Finite Group Databasebeta

← #11233699

Subgroups — #11233699

Order 768.

2406 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5579C211yesmaximal, characteristic, complemented
2C2×Q8×D12C211yesmaximal, characteristic, complemented
3C3×#2399C211yesmaximal, characteristic, complemented
4#82501C211yesmaximal, characteristic, complemented
5#82501C211yesmaximal, characteristic, complemented
6#77986C214yesmaximal, complemented
7#77952C214yesmaximal, complemented
8#77215C214yesmaximal, complemented
9#77172C214yesmaximal, complemented
10#73713C212yesmaximal, complemented
11#73677C212yesmaximal, complemented
12#73675C212yesmaximal, complemented
13#73653C212yesmaximal
14#73391C211yesmaximal, characteristic, complemented
15#73389C211yesmaximal, characteristic
16#45357—31nomaximal, Sylow
17C22×D3×Q8C2212yes—
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×#946C2212yes—
21#5579—28no—
22Q8×D12—28no—
23C2×#897C2214yes—
24C2×#897C2212yes—
25C2×#896C2214yes—
26C2×#896C2214yes—
27C2×#893C2214yes—
28C2×D3×C4⋊C4C2212yes—
29C2×#874C2211yescharacteristic
30C2×C4×D12C2211yescharacteristic
31C2×C4×D12C2212yes—
32C2×C4×D12C2211yescharacteristic
33C2×C4×D12C2212yes—
34C2×#870C2212yes—
35C2×#869C2212yes—
36C2×#869C2211yescharacteristic
37C3×#398C2212yes—
38C3×#398C2212yes—
39C3×#395C2212yes—
40C3×#395C2212yes—
41C3×C4×C4⋊C4C2212yes—
42C3×C4×C4⋊C4C2212yes—
43C3×#376C2211yescharacteristic
44#5231C2212yes—
45#5231C2212yes—
46#4994C2214yes—
47#4993C2212yes—
48#4993C2212yes—
49#4993C2214yes—
50#4989C2214yes—
51#4989C2214yes—
52#4989C2214yes—
53#4988C2214yes—
54#4982C2214yes—
55#4979C2214yes—
56#4939C2212yes—
57#4939C2212yes—
58#4939C2211yescharacteristic
59#4939C2211yescharacteristic
60#4936C2211yescharacteristic
61#4936C2211yescharacteristic
62C4×C4⋊Dic3C2212yes—
63#4928C2212yes—
64#4928C2212yes—
65#4928C2212yes—
66#4676C2214yes—
67#4673C2214yes—
68#4668C2214yes—
69#4667C2214yes—
70#4665C2214yes—
71#4665C2214yes—
72#4665C2214yes—
73#4665C2214yes—
74#4654C2214yes—
75#4647C2214yes—
76#4646C2214yes—
77#4646C2214yes—
78C2×#554—31no—
79C2×D4×Q8—31no—
80#2732—34no—
81#2692—34no—
82#2580—31no—
83#2580—31no—
84#2550—32no—
85#2454—32no—
86#2452—32no—
87#2449—34no—
88#2409—34no—
89#2399D311yescharacteristic, complemented
90#2366—32no—
91#2331—31no—
92#2328—31no—
93C22×C3×Q8C2311yescharacteristic
94C22×C3×Q8C2311yescharacteristic
95C2×D3×Q8—28no—
96C2×D3×Q8—48no—
97C22×D12C2311yescharacteristic
98C22×D12C2311yescharacteristic
99C22×D12C2314yes—
100C22×C4×D3C2314yes—
101C22×C4×D3C2312yes—
102C22×C4×D3C2312yes—
103C22×C4×D3C2314yes—
104C22×Dic6C2314yes—
105C22×Dic6C2312yes—
106C3×C4×Q8—24no—
107C3×C4×Q8—24no—
