Finite Group Databasebeta

← #10764139

Subgroups — #10764139

Order 768.

924 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86205C211yesmaximal, characteristic
2#79294C211yesmaximal, characteristic, complemented
3#79051C211yesmaximal, characteristic, complemented
4C3×#1549C211yesmaximal, characteristic, complemented
5#71773C211yesmaximal, characteristic, complemented
6#70773C211yesmaximal, characteristic, complemented
7#70505C211yesmaximal, characteristic, complemented
8#68430C212yesmaximal, complemented
9#68426C212yesmaximal, complemented
10#68421C212yesmaximal, complemented
11#68418C212yesmaximal, complemented
12#17781—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14#5817—22no—
15#5734—21no—
16#5733C414yes—
17#5732—21no—
18#5730—21no—
19#5728C2212yes—
20#5726—21no—
21#5725C2212yes—
22#5514—22no—
23#5510C412yes—
24#5504C412yes—
25#5477C2211yescharacteristic
26C4×#235C2211yescharacteristic
27C3×#431C2211yescharacteristic
28C3×#423C2211yescharacteristic
29#5127—22no—
30#4925C2211yescharacteristic
31#4924C2211yescharacteristic
32#4837C2212yes—
33#4835C2212yes—
34#4834—21no—
35#4833C2212yes—
36D3×C4⋊C8C2212yes—
37#4831—21no—
38#4729—22no—
39#4688—21no—
40C8×Dic6—21no—
41C3×#332C2212yes—
42C3×#332C2212yes—
43#4547C2212yes—
44#4547C2212yes—
45#4485C2211yescharacteristic
46#4485C2212yes—
47#4485C2211yescharacteristic
48#4481C2211yescharacteristic
49#4481C2212yes—
50#4481C2211yescharacteristic
51#3511—31no—
52#2991—31no—
53#2990—31no—
54#1560—31no—
55#1559—32no—
56#1553—32no—
57#1549D311yescharacteristic, complemented
58#1547—32no—
59#1546—31no—
60#1543—31no—
61#1542—32no—
62C22×C3×Q8—21no—
63C2×#239—22no—
64C2×#235C2×C411yescharacteristic
65C2×#235C2×C412yes—
66C2×C3×M16—21no—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8—21no—
69C3×C4⋊Q8C2×C412yes—
70C3×C4⋊Q8C2312yes—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2×C412yes—
73C2×C3×C42C2311yescharacteristic
74#947—24no—
75#947—22no—
76#946—22no—
77#946—24no—
78Q8×Dic3—22no—
79#930—21no—
80#930—21no—
81#930—24no—
82C4×#67—24no—
83C4×#67D412yes—
84C4×#67D412yes—
85#926C2×C411yescharacteristic
86C2×#208—21no—
87C2×ΓL(1,𝔽25)—22no—
88C2×D3×C8—22no—
89#898—24no—
90#897—22no—
91#897C2×C414yes—
92#895—22no—
93#895—22no—
94#895C2×C414yes—
95#894—24no—
96#893—22no—
97#893C2×C414yes—
98#892—22no—
99#892—22no—
100D3×C4⋊C4—22no—
101#890—22no—
102#887C2×C414yes—
103#875C2×C412yes—
104#874C2311yescharacteristic
105C4×D12—21no—
106C4×D12—21no—
107C4×D12D411yescharacteristic
108C4×D12D411yescharacteristic
109#872C2×C412yes—
110C42×D3C2312yes—
111#869C2311yescharacteristic
112C4×Dic6D411yescharacteristic
113C4×Dic6D411yescharacteristic
114C3×C4⋊C8—21no—
115C3×C4⋊C8C2312yes—
