Finite Group Databasebeta

← #10956219

Subgroups — #10956219

Order 768.

5050 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#1010C211yesmaximal, characteristic
2C2×#5516C211yesmaximal, characteristic, complemented
3C22×C4×Dic6C211yesmaximal, characteristic, complemented
4C3×#2297C211yesmaximal, characteristic, complemented
5#84087C211yesmaximal, characteristic, complemented
6#82270C211yesmaximal, characteristic, complemented
7#82070C211yesmaximal, characteristic, complemented
8#76614C214yesmaximal, complemented
9#76301C214yesmaximal, complemented
10#74662C214yesmaximal, complemented
11#74658C214yesmaximal, complemented
12#73454C214yesmaximal, complemented
13C22⋊C4×Dic6C214yesmaximal, complemented
14#34257—31nomaximal, Sylow
15C22×C3×#41C2211yescharacteristic
16C2×#1010—28no—
17C2×#1010—28no—
18C2×#1010C4116yescomplemented
19C22×D3×Q8C2212yes—
20C22×#237C2214yes—
21C22×#237C2212yes—
22C22×#235C2211yescharacteristic
23C22×#235C2212yes—
24C23×Dic6C2212yes—
25C23×Dic6C2211yescharacteristic
26C2×C3×#117C2211yescharacteristic
27C22×C3×C42C2211yescharacteristic
28C2×#926C2211yescharacteristic
29C22×C4⋊Dic3C2211yescharacteristic
30C22×Dic3⋊C4C2212yes—
31C22×C4×Dic3C2212yes—
32#5516—28no—
33C2×#887C2214yes—
34C2×#877C2214yes—
35C2×C4×Dic6—28no—
36C2×C4×Dic6C2214yes—
37C2×C4×Dic6C2214yes—
38C2×C4×Dic6C2214yes—
39C3×#391C2212yes—
40C3×#390C2212yes—
41C3×#386C2212yes—
42C3×#386C2212yes—
43C3×C4×C22⋊C4C2214yes—
44#5229C2212yes—
45#5221C2212yes—
46#5220C2214yes—
47#5213C2212yes—
48#5213C2212yes—
49#5212C2214yes—
50#4987C2214yes—
51#4958C2214yes—
52#4957C2218yes—
53#4957C2218yes—
54#4942C2218yes—
55Dic3×C22⋊C4C2218yes—
56#4939C2212yes—
57#4939C2212yes—
58C4×(C2×D3)⋊C4C2214yes—
59#4929C2212yes—
60#4929C2212yes—
61#4667C2218yes—
62#4666C2218yes—
63#4646C2218yes—
64#4646C2218yes—
65C22×2−1+4—31no—
66C2×#519—31no—
67C22×C4×Q8—31no—
68#2341—34no—
69#2341—34no—
70Q8×C22⋊C4—34no—
71Q8×C22⋊C4—34no—
72#2311—34no—
73#2311—31no—
74#2311—31no—
75#2297—31no—
76#2297—34no—
77#2297D311yescharacteristic, complemented
78C2×C3×#41—24no—
79C2×C3×#41—24no—
80C2×C3×#41C2×C418yes—
81C22×C3×Q8C2311yescharacteristic
82C22×C3×D4C2312yes—
83C22×C3×D4C2311yescharacteristic
84C23×C3×C4C2311yescharacteristic
85C23×C3×C4C2312yes—
86C23×C3×C4C2311yescharacteristic
87C23×C3×C4C2311yescharacteristic
88C22×#67C2314yes—
89C22×#67C2312yes—
90C23×Dic3C2314yes—
91C23×Dic3C2312yes—
92C23×Dic3C2312yes—