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C4⋊C4C2312yes—
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C4⋊C4C2312yes—
112C2×C3×C4⋊C4C2311yescharacteristic
113C2×C3×C4⋊C4C2311yescharacteristic
114C2×C3×C4⋊C4C2312yes—
115C2×C3×C4⋊C4C2312yes—
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2312yes—
118C2×C3×C42C2312yes—
119C2×C3×C42C2311yescharacteristic
120C2×C3×C42C2311yescharacteristic
121#946—28no—
122C2×(C2×D3)⋊C4C2312yes—
123C2×(C2×D3)⋊C4C2312yes—
124C2×(C2×D3)⋊C4C2314yes—
125C2×(C2×D3)⋊C4C2314yes—
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2314yes—
129C2×(C2×D3)⋊C4C2314yes—
130C2×(C2×D3)⋊C4C2314yes—
131C2×(C2×D3)⋊C4C2314yes—
132C2×C4⋊Dic3C2314yes—
133C2×C4⋊Dic3C2312yes—
134C2×C4⋊Dic3C2311yescharacteristic
135C2×C4⋊Dic3C2311yescharacteristic
136C2×C4⋊Dic3C2312yes—
137C2×C4⋊Dic3C2312yes—
138C2×Dic3⋊C4C2314yes—
139C2×Dic3⋊C4C2314yes—
140C2×Dic3⋊C4C2314yes—
141C2×Dic3⋊C4C2314yes—
142C2×Dic3⋊C4C2314yes—
143C2×C4×Dic3C2314yes—
144#897—28no—
145#897—216no—
146#896—216no—
147#896—48no—
148#893—48no—
149D3×C4⋊C4—28no—
150#874Q818yescomplemented
151C4×D12—24no—
152C4×D12—24no—
153C4×D12—28no—
154C4×D12—28no—
155#870—28no—
156#869—28no—
157#869D418yescomplemented
158C3×#95C2312yes—
159C3×#95C2312yes—
160C3×#95C2312yes—
161C3×#95C2312yes—
162C3×#95C2312yes—
163C3×#95C2312yes—
164C3⋊#95C2312yes—
165C3⋊#95C2314yes—
166C3⋊#95C2314yes—
167C3⋊#95C2314yes—
168C3⋊#95C2314yes—
169C3⋊#95C2314yes—
170C3⋊#95C2312yes—
171C23×Q8—32no—
172#554—68no—
173D4×Q8—68no—
174C2×C4⋊Q8—31no—
175C2×C4⋊Q8—32no—
176C2×#127—31no—
177C2×#125—32no—
178C2×C22⋊Q8—32no—
179C2×C22⋊Q8—34no—
180C2×C22⋊Q8—34no—
181C2×C22⋊Q8—32no—
182C2×C22⋊Q8—34no—
183C2×C4×Q8C2×D311yescharacteristic
184C2×C4×Q8C2×D311yescharacteristic
185C2×C4×D4—34no—
186C2×C4×D4—31no—
187C2×C4×D4—31no—
188C2×C4×D4—32no—
189C2×C4×D4—32no—
190C22×C4⋊C4—32no—
191#399—32no—
192#399—32no—
193#399—34no—
194#398—34no—
195#398C2×D312yes—
196#398C2×D312yes—
197#396—34no—
198#396—34no—
199#395—34no—
200#395C2×D312yes—
201#395C2×D312yes—
202#393—32no—
203#393—32no—
204#393—34no—
205#391—34no—
206#391—34no—
207#390—34no—
208#390—32no—
209#390—32no—
210#390—31no—
211#390—31no—
212#389—32no—
213#389—32no—
214#389—32no—
215#389—34no—
216#387—34no—
217#386—34no—
218#386—34no—
219#386—34no—
220#386—34no—
221#385—34no—
222#385—34no—
223#385—34no—
224#385—34no—
225#382—31no—
226#382—31no—
227#380—34no—
228C4×C4⋊C4—32no—
229C4×C4⋊C4C2×D312yes—
230C4×C4⋊C4C2×D312yes—
231#376C2×D311yescharacteristic
232C23×D3C2412yes—
233C23×D3C2412yes—
234C2×C3×Q8—22no—
235C2×C3×Q8—22no—
236C2×C3×Q8—24no—
237C2×C3×Q8—42no—
238C22×C3×C4C2412yes—
239C22×C3×C4C2412yes—
240C22×C3×C4C2412yes—
241C22×C3×C4C2412yes—
