Finite Group Databasebeta

← #10690280

Subgroups — #10690280

Order 768.

482 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79039C211yesmaximal, characteristic
2#73668C211yesmaximal, characteristic
3#70782C211yesmaximal, characteristic
4#70770C211yesmaximal, characteristic, complemented
5#70500C211yesmaximal, characteristic, complemented
6C3×#1303C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#9034—31nomaximal, Sylow
9C2×#872C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#5126—21no—
15Dic3×M16—21no—
16#4994—21no—
17#4981—21no—
18C4×(C2×D3)⋊C4—21no—
19C4×C4⋊Dic3—21no—
20#4925C2211yescharacteristic
21#4924C2211yescharacteristic
22#4840—22no—
23#4840—22no—
24#4677C411yescharacteristic
25#4673C411yescharacteristic
26#4659C411yescharacteristic
27#4656C411yescharacteristic
28#4484—22no—
29#4469—21no—
30#4468—21no—
31#4456—22no—
32#2921—31no—
33#2396—31no—
34#1487—31no—
35#1487—31no—
36#1458—31no—
37#1303—31no—
38#1303D311yescharacteristic, complemented
39C22×C4×D3D411yescharacteristic
40C22×Dic6C2×C411yescharacteristic
41C2×C3×M16—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C42—21no—
44C2×C3×C42C2311yescharacteristic
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4C2×C411yescharacteristic
48C2×(C2×D3)⋊C4D411yescharacteristic
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3C2×C411yescharacteristic
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4C2×C411yescharacteristic
54C2×Dic3⋊C4D411yescharacteristic
55C2×C4×Dic3—21no—
56C2×C4×Dic3D411yescharacteristic
57C2×#208—21no—
58#872—21no—
59#872—21no—
60#872—21no—
61#872—21no—
62C4×Dic6—21no—
63C4×Dic6—21no—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8D412yes—
69C3×C22⋊C8—21no—
70C3×C4×C8—21no—
71C3×#95C2×C411yescharacteristic
72C3×#95C2×C411yescharacteristic
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77#830—21no—
78(C2×D3)⋊C8—24no—
79#815—22no—
80C8×Dic3—22no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)D412yes—
84C4×C3⋊C8—21no—
85C2×C4×Q8—31no—
86C2×#117—31no—
87#433—62no—
88#433—62no—
89#431—31no—
90#431C2×D311yescharacteristic
91#423—31no—
92#423C2×D311yescharacteristic
93#407—61no—
94C4×M16—61no—
95#400—31no—
96#399—31no—
97#398—61no—
98#396—61no—
99#395—31no—
100#393—31no—
101C4×C4⋊C4—61no—
102C4×C4⋊C4C2×D311yescharacteristic
103C4×C22⋊C4—61no—
104Q8⋊C8—62no—
105Q8⋊C8—62no—
106#324—61no—
107#323—61no—
108C23×D3C22⋊C411yescharacteristic
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4C22×C411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3C22⋊C411yescharacteristic
117C22×Dic3C22⋊C411yescharacteristic
118C22×Dic3C22⋊C411yescharacteristic
119C2×C4×D3—21no—
120C2×C4×D3—21no—
121C2×C4×D3—21no—
122C2×C4×D3—21no—
123C2×C4×D3—21no—
124C2×C4×D3—21no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×Dic6—42no—
128C3×M16—41no—
129C3×M16—41no—
130C2×C3×C8—22no—
131C2×C3×C8—41no—
132C3×C4⋊C4—42no—
133C3×C42—41no—
134C3×C42—41no—
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138C3×C42#4111yescharacteristic
139(C2×D3)⋊C4—21no—
140(C2×D3)⋊C4—21no—
141(C2×D3)⋊C4—21no—
142(C2×D3)⋊C4—21no—
143(C2×D3)⋊C4—42no—
144(C2×D3)⋊C4—42no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—42no—
148C4⋊Dic3—42no—
149Dic3⋊C4—21no—
150Dic3⋊C4—21no—
151Dic3⋊C4—21no—
152Dic3⋊C4—21no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155C4×Dic3—21no—
156C4×Dic3—21no—
157C4×Dic3—41no—
158C4×Dic3—41no—
159#208—41no—
160#208—41no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—41no—
163C22×Q8—31no—
164C23×C4—31no—
165C2×M16—21no—
166C2×M16—61no—
167C4×Q8—61no—
168C4×Q8—61no—
169C4×Q8—61no—
170C4×Q8—61no—
171#117—61no—
172#117—61no—
173#117—61no—
174#117—61no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C22⋊C4—31no—
183C2×C22⋊C4—31no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C42—21no—
187C2×C42—31no—
188C2×C42—61no—
189C2×C42C22×D311yescharacteristic
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—32no—
193C4⋊C8—61no—
194C4⋊C8—61no—
195C4⋊C8D1212yes—
196C22⋊C8—21no—
197C22⋊C8—64no—
198C22⋊C8—61no—
199#97—62no—
200C4×C8—21no—
201C4×C8—62no—
202C4×C8—61no—
203#95—61no—
