Finite Group Databasebeta

← #10823677

Subgroups — #10823677

Order 768.

1038 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3461C211yesmaximal, characteristic
2C4×#951C212yesmaximal, complemented
3#82456C211yesmaximal, characteristic, complemented
4#71905C211yesmaximal, characteristic, complemented
5#71901C211yesmaximal, characteristic, complemented
6#71881C211yesmaximal, characteristic
7#71877C211yesmaximal, characteristic
8#71868C212yesmaximal, complemented
9#71863C212yesmaximal, complemented
10#71643C212yesmaximal, complemented
11C4×#836C211yesmaximal, characteristic, complemented
12#23594—31nomaximal, Sylow
13C3×#544C2211yescharacteristic
14C3×#542—22no—
15C3×#541—22no—
16C3×#539—22no—
17C3×#537C2211yescharacteristic
18C3×#536C2212yes—
19C3×#533—22no—
20C3×#529—22no—
21C3×#529—22no—
22C3×C4×#41C2212yes—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×#951C2212yes—
25C2×#944C2211yescharacteristic
26C2×C4×Dic6C2211yescharacteristic
27C2×#837C2212yes—
28C2×#837C2212yes—
29#5232C2212yes—
30#5229C2211yescharacteristic
31C2×#836C2212yes—
32#5030—22no—
33C4×#216—22no—
34#5018—22no—
35C4×#214—22no—
36#5004C2212yes—
37#4995C2212yes—
38#4978—22no—
39#4977C2211yescharacteristic
40#4973C2211yescharacteristic
41#4933—22no—
42#4932C2211yescharacteristic
43C4×#208C2212yes—
44#4532C2212yes—
45#4532C2212yes—
46#4530C2211yescharacteristic
47#4530C2211yescharacteristic
48#3461D311yescharacteristic
49C4×#137—32no—
50#2327—31no—
51#2061—31no—
52#2040—31no—
53#2038—32no—
54#2028—31no—
55#2007—31no—
56#2005—32no—
57#1759—32no—
58C4×#101—31no—
59C2×C3×#41C2312yes—
60C22×C3×Q8C2311yescharacteristic
61C22×Dic6C2311yescharacteristic
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8—22no—
64C3×#126—24no—
65C3×#126—24no—
66C3×#125—22no—
67C3×#125—24no—
68C3×#125—22no—
69C3×#124—22no—
70C3×#124—22no—
71C3×#123—24no—
72C3×#123—24no—
73C3×C22⋊Q8—24no—
74C3×C22⋊Q8—22no—
75C3×C22⋊Q8—22no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—22no—
78C3×#121—22no—
79C3×#121—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×Q8D412yes—
84C3×C4×Q8D412yes—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×C4×D4D412yes—
89C3×C4×D4D412yes—
90C3×#117—24no—
91C3×#117—22no—
92C3×#117C2312yes—
93C2×C3×C4⋊C4D412yes—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C42—22no—
96C2×C3×C42D412yes—
97C2×C3×C42C2311yescharacteristic
98#951—24no—
99#951—24no—
100#944—22no—
101#944—22no—
102C2×#216—22no—
103C2×#214—22no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—22no—
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3C2311yescharacteristic
109C2×C4×Dic3C2311yescharacteristic
110C2×#208C2312yes—
111C2×#208C2312yes—
112C4×Dic6—22no—
113C4×Dic6—22no—
114#837—24no—
115#837—24no—
116#836—24no—
117#835—22no—
118#832—22no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—22no—
122C3⋊#95—22no—
123C3⋊#95—22no—
124#810—22no—
125#810—22no—
126#808—22no—
127#807—22no—
128#803—22no—
129C4×C3⋊C8—22no—
130C2×#137—32no—
131#544C2×D311yescharacteristic
132#542—22no—
133#541—22no—
134#539—22no—
135#537C2×D311yescharacteristic
136#536C2×D312yes—
137#533—22no—
138C2×C4⋊Q8—31no—
139#529—22no—
140#529—22no—
141C4×#41C2×D312yes—
142C2×C4×Q8—31no—
143C2×C4×Q8C2×D311yescharacteristic
144#441—62no—
145#440—62no—
146C4×Q16—62no—
147C4×SD16—62no—