93#1010—2128no—
94C2×D3×Q8—28no—
95C2×D3×Q8—28no—
96C2×D3×Q8C2×C4116yescomplemented
97C2×#237—216no—
98C2×#237—216no—
99C2×#237—28no—
100C2×#237—28no—
101C2×#237C2×C4132yes—
102C2×#237C2×C4116yescomplemented
103C2×#235—24no—
104C2×#235—24no—
105C2×#235—28no—
106C2×#235—28no—
107C2×#235C2×C418yes—
108C2×#235C2×C4116yescomplemented
109C22×D12C2311yescharacteristic
110C22×C4×D3C2314yes—
111C22×C4×D3C2312yes—
112C22×Dic6—28no—
113C22×Dic6—28no—
114C22×Dic6C2×C418yes—
115C22×Dic6C2×C4116yescomplemented
116C22×Dic6D418yes—
117C22×Dic6D418yes—
118C22×Dic6C2314yes—
119C22×Dic6C2311yescharacteristic
120C22×Dic6C2312yes—
121C22×Dic6C2312yes—
122C3×#117—24no—
123C2×C3×C4⋊C4C2312yes—
124C2×C3×C22⋊C4C2314yes—
125C2×C3×C22⋊C4C2312yes—
126C2×C3×C22⋊C4C2314yes—
127C2×C3×C42—24no—
128C2×C3×C42C2312yes—
129C2×C3×C42C2312yes—
130C2×C3×C42C2312yes—
131C2×#217C2312yes—
132C2×#217C2318yes—
133C2×(C2×D3)⋊C4C2314yes—
134C2×(C2×D3)⋊C4C2318yes—
135C2×(C2×D3)⋊C4C2314yes—
136#926—24no—
137C2×C4⋊Dic3—24no—
138C2×C4⋊Dic3C2312yes—
139C2×C4⋊Dic3C2312yes—
140C2×C4⋊Dic3C2312yes—
141C2×Dic3⋊C4—28no—
142C2×Dic3⋊C4C2318yes—
143C2×Dic3⋊C4C2318yes—
144C2×C4×Dic3—28no—
145C2×C4×Dic3C2318yes—
146C2×C4×Dic3C2318yes—
147C2×C4×Dic3C2312yes—
148#887—216no—
149#877—216no—
150C4×Dic6—216no—
151C4×Dic6—216no—
152C4×Dic6—216no—
153C3×#95C2314yes—
154C3×#95C2314yes—
155C3⋊#95C2318yes—
156C3⋊#95C2314yes—
157C3⋊#95C2314yes—
158C3⋊#95C2318yes—
159C2×2−1+4—316no—
160C2×2−1+4—68no—
161C2×2−1+4—68no—
162C22×#41—34no—
163C22×#41—32no—
164C22×#41—31no—
165C22×#41—32no—
166C22×#41C2×D311yescharacteristic
167C23×Q8—32no—
168C23×Q8—31no—
169C23×Q8—32no—
170#519—68no—
171C2×C4×Q8—34no—
172C2×C4×Q8—34no—
173C2×C4×Q8—34no—
174C2×C4×Q8—34no—
175C2×C4×Q8—68no—
176C2×#117—31no—
177C2×#117—34no—
178C2×#117C2×D311yescharacteristic
179C22×C4⋊C4—32no—
180C22×C4⋊C4—31no—
181C22×C42—32no—
182C22×C42C2×D311yescharacteristic
183#391—32no—
184#391—32no—
185#391—32no—
186#391—38no—
187#391—38no—
188#391C2×D312yes—
189#390—32no—
190#390—38no—
191#390—38no—
192#390—32no—
193#390—32no—
194#390C2×D312yes—
195#386—34no—
196#386—32no—
197#386—32no—
198#386—34no—
199#386C2×D312yes—
200#386C2×D312yes—
201#380—34no—
202#380—34no—
203#380—38no—
204#380—38no—
205C4×C22⋊C4—38no—
206C4×C22⋊C4—34no—
207C4×C22⋊C4—38no—
208C4×C22⋊C4C2×D314yes—
209C24×C3C2412yes—
210C24×C3C2411yescharacteristic
211C23×D3C2412yes—
212C3×#41—232no—
213C2×C3×Q8—22no—
214C2×C3×Q8—22no—