204#95—61no—
205#95—61no—
206#95—61no—
207#95C4×D311yescharacteristic
208#95C4×D311yescharacteristic
209C23×C3C2×C22⋊C411yescharacteristic
210C22×D3—21no—
211C22×D3—21no—
212C22×D3—21no—
213C22×D3—41no—
214C22×D3—41no—
215C2×C3×C4—22no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4C4×D411yescharacteristic
224C2×C3×C4C4×D411yescharacteristic
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12311yescharacteristic
229C2×C3×C4#13511yescharacteristic
230C2×C3×C4#13511yescharacteristic
231C2×C3×C4#13711yescharacteristic
232C2×C3×C4#13711yescharacteristic
233C2×Dic3—22no—
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—21no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C4×D3—41no—
247C4×D3—41no—
248C4×D3—41no—
249C4×D3—41no—
250Dic6—22no—
251Dic6—22no—
252Dic6—41no—
253Dic6—41no—
254C3×C8—42no—
255C3×C8—81no—
256C3⋊C8—42no—
257C3⋊C8—81no—
258C24—31no—
259C2×Q8—61no—
260C2×Q8—61no—
261C2×Q8—122no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—31no—
265C22×C4—31no—
266C22×C4—31no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4C2×C4×D311yescharacteristic
276C22×C4D3×D411yescharacteristic
277C22×C4D3×D411yescharacteristic
278M16—41no—
279M16—41no—
280M16—121no—
281M16—121no—
282C2×C8—22no—
283C2×C8—41no—
284C2×C8—62no—
285C2×C8—121no—
286C4⋊C4—42no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—122no—
294C4⋊C4—122no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C22⋊C4—61no—
298C22⋊C4—61no—
299C22⋊C4—61no—
300C22⋊C4—61no—
301C22⋊C4—122no—
302C22⋊C4—122no—
303C42—41no—
304C42—41no—
305C42—61no—
306C42—61no—
307C42—121no—
308C42—121no—
309C42C2×D1211yescharacteristic
310C42#23511yescharacteristic
311C42D3×D411yescharacteristic
312C42#23911yescharacteristic
313C22×C3—21no—
314C22×C3—21no—
315C22×C3#38611yescharacteristic
316C22×C3#41611yescharacteristic
317C22×C3#41911yescharacteristic
318C2×D3—42no—
319C2×D3—41no—
320C2×D3—41no—
321C2×D3—41no—
322C2×D3—81no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—42no—
328C3×C4—81no—
329C3×C4#45211yescharacteristic
330C3×C4#45211yescharacteristic
331C3×C4#48511yescharacteristic
332C3×C4#48511yescharacteristic
333Dic3—41no—
334Dic3—41no—
335Dic3—42no—
336Dic3—41no—
337Dic3—81no—
338C23—61no—
339C23—61no—
340C23—61no—
341C23—121no—
342C23—121no—
343C23D3×C22⋊C411yescharacteristic
344Q8—62no—
345Q8—62no—
346Q8—121no—
347Q8—121no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—62no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4C4×D1211yescharacteristic
374C2×C4#88211yescharacteristic
375C2×C4#88411yescharacteristic
376C2×C4#89311yescharacteristic
377C2×C4#89411yescharacteristic
378C2×C4#89511yescharacteristic
379C2×C4#91211yescharacteristic
380C2×C4#91511yescharacteristic
381C2×C4#91611yescharacteristic
382C2×C4#91811yescharacteristic
383C8—42no—
384C8—81no—
385C8—122no—
386C8—241no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3#187311yescharacteristic
390C2×C3#188411yescharacteristic
391C2×C3#189111yescharacteristic
392D3—81no—
393C22—21no—
394C22—21no—
395C22—122no—
396C22—121no—
397C22—121no—
398C22—121no—
399C22—241no—
400C22#466511yescharacteristic
401C22#477211yescharacteristic
402C22#480211yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—42no—
408C4—81no—
409C4—121no—
410C4—121no—
411C4—122no—
412C4—121no—
413C4—241no—
414C4#478311yescharacteristic
415C4#480811yescharacteristic
416C4#486811yescharacteristic
417C4#488811yescharacteristic
418C3#903411yescharacteristic, complemented, Sylow
419C2—21no—
420C2—21no—
421C2—241no—
422C2#6877711yescharacteristic
423C2#6887911yescharacteristic
424C2#6903611yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1303C2
socleC22×C3#386
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1303D3
hypocenterC3#9034
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
4. #70770C2semidirect
5. #70500C2semidirect
6. C3×#1303C2semidirect
7. #67620C2semidirect
38. #1303D3semidirect
418. C3#9034semidirect

Maximal subgroups

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

subgroupindex
#790392
#736682
#707822
#707702
#705002
C3×#13032
#676202
#90343