116C3×C4⋊C8C2311yescharacteristic
117C3×C4⋊C8C2311yescharacteristic
118C3×C22⋊C8—21no—
119C3×C4×C8—21no—
120#830—21no—
121(C2×D3)⋊C8—22no—
122(C2×D3)⋊C8—22no—
123#815—22no—
124Dic3⋊C8—22no—
125Dic3⋊C8—22no—
126C8×Dic3—22no—
127C4⋊(C3⋊C8)—21no—
128C4⋊(C3⋊C8)C2312yes—
129C4⋊(C3⋊C8)C2311yescharacteristic
130C4⋊(C3⋊C8)C2311yescharacteristic
131C4×C3⋊C8—21no—
132#556—61no—
133#555—61no—
134#554—61no—
135#553—62no—
136#548—62no—
137#547—61no—
138#542—34no—
139#537—32no—
140#533—32no—
141#532—32no—
142#532—31no—
143C2×C4⋊Q8C2×D311yescharacteristic
144#520—32no—
145C4×#41—31no—
146#446—61no—
147C8×Q8—61no—
148#436—61no—
149#436—62no—
150#435—62no—
151C8×D4—61no—
152#432—32no—
153#431—31no—
154#431—32no—
155#431C2×D311yescharacteristic
156#423—31no—
157#423—32no—
158#423C2×D311yescharacteristic
159C2×C4⋊C8—32no—
160#333—31no—
161#333—32no—
162#333—31no—
163#332—32no—
164#332—32no—
165#332C2×D312yes—
166#332C2×D312yes—
167#331—31no—
168#331—31no—
169#331—32no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—81no—
173C2×C3×Q8—81no—
174C22×C3×C4—22no—
175C22×C3×C4C22×C411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C2×#67—22no—
179C2×#67C22⋊C412yes—
180#239—82no—
181#239—82no—
182#235—42no—
183#235—44no—
184C2×D12—21no—
185C2×D12—21no—
186C2×D12—24no—
187C2×D12C22⋊C411yescharacteristic
188C2×D12C22×C412yes—
189C2×C4×D3—22no—
190C2×C4×D3—24no—
191C2×C4×D3C22×C412yes—
192C2×C4×D3C2×D412yes—
193C2×C4×D3C2×D412yes—
194C2×Dic6C22⋊C411yescharacteristic
195C2×Dic6C22×C412yes—
196C3×M16—81no—
197C2×C3×C8—21no—
198C2×C3×C8—21no—
199C2×C3×C8—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4C22×C414yes—
204C3×C42C22×C411yescharacteristic
205C3×C42C2×D411yescharacteristic
206C3×C42C2×D411yescharacteristic
207C3×C42C2411yescharacteristic
208#217C22⋊C412yes—
209(C2×D3)⋊C4—24no—
210(C2×D3)⋊C4—22no—
211(C2×D3)⋊C4—22no—
212(C2×D3)⋊C4—24no—
213(C2×D3)⋊C4—24no—
214(C2×D3)⋊C4C22⋊C412yes—
215(C2×D3)⋊C4C22⋊C412yes—
216C4⋊Dic3—21no—
217C4⋊Dic3—21no—
218C4⋊Dic3—24no—
219C4⋊Dic3C22⋊C411yescharacteristic
220C4⋊Dic3C22⋊C411yescharacteristic
221Dic3⋊C4—24no—
222Dic3⋊C4C22⋊C412yes—
223Dic3⋊C4C22⋊C412yes—
224C4×Dic3—24no—
225C4×Dic3C22×C412yes—
226C4×Dic3C2×D412yes—
227C4×Dic3C2×D412yes—
228#208—81no—
229C2×C3⋊C8—21no—
230C2×C3⋊C8—21no—
231C2×C3⋊C8—22no—
232ΓL(1,𝔽25)—82no—
233D3×C8—82no—
234C2×#41—31no—
235C2×#41—32no—
236C2×#41—62no—
237C22×Q8—21no—
238C2×M16—21no—
239C2×M16—62no—
240C2×M16—61no—
241C22×C8—62no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—31no—
245C4⋊Q8C4×D312yes—
246C4⋊Q8C22×D312yes—
247#127—31no—
248#126—64no—
249#125—62no—
250#124—32no—
251#124—62no—
252#124—64no—
253#123—64no—
254C22⋊Q8—34no—
255C22⋊Q8—62no—
256C22⋊Q8—64no—