215C2×C3×Q8C22×C414yes—
216C2×C3×D4—24no—
217C2×C3×D4—24no—
218C2×C3×D4—22no—
219C2×C3×D4—22no—
220C2×C3×D4C22×C418yes—
221C2×C3×D4C22×C414yes—
222C22×C3×C4—22no—
223C22×C3×C4—24no—
224C22×C3×C4—24no—
225C22×C3×C4—24no—
226C22×C3×C4—22no—
227C22×C3×C4—42no—
228C22×C3×C4C22×C418yes—
229C22×C3×C4C22×C414yes—
230C22×C3×C4C2×D414yes—
231C22×C3×C4C2×D414yes—
232C22×C3×C4C2414yes—
233C22×C3×C4C2412yes—
234C22×C3×C4C2412yes—
235C22×C3×C4C2412yes—
236C22×C3×C4C2411yescharacteristic
237C22×C3×C4C2411yescharacteristic
238C2×#67—24no—
239C2×#67—24no—
240C2×#67—28no—
241C2×#67—28no—
242C2×#67C22×C418yes—
243C2×#67C22×C4116yes—
244C22×Dic3—28no—
245C22×Dic3—28no—
246C22×Dic3—24no—
247C22×Dic3—44no—
248C22×Dic3C22×C4116yes—
249C22×Dic3C22×C418yes—
250C22×Dic3C2×D418yes—
251C22×Dic3C2×D418yes—
252C22×Dic3C2414yes—
253C22×Dic3C2414yes—
254C22×Dic3C2412yes—
255C22×Dic3C2414yes—
256D3×Q8—264no—
257#237—2128no—
258#237—264no—
259#235—232no—
260#235—264no—
261C2×D12—22no—
262C2×D12—22no—
263C2×D12C22×C414yes—
264C2×C4×D3—24no—
265C2×C4×D3—24no—
266C2×C4×D3—28no—
267C2×C4×D3—28no—
268C2×C4×D3C22×C418yes—
269C2×C4×D3C22×C4116yes—
270C2×Dic6—22no—
271C2×Dic6—22no—
272C2×Dic6—28no—
273C2×Dic6—28no—
274C2×Dic6—264no—
275C2×Dic6C22⋊C4164yescomplemented
276C2×Dic6C22×C414yes—
277C2×Dic6C22×C4116yes—
278C2×Dic6C22×C418yes—
279C2×Dic6C22×C418yes—
280C2×Dic6C2×D418yes—
281C2×Dic6C2×D418yes—
282C2×Dic6C2×D418yes—
283C2×Dic6C2×D418yes—
284C3×C4⋊C4—24no—
285C3×C22⋊C4—24no—
286C3×C22⋊C4—28no—
287C3×C22⋊C4—44no—
288C3×C42—24no—
289C3×C42—24no—
290C3×C42—24no—
291#217—24no—
292#217—48no—
293(C2×D3)⋊C4—28no—
294(C2×D3)⋊C4—44no—
295(C2×D3)⋊C4—48no—
296C4⋊Dic3—24no—
297C4⋊Dic3—24no—
298C4⋊Dic3—24no—
299Dic3⋊C4—216no—
300Dic3⋊C4—216no—
301C4×Dic3—24no—
302C4×Dic3—216no—
303C4×Dic3—216no—
3042−1+4—6128no—
305C2×#41—24no—
306C2×#41—24no—
307C2×#41—332no—
308C2×#41—316no—
309C2×#41—38no—
310C2×#41—316no—
311C2×#41—68no—
312C2×#41—68no—
313C2×#41—616no—
314C2×#41—616no—
315C2×#41—68no—
316C2×#41—68no—
317C2×#41—64no—
318C2×#41—64no—
319C2×#41C4×D318yes—
320C22×Q8—316no—
321C22×Q8—38no—
322C22×Q8—316no—
323C22×Q8—38no—
324C22×Q8—38no—
325C22×Q8—32no—
326C22×Q8—32no—
327C22×Q8—34no—
328C22×Q8—31no—
329C22×Q8—68no—
330C22×Q8—68no—
331C22×Q8—68no—
332C22×Q8—68no—
333C22×Q8C22×D311yescharacteristic
334C22×D4—32no—
335C22×D4—34no—
336C22×D4—31no—
337C22×D4C22×D312yes—
338C22×D4C22×D311yescharacteristic
339C23×C4—32no—