242C22×C3×C4C2411yescharacteristic
243C22×C3×C4C2411yescharacteristic
244C22×C3×C4C2412yes—
245C22×C3×C4C2412yes—
246C22×C3×C4C2411yescharacteristic
247C22×Dic3C2414yes—
248C22×Dic3C2412yes—
249C22×Dic3C2412yes—
250C22×Dic3C2414yes—
251D3×Q8—432no—
252C2×D12—22no—
253C2×D12—22no—
254C2×D12—22no—
255C2×D12—22no—
256C2×D12—44no—
257C2×D12—44no—
258C2×D12C2×Q814yes—
259C2×D12C2×Q814yescomplemented
260C2×D12C2×Q8116yescomplemented
261C2×C4×D3—24no—
262C2×C4×D3—24no—
263C2×C4×D3—28no—
264C2×C4×D3—28no—
265C2×C4×D3—28no—
266C2×C4×D3—44no—
267C2×C4×D3—44no—
268C2×C4×D3—44no—
269C2×C4×D3—44no—
270C2×C4×D3—44no—
271C2×Dic6—28no—
272C2×Dic6—28no—
273C2×Dic6—44no—
274C2×Dic6—44no—
275C2×Dic6C2×D418yes—
276C3×C4⋊C4—22no—
277C3×C4⋊C4—24no—
278C3×C4⋊C4—24no—
279C3×C4⋊C4—22no—
280C3×C4⋊C4—24no—
281C3×C4⋊C4—24no—
282C3×C4⋊C4—42no—
283C3×C4⋊C4—42no—
284C3×C42—24no—
285C3×C42—24no—
286C3×C42—22no—
287C3×C42C2×D414yes—
288C3×C42C2×Q814yescomplemented
289(C2×D3)⋊C4—24no—
290(C2×D3)⋊C4—24no—
291(C2×D3)⋊C4—28no—
292(C2×D3)⋊C4—28no—
293(C2×D3)⋊C4—28no—
294(C2×D3)⋊C4—44no—
295(C2×D3)⋊C4—44no—
296(C2×D3)⋊C4—44no—
297(C2×D3)⋊C4—44no—
298(C2×D3)⋊C4—44no—
299C4⋊Dic3—22no—
300C4⋊Dic3—22no—
301C4⋊Dic3—24no—
302C4⋊Dic3—28no—
303C4⋊Dic3C2×D418yes—
304C4⋊Dic3C2×D418yes—
305Dic3⋊C4—28no—
306Dic3⋊C4—28no—
307Dic3⋊C4—44no—
308Dic3⋊C4—44no—
309Dic3⋊C4—44no—
310C4×Dic3—44no—
311C22×Q8—32no—
312C22×Q8—34no—
313C22×Q8—68no—
314C22×Q8—128no—
315C22×Q8C22×D311yescharacteristic
316C22×Q8C22×D311yescharacteristic
317C22×D4—31no—
318C22×D4—31no—
319C22×D4—34no—
320C23×C4—34no—
321C23×C4—34no—
322C23×C4—32no—
323C23×C4—32no—
324C4⋊Q8—38no—
325C4⋊Q8—68no—
326#127—38no—
327#125—68no—
328C22⋊Q8—616no—
329C22⋊Q8—68no—
330C22⋊Q8—616no—
331C22⋊Q8—68no—
332C22⋊Q8—128no—
333C4×Q8—24no—
334C4×Q8—24no—
335C4×D4—68no—
336C4×D4—68no—
337C4×D4—64no—
338C4×D4—64no—
339C4×D4—128no—
340C2×C4⋊C4—32no—
341C2×C4⋊C4—32no—
342C2×C4⋊C4—34no—
343C2×C4⋊C4—31no—
344C2×C4⋊C4—31no—
345C2×C4⋊C4—34no—
346C2×C4⋊C4—34no—
347C2×C4⋊C4—32no—
348C2×C4⋊C4—34no—
349C2×C4⋊C4—34no—
350C2×C4⋊C4—34no—
351C2×C4⋊C4—68no—
352C2×C4⋊C4C22×D311yescharacteristic
353C2×C4⋊C4C22×D312yes—
354C2×C4⋊C4C22×D312yes—
355C2×C4⋊C4C22×D312yes—
356C2×C4⋊C4C22×D311yescharacteristic
357C2×C4⋊C4C22×D312yes—
358C2×C4⋊C4C22×D312yes—
359C2×C4⋊C4C22×D312yes—
360C2×C22⋊C4—32no—
361C2×C22⋊C4—32no—
362C2×C22⋊C4—34no—
363C2×C22⋊C4—34no—
364C2×C22⋊C4—34no—
365C2×C22⋊C4—34no—
366C2×C22⋊C4—34no—
367C2×C22⋊C4—34no—
368C2×C22⋊C4—34no—
369C2×C22⋊C4—34no—
370C2×C42—34no—
371C2×C42C22×D312yes—
372C2×C42C22×D312yes—
373C2×C42C22×D311yescharacteristic
374C2×C42C22×D311yescharacteristic
375C2×C42C22×D311yescharacteristic
376#95—34no—
377#95—34no—
