Finite Group Databasebeta

← #10772476

Subgroups — #10772476

Order 768.

940 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3511C211yesmaximal, characteristic
2#83317C211yesmaximal, characteristic, complemented
3C2×#5079C211yesmaximal, characteristic, complemented
4#82508C211yesmaximal, characteristic, complemented
5#71929C214yesmaximal, complemented
6#71925C214yesmaximal, complemented
7#71802C211yesmaximal, characteristic, complemented
8#71774C211yesmaximal, characteristic, complemented
9#71638C211yesmaximal, characteristic, complemented
10#18637—31nomaximal, Sylow
11C3×#556—21no—
12C3×#555C2212yes—
13C3×#554C2212yes—
14C3×#553—22no—
15C3×#548—22no—
16C3×#547—21no—
17C3×#542—22no—
18C3×#537—21no—
19C3×#533—21no—
20C3×#532—21no—
21C3×#532C2211yescharacteristic
22C2×C3×C4⋊Q8C2211yescharacteristic
23C3×#520—21no—
24C3×C4×#41C2211yescharacteristic
25C2×#869C2211yescharacteristic
26#5232—21no—
27C2×#835—22no—
28#5093—21no—
29#5079—22no—
30#5079C2214yes—
31#5076—21no—
32#5040—22no—
33#5021—21no—
34#5020C2212yes—
35#5019C2214yes—
36#5008—21no—
37#5007C2212yes—
38#4995—21no—
39#4925C2211yescharacteristic
40#4924C2211yescharacteristic
41C2×C4×C3⋊C8C2211yescharacteristic
42#4546C2214yes—
43#4535C2212yes—
44D4⋊(C3⋊C8)C2214yes—
45D4⋊(C3⋊C8)C2212yes—
46#3511D311yescharacteristic
47#3206—31no—
48C2×#487—31no—
49#3079—31no—
50#1646—34no—
51#1643—34no—
52#1548—31no—
53#1492—31no—
54#1476—31no—
55C2×C3×#41—22no—
56C2×C3×#41—21no—
57C2×C3×#41D411yescharacteristic
58C22×C3×Q8D412yes—
59C22×Dic6—21no—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8D412yes—
63C3×C4⋊Q8D411yescharacteristic
64C3×C4⋊Q8C2312yes—
65C3×#127D411yescharacteristic
66C3×#126—24no—
67C3×#125—22no—
68C3×#124—22no—
69C3×#124—24no—
70C3×#124—21no—
71C3×#123—24no—
72C3×C22⋊Q8—22no—
73C3×C22⋊Q8—22no—
74C3×C22⋊Q8—22no—
75C3×C22⋊Q8—24no—
76C3×#121—22no—
77C3×#121—24no—
78C3×#121—22no—
79C3×#121—22no—
80C3×#121—21no—
81C3×#121D412yes—
82C3×C4×Q8—22no—
83C3×C4×Q8—22no—
84C3×C4×Q8D411yescharacteristic
85C3×C4×Q8C2311yescharacteristic
86C3×C4×D4—22no—
87C3×C4×D4—24no—
88C3×C4×D4—22no—
89C3×C4×D4—21no—
90C3×C4×D4—21no—
91C3×C4×D4—21no—
92C3×C4×D4D411yescharacteristic
93C3×C4×D4C2311yescharacteristic
94C3×C4×D4C2312yes—
95C3×#117—22no—
96C3×#117—22no—
97C3×#117—21no—
98C3×#117D411yescharacteristic
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—22no—
102C2×C3×C42—21no—
103C2×C3×C42C2311yescharacteristic
104C2×#214—24no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×#208—21no—
108C22×C3⋊C8—21no—
109#869—21no—
110#869—21no—
111#869—21no—
112#869C2312yes—
113#835—28no—
114#835—22no—
115#832—22no—
116#830—21no—
117#810—24no—
118#808—22no—
119#807—22no—
120C4⋊(C3⋊C8)—21no—
121C4⋊(C3⋊C8)C2312yes—
122C4⋊(C3⋊C8)C2312yes—
123C4×C3⋊C8—21no—
124C4×C3⋊C8—21no—
125C4×C3⋊C8C2312yes—
126#556—21no—
127#555C2×D312yes—
128#554C2×D312yes—
129#553—22no—
130#548—22no—
131#547—21no—
132#542—22no—
133#537—21no—
134#533—21no—
135#532—21no—
136#532C2×D311yescharacteristic
137C2×C4⋊Q8—31no—
138C2×C4⋊Q8C2×D311yescharacteristic
139#520—21no—
140C4×#41C2×D311yescharacteristic
141#492—61no—
142#487—34no—
143#487—62no—
144#487—61no—
145#477—61no—
146#476—61no—
147#475—32no—
148#474—32no—
149#461—34no—
150#448—62no—
151#431—31no—
152#428—61no—
153#423—31no—
154#417—61no—
155C2×#103—62no—
156C2×C4×C8—31no—
157#333—34no—
158Q8⋊C8—32no—
159D4⋊C8—34no—
160D4⋊C8—32no—
161C3×#41—22no—