340C23×C4—34no—
341C23×C4—34no—
342C23×C4—32no—
343C23×C4—32no—
344C23×C4C22×D311yescharacteristic
345C23×C4C22×D312yes—
346C23×C4C22×D311yescharacteristic
347C23×C4C22×D311yescharacteristic
348C4×Q8—616no—
349C4×Q8—616no—
350C4×Q8—616no—
351C4×Q8—616no—
352#117—24no—
353#117—616no—
354#117—64no—
355C2×C4⋊C4—38no—
356C2×C4⋊C4—32no—
357C2×C4⋊C4—32no—
358C2×C4⋊C4—38no—
359C2×C4⋊C4—32no—
360C2×C4⋊C4—68no—
361C2×C4⋊C4—64no—
362C2×C4⋊C4C22×D312yes—
363C2×C22⋊C4—32no—
364C2×C22⋊C4—34no—
365C2×C22⋊C4—38no—
366C2×C22⋊C4—34no—
367C2×C22⋊C4—38no—
368C2×C22⋊C4C22×D312yes—
369C2×C22⋊C4C22×D314yes—
370C2×C22⋊C4C22×D314yes—
371C2×C42—24no—
372C2×C42—38no—
373C2×C42—32no—
374C2×C42—38no—
375C2×C42—68no—
376C2×C42C22×D312yes—
377C2×C42C22×D312yes—
378C2×C42C22×D312yes—
379#95—34no—
380#95—34no—
381#95—38no—
382#95—38no—
383#95C22×D314yes—
384#95C22×D314yes—
385C23×C3—22no—
386C23×C3—22no—
387C23×C3—22no—
388C23×C3—22no—
389C23×C3—24no—
390C23×C3—24no—
391C23×C3C23×C414yes—
392C23×C3C23×C412yes—
393C23×C3C22×D412yes—
394C23×C3C22×D412yes—
395C23×C3C2511yescharacteristic
396C22×D3—22no—
397C22×D3—22no—
398C22×D3—24no—
399C22×D3—24no—
400C22×D3C23×C414yes—
401C3×Q8—28no—
402C3×D4—216no—
403C3×D4—28no—
404C2×C3×C4—28no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—216no—
410C2×C3×C4—24no—
411C2×C3×C4—22no—
412C2×C3×C4—24no—
413C2×C3×C4—24no—
414C2×C3×C4—22no—
415C2×C3×C4—22no—
416C2×C3×C4—44no—
417C2×C3×C4C2×C22⋊C4116yes—
418C2×C3×C4C23×C414yes—
419C2×C3×C4C23×C412yes—
420C2×C3×C4C23×C412yes—
421C2×C3×C4C22×D412yes—
422C2×C3×C4C22×D412yes—
423C2×C3×C4C22×D412yes—
424C2×C3×C4C22×D412yes—
425C2×C3×C42−1+414yescomplemented
426#67—216no—
427#67—232no—
428C2×Dic3—22no—
429C2×Dic3—22no—
430C2×Dic3—232no—
431C2×Dic3—24no—
432C2×Dic3—24no—
433C2×Dic3—24no—
434C2×Dic3—24no—
435C2×Dic3—28no—
436C2×Dic3—48no—
437C2×Dic3C2×C22⋊C4132yes—
438C2×Dic3C23×C414yes—
439C2×Dic3C23×C418yes—
440C2×Dic3C22×D418yes—
441C2×Dic3C22×D418yes—
442D12—28no—
443C4×D3—216no—
444C4×D3—232no—
445Dic6—28no—
446Dic6—232no—
447Dic6C2×C22⋊C4132yes—
448Dic6C2×C22⋊C4132yes—
449C24—32no—
450C24C23×D312yes—
451C24C23×D311yescharacteristic
452#41—232no—
453#41—664no—
454#41—6128no—
455#41—664no—
456#41—632no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—364no—
460C2×Q8—38no—
461C2×Q8—38no—
462C2×Q8—316no—
463C2×Q8—34no—
464C2×Q8—38no—
465C2×Q8—38no—
466C2×Q8—38no—
467C2×Q8—38no—
468C2×Q8—68no—
469C2×Q8—68no—