378#95—32no—
379#95—32no—
380#95—34no—
381#95—34no—
382#95—34no—
383#95C22×D312yes—
384#95C22×D312yes—
385#95C22×D312yes—
386#95C22×D312yes—
387#95C22×D312yes—
388#95C22×D312yes—
389C23×C3C2511yescharacteristic
390C22×D3—22no—
391C22×D3—22no—
392C22×D3—24no—
393C22×D3—24no—
394C22×D3—22no—
395C22×D3—22no—
396C22×D3—24no—
397C22×D3—24no—
398C22×D3C22×Q814yes—
399C22×D3C22×Q814yes—
400C3×Q8—28no—
401C3×Q8—44no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—22no—
415C2×C3×C4—22no—
416C2×C3×C4—22no—
417C2×C3×C4—22no—
418C2×C3×C4—22no—
419C2×C3×C4—22no—
420C2×C3×C4C22×D412yes—
421C2×C3×C4C22×D412yes—
422C2×C3×C4C22×D412yes—
423C2×C3×C4C22×Q814yes—
424C2×C3×C4C22×Q812yes—
425C2×C3×C4H(5,𝔽2)14yes—
426C2×C3×C42−1+412yes—
427C2×Dic3—24no—
428C2×Dic3—24no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—24no—
432C2×Dic3—24no—
433C2×Dic3—24no—
434C2×Dic3—24no—
435C2×Dic3—24no—
436C2×Dic3C22×D418yes—
437D12—28no—
438D12—28no—
439D12—416no—
440C4×D3—416no—
441C4×D3—48no—
442C4×D3—48no—
443C4×D3—88no—
444Dic6—48no—
445Dic6—416no—
446C24—32no—
447C24—32no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—24no—
451C2×Q8—38no—
452C2×Q8—42no—
453C2×Q8—68no—
454C2×Q8—68no—
455C2×Q8—124no—
456C2×Q8—124no—
457C2×Q8—1232no—
458C2×D4—34no—
459C2×D4—34no—
460C2×D4—316no—
461C2×D4—62no—
462C2×D4—62no—
463C2×D4—62no—
464C2×D4—62no—
465C2×D4—124no—
466C2×D4—124no—
467C22×C4—34no—
468C22×C4—34no—
469C22×C4—32no—
470C22×C4—32no—
471C22×C4—68no—
472C22×C4—68no—
473C22×C4—64no—
474C22×C4—64no—
475C22×C4—68no—
476C22×C4—124no—
477C22×C4—124no—
478C22×C4—124no—
479C22×C4—124no—
480C22×C4—124no—
481C22×C4C23×D312yes—
482C22×C4C23×D312yes—
483C22×C4C23×D312yes—
484C22×C4C23×D312yes—
485C22×C4C23×D311yescharacteristic
486C22×C4C23×D311yescharacteristic
487C22×C4C23×D312yes—
488C22×C4C23×D312yes—
489C22×C4C23×D311yescharacteristic
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—38no—
497C4⋊C4—38no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—68no—
501C4⋊C4—68no—
502C4⋊C4—68no—
503C4⋊C4—62no—
504C4⋊C4—62no—
505C4⋊C4—64no—
506C4⋊C4—124no—
507C4⋊C4—124no—
508C4⋊C4—124no—
509C22⋊C4—68no—
510C22⋊C4—68no—
511C22⋊C4—64no—
512C22⋊C4—64no—
513C22⋊C4—68no—
514C22⋊C4—124no—
515C22⋊C4—124no—
516C22⋊C4—124no—
517C22⋊C4—124no—
518C22⋊C4—124no—
519C42—24no—
520C42—24no—
521C42—22no—
522C42—124no—
523C42D3×D414yes—
524C42D3×Q814yescomplemented
525C22×C3C23×D411yescharacteristic
526C22×C3C23×Q811yescharacteristic
527C22×C3C2×H(5,𝔽2)12yes—
528C22×C3C2×2−1+411yescharacteristic
529C22×C3#58512yes—
530C2×D3—24no—
531C2×D3—24no—
532C2×D3—28no—
533C2×D3—28no—
534C2×D3—42no—
535C2×D3—42no—
536C2×D3—42no—
537C2×D3—42no—
538C2×D3—44no—
539C2×D3—44no—