162C3×#41—22no—
163C3×#41—42no—
164C3×#41—42no—
165C3×#41—84no—
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—44no—
169C2×C3×Q8C2×D411yescharacteristic
170C2×C3×Q8C2×D414yes—
171C2×C3×D4—21no—
172C2×C3×D4—22no—
173C2×C3×D4—24no—
174C2×C3×D4—21no—
175C2×C3×D4—21no—
176C2×C3×D4C2×D411yescharacteristic
177C2×C3×D4C2×D412yes—
178C22×C3×C4—24no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—22no—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×Dic3—41no—
188C2×Dic6—22no—
189C2×Dic6—41no—
190C2×Dic6—82no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4C2×D411yescharacteristic
201C3×C4⋊C4C2×D411yescharacteristic
202C3×C4⋊C4C2×D412yes—
203C3×C22⋊C4—24no—
204C3×C22⋊C4—24no—
205C3×C22⋊C4—24no—
206C3×C22⋊C4—21no—
207C3×C22⋊C4—21no—
208C3×C22⋊C4—22no—
209C3×C22⋊C4—21no—
210C3×C22⋊C4—22no—
211C3×C42—24no—
212C3×C42—21no—
213C3×C42—21no—
214C3×C42—21no—
215C3×C42C2×D411yescharacteristic
216C3×C42C2×D411yescharacteristic
217C3×C42C2×D411yescharacteristic
218C3×C42C2411yescharacteristic
219#214—84no—
220C4⋊Dic3—22no—
221C4⋊Dic3—22no—
222C4⋊Dic3—41no—
223C4⋊Dic3—41no—
224#208—42no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—42no—
229C2×#41—22no—
230C2×#41—21no—
231C2×#41#6711yescharacteristic
232C22×Q8—61no—
233C22×Q8#6712yes—
234C2×SD16—64no—
235C2×M16—61no—
236C22×C8—61no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—32no—
240C4⋊Q8—61no—
241C4⋊Q8—61no—
242C4⋊Q8—61no—
243C4⋊Q8#6712yes—
244C4⋊Q8#6711yescharacteristic
245C4⋊Q8C22×D312yes—
246#127#6711yescharacteristic
247#126—24no—
248#125—22no—
249#124—22no—
250#124—24no—
251#124—21no—
252#123—24no—
253C22⋊Q8—22no—
254C22⋊Q8—22no—
255C22⋊Q8—22no—
256C22⋊Q8—24no—
257#121—22no—
258#121—24no—
259#121—22no—
260#121—22no—
261#121—21no—
262#121#6712yes—
263C4×Q8—22no—
264C4×Q8—22no—
265C4×Q8#6711yescharacteristic
266C4×Q8C22×D311yescharacteristic
267C4×D4—22no—
268C4×D4—24no—
269C4×D4—22no—
270C4×D4—21no—
271C4×D4—21no—
272C4×D4—21no—
273C4×D4#6711yescharacteristic
274C4×D4C22×D311yescharacteristic
275C4×D4C22×D312yes—
276#117—22no—
277#117—22no—
278#117—21no—
279#117#6711yescharacteristic
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C42—21no—
286C2×C42C22×D311yescharacteristic
287#107—62no—
288#106—62no—
289C4⋊C8—32no—
290C4⋊C8—32no—
291C4⋊C8—61no—
292#103—68no—
293#103—62no—
294#103—64no—
295#102—62no—
296C22⋊C8—61no—
297C4×C8—32no—
298C4×C8—61no—
299C4×C8—61no—
300C23×C3—21no—
301C23×C3—22no—
302C23×C3#12011yescharacteristic
303C3×Q8—21no—
304C3×Q8—21no—
305C3×Q8—44no—
306C3×Q8—42no—
307C3×Q8—44no—
308C3×D4—21no—
309C3×D4—21no—
310C3×D4—24no—
311C3×D4—42no—
312C3×D4—44no—
313C3×D4—42no—
314C3×D4—42no—
315C3×D4—84no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—22no—
321C2×C3×C4—21no—
322C2×C3×C4—24no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—82no—
329C2×C3×C4—84no—
330C2×C3×C4#12011yescharacteristic
331C2×C3×C4#12011yescharacteristic
332C2×C3×C4#12011yescharacteristic
333C2×C3×C4#13512yes—
334C2×C3×C4#13712yes—
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C22×D411yescharacteristic
338C2×Dic3—42no—
339C2×Dic3—82no—
340Dic6—44no—
341Dic6—82no—
342C3⋊C8—42no—
343C3⋊C8—42no—
344C3⋊C8—42no—
345#41—22no—
346#41—22no—
347#41—42no—
348#41—42no—
349#41—84no—
350C2×Q8—21no—
351C2×Q8—22no—
352C2×Q8—44no—
353C2×Q8—62no—
354C2×Q8—121no—
355C2×Q8—242no—
356C2×Q8C2×#6711yescharacteristic
357C2×Q8C2×#6714yes—
358C2×D4—21no—