148#428—32no—
149C2×#104—32no—
150C2×#104—32no—
151#419—32no—
152#417—32no—
153C2×#101—32no—
154C4×M16—32no—
155#391—31no—
156#391—31no—
157#387—62no—
158#387—62no—
159#382—31no—
160#376—31no—
161#343—31no—
162#343—31no—
163#339—32no—
164#339—32no—
165C3×#41—22no—
166C3×#41—22no—
167C3×#41—22no—
168C3×#41—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8C2×D412yes—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4C2×D412yes—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2411yescharacteristic
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C2×Dic6—22no—
191C2×Dic6—22no—
192C2×Dic6—21no—
193C2×Dic6#4112yes—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4C2×D412yes—
204C3×C4⋊C4C2×D412yes—
205C3×C22⋊C4—24no—
206C3×C22⋊C4—24no—
207C3×C22⋊C4—24no—
208C3×C22⋊C4—22no—
209C3×C22⋊C4—22no—
210C3×C22⋊C4—22no—
211C3×C42—22no—
212C3×C42—22no—
213C3×C42—22no—
214C3×C42—22no—
215C3×C42—22no—
216C3×C42C2×D412yes—
217C3×C42C2×D412yes—
218#216—28no—
219#214—28no—
220C4⋊Dic3—22no—
221Dic3⋊C4—28no—
222Dic3⋊C4—41no—
223Dic3⋊C4—41no—
224C4×Dic3—21no—
225C4×Dic3—22no—
226C4×Dic3—42no—
227C4×Dic3#4112yes—
228#208—22no—
229#208—22no—
230#208—22no—
231#208#4114yes—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234C2×#41C22×D312yes—
235C22×Q8—31no—
236C22×Q8C22×D311yescharacteristic
237#137—64no—
238#137—64no—
239C2×Q16—62no—
240C2×SD16—62no—
241C2×M16—32no—
242C2×M16—32no—
243C4⋊Q8—22no—
244C4⋊Q8—22no—
245C4⋊Q8—62no—
246C4⋊Q8—62no—
247#126—24no—
248#126—24no—
249#125—22no—
250#125—24no—
251#125—22no—
252#124—22no—
253#124—22no—
254#123—24no—
255#123—24no—
256C22⋊Q8—24no—
257C22⋊Q8—22no—
258C22⋊Q8—22no—
259C22⋊Q8—22no—
260C22⋊Q8—22no—
261#121—22no—
262#121—22no—
263C4×Q8—22no—
264C4×Q8—22no—
265C4×Q8—22no—
266C4×Q8—62no—
267C4×Q8—62no—
268C4×Q8#6712yes—
269C4×Q8#6712yes—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4#6712yes—
274C4×D4#6712yes—
275#117—24no—
276#117—22no—
277#117C22×D312yes—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—62no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4#6712yes—
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C42—22no—
284C2×C42—31no—
285C2×C42—31no—
286C2×C42—61no—
287C2×C42#6712yes—
288C2×C42C22×D311yescharacteristic
289#107—62no—
290#106—62no—
291#104—64no—
292#104—64no—
293#103—62no—
294#103—62no—
295#103—62no—
296#102—62no—
297#101—64no—
298#97—62no—
299C4×C8—62no—
300#95—61no—
301#95—61no—
302#95—62no—
303#95—62no—
304#95—62no—
305C23×C3—22no—
306C23×C3C22×D411yescharacteristic
307C3×Q8—22no—
308C3×Q8—22no—
309C3×Q8—42no—
310C3×Q8—42no—
311C3×Q8—82no—
312C3×D4—22no—
313C3×D4—22no—
314C3×D4—42no—
315C3×D4—42no—
316C3×D4—82no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—44no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—82no—
331C2×C3×C4#12012yes—
332C2×C3×C4#12011yescharacteristic
333C2×C3×C4#12011yescharacteristic
334C2×C3×C4C22×D412yes—
335C2×C3×C4C2×#4111yescharacteristic
336C2×C3×C4C2×#4112yes—
337C2×C3×C4C2×#4111yescharacteristic
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—41no—
341C2×Dic3—41no—
342C2×Dic3—41no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—42no—
347C2×Dic3—41no—
348C2×Dic3—41no—
349Dic6—24no—
350Dic6—42no—
351C3⋊C8—24no—
352C3⋊C8—42no—
353#41—22no—
354#41—22no—
355#41—22no—