257C22⋊Q8—62no—
258#121—34no—
259#121—62no—
260#121—62no—
261#121—64no—
262#121—61no—
263#121—61no—
264C4×Q8—31no—
265C4×Q8—31no—
266C4×Q8—34no—
267C4×Q8—62no—
268C4×D4—32no—
269C4×D4—32no—
270C4×D4—34no—
271C4×D4—31no—
272C4×D4—31no—
273C4×D4—62no—
274C4×D4—64no—
275C4×D4—61no—
276C4×D4—61no—
277#117—31no—
278#117—32no—
279#117—62no—
280#117—62no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—62no—
283C2×C4⋊C4C4×D312yes—
284C2×C42—32no—
285C2×C42C22×D311yescharacteristic
286C4⋊C8—21no—
287C4⋊C8—31no—
288C4⋊C8—32no—
289C4⋊C8—31no—
290C4⋊C8—62no—
291C4⋊C8—62no—
292C4⋊C8—61no—
293C4⋊C8C22×D312yes—
294C4⋊C8C22×D311yescharacteristic
295C4⋊C8C22×D311yescharacteristic
296C22⋊C8—21no—
297C22⋊C8—62no—
298C22⋊C8—62no—
299C22⋊C8—61no—
300#97—62no—
301C4×C8—21no—
302C4×C8—62no—
303C4×C8—61no—
304C23×C3C2×C22⋊C411yescharacteristic
305C22×D3—22no—
306C22×D3C2×C22⋊C412yes—
307C3×Q8—42no—
308C3×Q8—42no—
309C3×Q8—82no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—82no—
317C2×C3×C4C2×C22⋊C411yescharacteristic
318C2×C3×C4C2×C22⋊C411yescharacteristic
319C2×C3×C4C2×C22⋊C411yescharacteristic
320C2×C3×C4C23×C411yescharacteristic
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C22×D411yescharacteristic
323#67—42no—
324#67—44no—
325C2×Dic3—22no—
326C2×Dic3C2×C22⋊C412yes—
327C2×Dic3C2×C22⋊C412yes—
328C2×Dic3C2×C22⋊C412yes—
329D12—22no—
330D12—42no—
331D12—41no—
332D12—41no—
333D12—41no—
334D12—41no—
335D12—84no—
336C4×D3—22no—
337C4×D3—22no—
338C4×D3—22no—
339C4×D3—22no—
340C4×D3—42no—
341C4×D3—44no—
342C4×D3—84no—
343Dic6—22no—
344Dic6—42no—
345C3×C8—41no—
346C3×C8—42no—
347C3×C8—41no—
348C3⋊C8—42no—
349C3⋊C8—41no—
350C3⋊C8—41no—
351#41—122no—
352#41—124no—
353#41—242no—
354#41—242no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—31no—
358C2×Q8—32no—
359C2×Q8—81no—
360C2×Q8—81no—
361C2×D4—32no—
362C2×D4—31no—
363C2×D4—32no—
364C2×D4—64no—
365C2×D4—62no—
366C2×D4—61no—
367C2×D4—61no—
368C22×C4—22no—
369C22×C4—32no—
370C22×C4—32no—
371C22×C4—32no—
372C22×C4—64no—
373C22×C4—62no—
374C22×C4C2×C4×D311yescharacteristic
375C22×C4D3×D411yescharacteristic
376C22×C4D3×D411yescharacteristic
377M16—81no—
378M16—242no—
379M16—241no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—22no—
383C2×C8—61no—
384C2×C8—61no—
385C2×C8—62no—
386C2×C8—242no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—31no—
391C4⋊C4—31no—
392C4⋊C4—32no—
393C4⋊C4—32no—
394C4⋊C4—64no—
395C4⋊C4—64no—
396C4⋊C4—61no—
397C4⋊C4—61no—
398C4⋊C4C2×C4×D314yes—
399C22⋊C4—32no—
400C22⋊C4—32no—
401C22⋊C4—32no—
402C22⋊C4—64no—
403C22⋊C4—64no—
404C22⋊C4—64no—
405C22⋊C4—62no—
406C22⋊C4—62no—
407C42—32no—
408C42—32no—
409C42—32no—
410C42—64no—
411C42C2×C4×D311yescharacteristic