540C3×C4—22no—
541C3×C4—24no—
542C3×C4—22no—
543C3×C4—24no—
544C3×C4—42no—
545C3×C4—42no—
546C3×C4—42no—
547C3×C4—42no—
548C3×C4D4×Q814yes—
549Dic3—28no—
550Dic3—42no—
551Dic3—42no—
552Dic3—44no—
553C23—34no—
554C23—34no—
555C23—62no—
556C23—62no—
557C23—62no—
558C23—62no—
559C23—64no—
560C23—64no—
561C23—64no—
562C23—64no—
563C23C24×D311yescharacteristic
564Q8—28no—
565Q8—44no—
566Q8—128no—
567Q8—1216no—
568D4—68no—
569D4—68no—
570D4—1216no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—38no—
590C2×C4—64no—
591C2×C4—64no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—64no—
595C2×C4—64no—
596C2×C4—64no—
597C2×C4—64no—
598C2×C4—64no—
599C2×C4—1216no—
600C2×C4—128no—
601C2×C4—128no—
602C2×C4—248no—
603C2×C4C2×D3×D412yes—
604C2×C4C2×D3×D412yes—
605C2×C4C2×D3×D412yes—
606C2×C4C2×D3×Q814yes—
607C2×C4C2×D3×Q812yes—
608C2×C4#100712yes—
609C2×C4#100914yes—
610C2×C3C2×D4×Q811yescharacteristic
611C2×C3#348411yescharacteristic
612C2×C3#348711yescharacteristic
613C2×C3#350112yes—
614C2×C3#355412yes—
615D3—44no—
616D3—44no—
617C22—68no—
618C22—68no—
619C22—64no—
620C22—64no—
621C22—122no—
622C22—122no—
623C22—124no—
624C22—124no—
625C22—122no—
626C22—122no—
627C22C22×D3×D411yescharacteristic
628C22C22×D3×Q811yescharacteristic
629C22C2×#100711yescharacteristic
630C22C2×#100912yes—
631C22#596412yes—
632C4—22no—
633C4—24no—
634C4—22no—
635C4—24no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—68no—
641C4—122no—
642C4—124no—
643C4—122no—
644C4#572714yes—
645C3#4535711yescharacteristic, complemented, Sylow
646C2—124no—
647C2—124no—
648C2#8602311yescharacteristic
649C2#8603911yescharacteristic
650C2C2×#572711yescharacteristic
651C2#8628012yes—
652C2#8628912yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2399C2
prime radicalC2×#874C22
hypercenter#2399D3
hypocenterC3#45357
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2399

Complements

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

NH ≅ G/Ntype
1. C2×#5579C2semidirect
2. C2×Q8×D12C2semidirect
3. C3×#2399C2semidirect
4. #82501C2semidirect
5. #82501C2semidirect
6. #77986C2semidirect
7. #77952C2semidirect
8. #77215C2semidirect
9. #77172C2semidirect
10. #73713C2semidirect
11. #73677C2semidirect
12. #73675C2semidirect
14. #73391C2semidirect
89. #2399D3semidirect
150. #874Q8semidirect
157. #869D4semidirect
259. C2×D12C2×Q8semidirect
260. C2×D12C2×Q8semidirect
288. C3×C42C2×Q8semidirect
524. C42D3×Q8semidirect
645. C3#45357semidirect

Maximal subgroups

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

subgroupindex
C2×#55792
C2×Q8×D122
C3×#23992
#825012
#779862
#779522
#772152
#771722
#737132
#736772
#736752
#736532
#733912
#733892
#453573