470C2×Q8—664no—
471C2×Q8—664no—
472C2×Q8—62no—
473C2×Q8—62no—
474C2×Q8C2×C4×D314yes—
475C2×D4—24no—
476C2×D4—24no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—316no—
480C2×D4—38no—
481C2×D4—34no—
482C2×D4—68no—
483C2×D4—68no—
484C2×D4—64no—
485C2×D4—64no—
486C2×D4—62no—
487C2×D4—62no—
488C2×D4C2×C4×D318yes—
489C2×D4C2×C4×D314yes—
490C22×C4—22no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—22no—
495C22×C4—38no—
496C22×C4—38no—
497C22×C4—38no—
498C22×C4—34no—
499C22×C4—316no—
500C22×C4—32no—
501C22×C4—316no—
502C22×C4—38no—
503C22×C4—34no—
504C22×C4—34no—
505C22×C4—42no—
506C22×C4—68no—
507C22×C4—68no—
508C22×C4—68no—
509C22×C4—68no—
510C22×C4—64no—
511C22×C4—64no—
512C22×C4—64no—
513C22×C4—124no—
514C22×C4C2×C4×D318yes—
515C22×C4C2×C4×D314yes—
516C22×C4D3×D414yes—
517C22×C4D3×D414yes—
518C22×C4C23×D314yes—
519C22×C4C23×D312yes—
520C22×C4C23×D312yes—
521C22×C4C23×D312yes—
522C22×C4C23×D311yescharacteristic
523C22×C4C23×D311yescharacteristic
524C4⋊C4—24no—
525C4⋊C4—616no—
526C4⋊C4—616no—
527C4⋊C4—64no—
528C4⋊C4—64no—
529C4⋊C4—64no—
530C22⋊C4—24no—
531C22⋊C4—28no—
532C22⋊C4—44no—
533C22⋊C4—64no—
534C22⋊C4—68no—
535C22⋊C4—128no—
536C22⋊C4—124no—
537C22⋊C4—128no—
538C42—24no—
539C42—24no—
540C42—24no—
541C42—616no—
542C42—616no—
543C42—64no—
544C22×C3—24no—
545C22×C3—22no—
546C22×C3—22no—
547C22×C3—24no—
548C22×C3—24no—
549C22×C3—28no—
550C22×C3—42no—
551C22×C3—42no—
552C22×C3—44no—
553C22×C3C22×C22⋊C414yes—
554C22×C3C24×C411yescharacteristic
555C22×C3C23×D411yescharacteristic
556C22×C3C23×D411yescharacteristic
557C22×C3C2×2−1+412yes—
558C22×C3#58512yes—
559C2×D3—24no—
560C2×D3—28no—
561C2×D3—42no—
562C2×D3—42no—
563C2×D3—44no—
564C3×C4—24no—
565C3×C4—24no—
566C3×C4—42no—
567C3×C4—44no—
568C3×C4C22×C22⋊C414yes—
569C3×C4C22×C22⋊C414yes—
570Dic3—24no—
571Dic3—44no—
572Dic3—44no—
573Dic3C22×C22⋊C4116yes—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—24no—
579C23—24no—
580C23—34no—
581C23—62no—
582C23—62no—
583C23—64no—
584C23—64no—
585C23C22×C4×D314yes—
586C23C22×C4×D312yes—
587C23C2×D3×D412yes—
588C23C2×D3×D412yes—
589C23C24×D311yescharacteristic
590Q8—28no—
591Q8—332no—
592Q8—332no—
593Q8—632no—
594Q8—68no—
595D4—216no—
596D4—28no—
597D4—632no—
598D4—616no—
599D4—68no—
600C2×C4—28no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—216no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—38no—
613C2×C4—38no—
614C2×C4—38no—
615C2×C4—332no—
