×D4—31no—
332C4×D4—31no—
333C4×D4—61no—
334C4×D4—61no—
335C4×D4C4×D311yescharacteristic
336C4×D4C4×D311yescharacteristic
337#117—21no—
338#117—31no—
339#117—61no—
340#117C22×D311yescharacteristic
341C2×C4⋊C4—61no—
342C2×C42—21no—
343C2×C42—31no—
344C2×C42—31no—
345C2×C42—61no—
346C2×C42C22×D311yescharacteristic
347#108—62no—
348#108—62no—
349#108—121no—
350#108—121no—
351#107—31no—
352#107—31no—
353#106—31no—
354#106—31no—
355C4⋊C8—21no—
356C4⋊C8—21no—
357C4⋊C8—61no—
358C4⋊C8—61no—
359C4⋊C8#6711yescharacteristic
360C4⋊C8#6711yescharacteristic
361C4⋊C8#6711yescharacteristic
362C4⋊C8#6711yescharacteristic
363#104—121no—
364#104—121no—
365#104—121no—
366#104—121no—
367#103—31no—
368#103—31no—
369#103—31no—
370#103—31no—
371#102—31no—
372#102—31no—
373#102—31no—
374#102—31no—
375C22⋊C8—21no—
376C22⋊C8—21no—
377C22⋊C8—21no—
378C22⋊C8—21no—
379C22⋊C8—61no—
380C22⋊C8—61no—
381C22⋊C8—61no—
382C22⋊C8—61no—
383#97—21no—
384#97—21no—
385#97—31no—
386#97—31no—
387#97—31no—
388#97—31no—
389#97—62no—
390#97—61no—
391#97—61no—
392C4×C8—21no—
393C4×C8—21no—
394C4×C8—62no—
395C4×C8—61no—
396C4×C8—61no—
397#95—61no—
398C23×C3—21no—
399C23×C3C2×#4111yescharacteristic
400C22×D3—21no—
401C3×Q8—21no—
402C3×Q8—21no—
403C3×D4—21no—
404C3×D4—21no—
405C3×D4—41no—
406C3×D4—41no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—42no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4—81no—
416C2×C3×C4C4×D411yescharacteristic
417C2×C3×C4C4×D411yescharacteristic
418C2×C3×C4C4×D411yescharacteristic
419C2×C3×C4C4×D411yescharacteristic
420C2×C3×C4C23×C411yescharacteristic
421C2×C3×C4C22×D411yescharacteristic
422C2×C3×C4C2×#4111yescharacteristic
423C2×C3×C4C2×#4111yescharacteristic
424C2×C3×C4H(5,𝔽2)11yescharacteristic
425C2×C3×C42−1+411yescharacteristic
426#67—41no—
427#67—41no—
428C2×Dic3—22no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436C2×Dic3—41no—
437D12—21no—
438D12—21no—
439C4×D3—42no—
440C4×D3—41no—
441C4×D3—41no—
442C4×D3—81no—
443Dic6—21no—
444Dic6—21no—
445C3×C8—21no—
446C3×C8—21no—
447C3×C8—21no—
448C3×C8—21no—
449C3×C8—41no—
450C3×C8—41no—
451C3⋊C8—21no—
452C3⋊C8—21no—
453C3⋊C8—41no—
454C3⋊C8—41no—
455C3⋊C8#11712yes—
456C3⋊C8#11712yes—
457#41—21no—
458#41—21no—
459#41—21no—
460#41—21no—
461#41—61no—
462#41—61no—
463#41—61no—
464#41—61no—
465C2×Q8—31no—
466C2×Q8C2×C4×D311yescharacteristic
467C2×D4—21no—
468C2×D4—31no—
469C2×D4—61no—
470C2×D4C2×C4×D311yescharacteristic
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4#23511yescharacteristic
480C22×C4#23511yescharacteristic
481C22×C4C23×D311yescharacteristic
482Q16—62no—
483SD16—62no—
484SD16—62no—
485D8—62no—
486M16—21no—
487M16—21no—
488M16—21no—
489M16—21no—
490M16—61no—
491M16—61no—
492M16—61no—
493M16—61no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—31no—
501C2×C8—31no—
502C2×C8—32no—
503C2×C8—32no—
504C2×C8—31no—
505C2×C8—31no—
506C2×C8—61no—
507C2×C8—61no—
508C2×C8—61no—
509C2×C8—61no—
510C2×C8C2×#6711yescharacteristic
511C2×C8C2×#6711yescharacteristic
512C2×C8C2×#6711yescharacteristic
513C2×C8C2×#6711yescharacteristic
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—31no—
517C4⋊C4—31no—
518C4⋊C4—61no—
519C4⋊C4—61no—
520C4⋊C4—122no—
521C4⋊C4C2×C4×D311yescharacteristic
522C4⋊C4C2×C4×D311yescharacteristic
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—61no—
527C22⋊C4—61no—
528C22⋊C4—61no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—61no—
533C42—61no—