412C42D3×D411yescharacteristic
413C42D3×D411yescharacteristic
414C42C23×D311yescharacteristic
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3C22×C22⋊C411yescharacteristic
419C2×D3—22no—
420C2×D3—22no—
421C2×D3—22no—
422C2×D3—42no—
423C2×D3—42no—
424C2×D3—42no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—44no—
429C3×C4#41312yes—
430C3×C4#57611yescharacteristic
431C3×C4#57711yescharacteristic
432C3×C4#57711yescharacteristic
433C3×C4#57811yescharacteristic
434Dic3—22no—
435Dic3—22no—
436Dic3—22no—
437Dic3—42no—
438C23—32no—
439C23—62no—
440C23D3×C22⋊C411yescharacteristic
441Q8—42no—
442Q8—42no—
443Q8—62no—
444Q8—82no—
445Q8—122no—
446D4—62no—
447D4—122no—
448D4—122no—
449D4—124no—
450D4—121no—
451D4—121no—
452D4—121no—
453D4—121no—
454D4—244no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—24no—
461C2×C4—32no—
462C2×C4—32no—
463C2×C4—32no—
464C2×C4—62no—
465C2×C4—62no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—62no—
469C2×C4—82no—
470C2×C4—122no—
471C2×C4—124no—
472C2×C4—244no—
473C2×C4D3×C22⋊C411yescharacteristic
474C2×C4D3×C22⋊C411yescharacteristic
475C2×C4D3×C22⋊C411yescharacteristic
476C2×C4C22×C4×D311yescharacteristic
477C2×C4C2×D3×D411yescharacteristic
478C2×C4C2×D3×D411yescharacteristic
479C8—42no—
480C8—41no—
481C8—41no—
482C8—121no—
483C8—121no—
484C8—122no—
485C2×C3—41no—
486C2×C3C2×#41311yescharacteristic
487C2×C3#289711yescharacteristic
488C2×C3#289811yescharacteristic
489D3—42no—
490D3—82no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—62no—
495C22—62no—
496C22—62no—
497C22—122no—
498C22—122no—
499C22—122no—
500C22C2×D3×C22⋊C411yescharacteristic
501C4—21no—
502C4—21no—
503C4—21no—
504C4—44no—
505C4—62no—
506C4—62no—
507C4—62no—
508C4—122no—
509C4#475112yes—
510C4#576211yescharacteristic
511C4#576611yescharacteristic
512C4#577411yescharacteristic
513C4#577511yescharacteristic
514C3#1778111yescharacteristic, complemented, Sylow
515C2—41no—
516C2—122no—
517C2—242no—
518C2C2×#475111yescharacteristic
519C2#7879011yescharacteristic
520C2#7892511yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1549C2
socleC22×C3C22×C22⋊C4
prime radical#86205C2
cosocleC3×C42C24
hypercenter#1549D3
hypocenterC3#17781
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
2. #79294C2semidirect
3. #79051C2semidirect
4. C3×#1549C2semidirect
5. #71773C2semidirect
6. #70773C2semidirect
7. #70505C2semidirect
8. #68430C2semidirect
9. #68426C2semidirect
10. #68421C2semidirect
11. #68418C2semidirect
57. #1549D3semidirect
514. C3#17781semidirect

Maximal subgroups

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

subgroupindex
#862052
#792942
#790512
C3×#15492
#717732
#707732
#705052
#684302
#684262
#684212
#684182
#177813