616C2×C4—34no—
617C2×C4—44no—
618C2×C4—632no—
619C2×C4—632no—
620C2×C4—62no—
621C2×C4—62no—
622C2×C4—616no—
623C2×C4—64no—
624C2×C4—64no—
625C2×C4—64no—
626C2×C4—64no—
627C2×C4—68no—
628C2×C4—128no—
629C2×C4D3×C22⋊C4116yes—
630C2×C4C22×C4×D314yes—
631C2×C4C22×C4×D312yes—
632C2×C4C22×C4×D312yes—
633C2×C4C2×D3×D412yes—
634C2×C4C2×D3×D412yes—
635C2×C4C2×D3×D412yes—
636C2×C4C2×D3×D412yes—
637C2×C4#101014yescomplemented
638C2×C3—24no—
639C2×C3—44no—
640C2×C3C23×C22⋊C411yescharacteristic
641C2×C3C4×2−1+412yes—
642C2×C3#348611yescharacteristic
643C2×C3#348711yescharacteristic
644C2×C3#349712yes—
645D3—44no—
646C22—22no—
647C22—22no—
648C22—24no—
649C22—24no—
650C22—28no—
651C22—24no—
652C22—42no—
653C22—42no—
654C22—44no—
655C22—68no—
656C22—64no—
657C22—122no—
658C22—122no—
659C22—124no—
660C22C2×D3×C22⋊C414yes—
661C22C23×C4×D311yescharacteristic
662C22C22×D3×D411yescharacteristic
663C22C22×D3×D411yescharacteristic
664C22C2×#101012yes—
665C22#596412yes—
666C4—24no—
667C4—24no—
668C4—316no—
669C4—42no—
670C4—44no—
671C4—64no—
672C4—124no—
673C4—124no—
674C4C2×D3×C22⋊C414yes—
675C4C2×D3×C22⋊C414yes—
676C3#3425711yescharacteristic, complemented, Sylow
677C2—24no—
678C2—44no—
679C2—124no—
680C2C22×D3×C22⋊C411yescharacteristic
681C2C4×#101012yes—
682C2#8602211yescharacteristic
683C2#8602311yescharacteristic
684C2#8622212yes—
subgroupG/H
center, FrattiniC23C24×D3
commutatorC22×C3C24×C4
FittingC3×#2297C2
socle, cosocleC23×C3C25
prime radicalC22×#1010C2
hypercenter#2297D3
hypocenterC3#34257
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2297

Complements

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

NH ≅ G/Ntype
2. C2×#5516C2semidirect
3. C22×C4×Dic6C2semidirect
4. C3×#2297C2semidirect
5. #84087C2semidirect
6. #82270C2semidirect
7. #82070C2semidirect
8. #76614C2semidirect
9. #76301C2semidirect
10. #74662C2semidirect
11. #74658C2semidirect
12. #73454C2semidirect
13. C22⋊C4×Dic6C2semidirect
18. C2×#1010C4semidirect
77. #2297D3semidirect
96. C2×D3×Q8C2×C4semidirect
102. C2×#237C2×C4semidirect
108. C2×#235C2×C4semidirect
115. C22×Dic6C2×C4semidirect
275. C2×Dic6C22⋊C4semidirect
425. C2×C3×C42−1+4semidirect
637. C2×C4#1010semidirect
676. C3#34257semidirect

Maximal subgroups

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

subgroupindex
C22×#10102
C2×#55162
C22×C4×Dic62
C3×#22972
#840872
#822702
#820702
#766142
#763012
#746622
#746582
#734542
C22⋊C4×Dic62
#342573