534C42—61no—
535C42—61no—
536C42—61no—
537C42—61no—
538C42—61no—
539C42C2×C4×D311yescharacteristic
540C42C2×C4×D311yescharacteristic
541C42C2×#6711yescharacteristic
542C42C2×#6711yescharacteristic
543C22×C3—21no—
544C22×C3—21no—
545C22×C3—41no—
546C22×C3—41no—
547C22×C3—41no—
548C22×C3C2×C4×D411yescharacteristic
549C22×C3#55511yescharacteristic
550C22×C3#55611yescharacteristic
551C2×D3—41no—
552C2×D3—41no—
553C2×D3—41no—
554C3×C4—21no—
555C3×C4—21no—
556C3×C4—21no—
557C3×C4—21no—
558C3×C4—41no—
559C3×C4—41no—
560C3×C4—41no—
561C3×C4C2×#11711yescharacteristic
562C3×C4#52011yescharacteristic
563C3×C4#54711yescharacteristic
564C3×C4#55011yescharacteristic
565Dic3—42no—
566Dic3—41no—
567Dic3—41no—
568Dic3—41no—
569Dic3—81no—
570C23—21no—
571C23—61no—
572C23C2×#23511yescharacteristic
573Q8—21no—
574Q8—21no—
575Q8—61no—
576Q8—61no—
577D4—21no—
578D4—21no—
579D4—41no—
580D4—41no—
581D4—61no—
582D4—61no—
583D4—121no—
584D4—121no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—62no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—81no—
598C2×C4—122no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—241no—
607C2×C4C4×#6711yescharacteristic
608C2×C4C4×#6711yescharacteristic
609C2×C4C4×#6711yescharacteristic
610C2×C4C4×#6711yescharacteristic
611C2×C4C22×C4×D311yescharacteristic
612C2×C4D3×#4111yescharacteristic
613C2×C4D3×#4111yescharacteristic
614C2×C4#100911yescharacteristic
615C2×C4#101011yescharacteristic
616C2×C4C22×#6711yescharacteristic
617C8—21no—
618C8—21no—
619C8—21no—
620C8—21no—
621C8—32no—
622C8—32no—
623C8—41no—
624C8—41no—
625C8—61no—
626C8—61no—
627C8—121no—
628C8—121no—
629C2×C3—21no—
630C2×C3—21no—
631C2×C3—81no—
632C2×C3#235411yescharacteristic
633C2×C3#295511yescharacteristic
634C2×C3#295611yescharacteristic
635D3—81no—
636C22—21no—
637C22—21no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—121no—
642C22—121no—
643C22—121no—
644C22#553211yescharacteristic
645C22#553311yescharacteristic
646C22C2×C4×#6711yescharacteristic
647C4—21no—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—41no—
652C4—41no—
653C4—41no—
654C4—122no—
655C4—121no—
656C4—121no—
657C4—121no—
658C4—241no—
659C4D3×#11711yescharacteristic
660C4#552311yescharacteristic
661C4#582411yescharacteristic
662C4#583411yescharacteristic
663C3#2280411yescharacteristic, complemented, Sylow
664C2—21no—
665C2—21no—
666C2—81no—
667C2—241no—
668C2#8224111yescharacteristic
669C2#8379211yescharacteristic
670C2#8381111yescharacteristic
subgroupG/H
centerC2×C4C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2925C2
socleC22×C3C2×C4×D4
prime radicalC2×#950C22
cosocleC22×C3×C4C24
hypercenter#2925D3
hypocenterC3#22804
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ #2925

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#2925C2semidirect
3. #79000C2semidirect
4. #71646C2semidirect
5. #70529C2semidirect
6. #70519C2semidirect
7. #70467C2semidirect
10. #69996C2semidirect
11. #69996C2semidirect
12. #69900C2semidirect
13. #69900C2semidirect
14. #69890C2semidirect
15. #69889C2semidirect
45. #5030C4semidirect
46. #5030C4semidirect
47. #5026C4semidirect
48. #5026C4semidirect
49. #5018C4semidirect
50. #5018C4semidirect
51. #5014C4semidirect
52. #5014C4semidirect
70. #2925D3semidirect
663. C3#22804semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#29252
#825062
#790002
#716462
#705292
#705192
#704672
#700002
#699992
#699962
#699002
#698902
#698892
#228043