356#41—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—22no—
362C2×Q8—32no—
363C2×Q8—61no—
364C2×Q8—62no—
365C2×Q8—62no—
366C2×Q8C2×#6712yes—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4C2×#6712yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—61no—
374C22×C4—61no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4C2×#6712yes—
378C22×C4C2×#6712yes—
379C22×C4C2×#6711yescharacteristic
380C22×C4C2×#6711yescharacteristic
381C22×C4C23×D311yescharacteristic
382Q16—68no—
383SD16—68no—
384M16—34no—
385M16—62no—
386M16—62no—
387M16—62no—
388C2×C8—62no—
389C2×C8—62no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—68no—
400C4⋊C4—62no—
401C4⋊C4—121no—
402C4⋊C4—121no—
403C4⋊C4C2×#6712yes—
404C4⋊C4C2×#6712yes—
405C22⋊C4—24no—
406C22⋊C4—24no—
407C22⋊C4—24no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—32no—
417C42—61no—
418C42—62no—
419C42—122no—
420C42C2×#6712yes—
421C42C2×#6712yes—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3C2×#12011yescharacteristic
428C22×C3#52511yescharacteristic
429C22×C3#53211yescharacteristic
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—22no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4—42no—
436C3×C4—42no—
437C3×C4—42no—
438C3×C4#52512yes—
439C3×C4#53212yes—
440Dic3—42no—
441Dic3—42no—
442Dic3—81no—
443Dic3—81no—
444C23—22no—
445C23C22×#6711yescharacteristic
446Q8—22no—
447Q8—22no—
448Q8—42no—
449Q8—42no—
450Q8—64no—
451Q8—82no—
452Q8—122no—
453D4—22no—
454D4—22no—
455D4—42no—
456D4—42no—
457D4—82no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—62no—
472C2×C4—62no—
473C2×C4—82no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4—121no—
478C2×C4—121no—
479C2×C4—121no—
480C2×C4—122no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4#95311yescharacteristic
484C2×C4#95311yescharacteristic
485C2×C4#95312yes—
486C2×C4D3×#4111yescharacteristic
487C2×C4D3×#4112yes—
488C2×C4D3×#4111yescharacteristic
489C2×C4C22×#6712yes—
490C8—64no—
491C8—122no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—82no—
495C2×C3#238811yescharacteristic
496C2×C3#302011yescharacteristic
497C2×C3#302211yescharacteristic
498C22—21no—
499C22—21no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22#582811yescharacteristic
504C22#583211yescharacteristic
505C22C2×#95311yescharacteristic
506C4—21no—
507C4—21no—
508C4—22no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—42no—
513C4—42no—
514C4—122no—
515C4—122no—
516C4—241no—
517C4—241no—
518C4#582812yes—
519C4#583212yes—
520C3#2359411yescharacteristic, complemented, Sylow
521C2—21no—
522C2—21no—
523C2—82no—
524C2#8409511yescharacteristic
525C2#8411611yescharacteristic
526C2#8412311yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3461C2
socleC22×C3C2×#120
prime radicalC3×#537C22
hypercenter#3461D3
hypocenterC3#23594
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3461

Complements

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

NH ≅ G/Ntype
2. C4×#951C2semidirect
3. #82456C2semidirect
4. #71905C2semidirect
5. #71901C2semidirect
8. #71868C2semidirect
9. #71863C2semidirect
10. #71643C2semidirect
11. C4×#836C2semidirect
520. C3#23594semidirect

Maximal subgroups

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

subgroupindex
C3×#34612
C4×#9512
#824562
#719052
#719012
#718812
#718772
#718682
#718632
#716432
C4×#8362
#235943