359C2×D4—22no—
360C2×D4—24no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4C2×#6711yescharacteristic
364C2×D4C2×#6712yes—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—121no—
372C22×C4C2×#6711yescharacteristic
373C22×C4C2×#6711yescharacteristic
374C22×C4C2×#6711yescharacteristic
375SD16—244no—
376M16—122no—
377C2×C8—62no—
378C2×C8—62no—
379C2×C8—62no—
380C2×C8—122no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—62no—
391C4⋊C4—62no—
392C4⋊C4—121no—
393C4⋊C4—121no—
394C4⋊C4C2×#6712yes—
395C4⋊C4C2×#6711yescharacteristic
396C4⋊C4C2×#6711yescharacteristic
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—21no—
401C22⋊C4—21no—
402C22⋊C4—22no—
403C22⋊C4—21no—
404C22⋊C4—22no—
405C42—24no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42C2×#6711yescharacteristic
410C42C2×#6711yescharacteristic
411C42C2×#6711yescharacteristic
412C42C23×D311yescharacteristic
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—41no—
416C22×C3—42no—
417C22×C3—41no—
418C22×C3—42no—
419C22×C3—41no—
420C22×C3—42no—
421C22×C3#45212yes—
422C22×C3C2×#12011yescharacteristic
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—41no—
426C3×C4—42no—
427C3×C4—41no—
428C3×C4—41no—
429C3×C4—44no—
430C3×C4#45212yes—
431C3×C4#45512yes—
432C3×C4C2×#13511yescharacteristic
433C3×C4C2×#13711yescharacteristic
434C3×C4#57712yes—
435Dic3—82no—
436C23—21no—
437C23—22no—
438C23#95311yescharacteristic
439Q8—21no—
440Q8—21no—
441Q8—44no—
442Q8—42no—
443Q8—44no—
444Q8—124no—
445Q8—242no—
446D4—21no—
447D4—21no—
448D4—24no—
449D4—42no—
450D4—44no—
451D4—42no—
452D4—42no—
453D4—84no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—24no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—82no—
467C2×C4—84no—
468C2×C4—122no—
469C2×C4—242no—
470C2×C4#95012yes—
471C2×C4#95112yes—
472C2×C4#95311yescharacteristic
473C2×C4#95311yescharacteristic
474C2×C4#95311yescharacteristic
475C2×C4C22×#6711yescharacteristic
476C2×C4C22×#6711yescharacteristic
477C2×C4C22×#6711yescharacteristic
478C8—122no—
479C8—122no—
480C8—122no—
481C2×C3—22no—
482C2×C3—81no—
483C2×C3—82no—
484C2×C3C2×#45211yescharacteristic
485C2×C3#301011yescharacteristic
486C2×C3C2×#45511yescharacteristic
487C22—21no—
488C22—21no—
489C22—41no—
490C22—42no—
491C22—41no—
492C22—42no—
493C22—41no—
494C22—42no—
495C22#523912yes—
496C22C2×#95311yescharacteristic
497C4—21no—
498C4—21no—
499C4—41no—
500C4—42no—
501C4—41no—
502C4—44no—
503C4—41no—
504C4—242no—
505C4#523912yes—
506C4#524212yes—
507C4C2×#95011yescharacteristic
508C4C2×#95111yescharacteristic
509C4#583712yes—
510C3#1863711yescharacteristic, complemented, Sylow
511C2—22no—
512C2—81no—
513C2—82no—
514C2C2×#523911yescharacteristic
515C2C2×#524211yescharacteristic
516C2#8411311yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
Fitting, prime radicalC3×#3511C2
socleC22×C3C2×#120
hypercenter#3511D3
hypocenterC3#18637
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3511

Complements

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

NH ≅ G/Ntype
2. #83317C2semidirect
3. C2×#5079C2semidirect
4. #82508C2semidirect
5. #71929C2semidirect
6. #71925C2semidirect
7. #71802C2semidirect
8. #71774C2semidirect
9. #71638C2semidirect
510. C3#18637semidirect

Maximal subgroups

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

subgroupindex
C3×#35112
#833172
C2×#50792
#825082
#719292
#719252
#718022
#717742
#716382
#186373