Finite Group Databasebeta

← #10691005

Subgroups — #10691005

Order 768.

472 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3115C211yesmaximal, characteristic
2#82519C211yesmaximal, characteristic, complemented
3#71812C211yesmaximal, characteristic, complemented
4#71797C211yesmaximal, characteristic, complemented
5#71643C211yesmaximal, characteristic, complemented
6#67753C211yesmaximal, characteristic, complemented
7#67748C211yesmaximal, characteristic, complemented
8#9162—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C3×C4×#41C2211yescharacteristic
11C3×#482—21no—
12C3×#475C411yescharacteristic, complemented
13C3×#474C411yescharacteristic, complemented
14C3×#460C411yescharacteristic, complemented
15C3×#459C411yescharacteristic, complemented
16C3×#449—21no—
17C3×#442—21no—
18C3×#440—21no—
19C3×#432C2211yescharacteristic
20C2×#837—21no—
21Q8×C3⋊C8—21no—
22#5012—21no—
23#5012—22no—
24#4932C2211yescharacteristic
25#4926—21no—
26#4926C2211yescharacteristic
27C4×#208—21no—
28C4×#208C2211yescharacteristic
29C4×#208C2211yescharacteristic
30#4540—21no—
31#4540—21no—
32#4537—22no—
33#4532—21no—
34#3115D311yescharacteristic
35#2971—31no—
36#1759—31no—
37#1525—31no—
38#1505—31no—
39#1300—31no—
40#1291—31no—
41C2×C3×#41—21no—
42C2×C3×#41D411yescharacteristic
43C2×C3×SD16—21no—
44C2×C3×D8—21no—
45C3×C4⋊Q8C2×C411yescharacteristic
46C3×#127C2×C411yescharacteristic
47C3×#124—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×C4×Q8C2×C411yescharacteristic
51C3×C4×Q8D411yescharacteristic
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4C2×C411yescharacteristic
57C3×C4×D4D411yescharacteristic
58C3×#117—21no—
59C3×#117D411yescharacteristic
60C2×C3×C42—21no—
61C2×C3×C42C2311yescharacteristic
62C2×C4×Dic3—21no—
63C2×#208—22no—
64C2×#208—21no—
65C2×#208—21no—
66C3×#107—21no—
67C3×#106—21no—
68C3×C4⋊C8C2×C411yescharacteristic
69C3×C4⋊C8C2×C411yescharacteristic
70C3×#103—21no—
71C3×#102—21no—
72C3×#102—21no—
73C3×#102—21no—
74C3×C22⋊C8—21no—
75C3×#97—21no—
76#837—24no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79#830—22no—
80#830D412yes—
81C4⋊(C3⋊C8)—22no—
82C4⋊(C3⋊C8)—21no—
83#803—21no—
84#803—21no—
85#803D411yescharacteristic
86#803D411yescharacteristic
87C4×C3⋊C8—22no—
88C4×C3⋊C8—21no—
89C4×C3⋊C8—21no—
90#532C2×D311yescharacteristic
91C4×#41C2×D311yescharacteristic
92#482—21no—
93#475Dic311yescharacteristic, complemented
94#474Dic311yescharacteristic, complemented
95#460Dic311yescharacteristic, complemented
96#459Dic311yescharacteristic, complemented
97#449—21no—
98C8×Q8—61no—
99#442—21no—
100#440—21no—
101#436—62no—
102#436—61no—
103#432—31no—
104#432—61no—
105#432C2×D311yescharacteristic
106C2×#104—61no—
107C4×M16—31no—
108C4×M16—31no—
109C4×M16—61no—
110#376—31no—
111#339—61no—
112#329—61no—
113#329—61no—
114#327—62no—
115C3×#41—22no—
116C3×#41—41no—
117C3×#41—82no—
118C2×C3×Q8—21no—
119C2×C3×Q8C22⋊C411yescharacteristic
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—41no—
125C2×C3×D4C22⋊C411yescharacteristic
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—41no—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C22×Dic3—41no—
134C3×SD16—42no—
135C3×D8—42no—
136C2×C3×C8—21no—
137C2×C3×C8—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4C22⋊C411yescharacteristic
143C3×C4⋊C4C22⋊C411yescharacteristic
144C3×C22⋊C4—21no—
145C3×C22⋊C4—21no—
146C3×C22⋊C4—21no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C42—21no—
150C3×C42—21no—
151C3×C42—21no—
152C3×C42C22×C411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42#4111yescharacteristic
156C4×Dic3—22no—
157C4×Dic3—41no—
158#208—22no—
159#208—42no—
160#208—44no—
161#208—41no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8—41no—
166C2×#41—21no—
167C2×#41#6711yescharacteristic
168C2×SD16—21no—
169C2×D8—21no—
170C2×M16—62no—
171C2×M16—61no—
172C2×M16—61no—
173C4⋊Q8C2×Dic311yescharacteristic
174#127C2×Dic311yescharacteristic
175#124—21no—
176#121—21no—
177#121—21no—
178C4×Q8C2×Dic311yescharacteristic
179C4×Q8#6711yescharacteristic
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4C2×Dic311yescharacteristic
185C4×D4#6711yescharacteristic
186#117—21no—
187#117#6711yescharacteristic
188C2×C42—21no—
189C2×C42—61no—
190C2×C42C22×D311yescharacteristic
191#107—21no—
192#106—21no—
193C4⋊C8—62no—
194C4⋊C8—61no—
195C4⋊C8C2×Dic311yescharacteristic
196C4⋊C8C2×Dic311yescharacteristic
197#104—64no—
198#103—21no—
199#102—21no—
200#102—21no—
201#102—21no—
202C22⋊C8—21no—
203C22⋊C8—32no—
204C22⋊C8—62no—
205#97—21no—
206#97—31no—
207#97—31no—
208#97—61no—
209#97—61no—
210C4×C8—62no—
211C4×C8—61no—
212C4×C8—61no—
213#95—61no—
214#95—61no—
215C23×C3—21no—
216C23×C3—41no—
217C23×C3#12111yescharacteristic
218C3×Q8—21no—
219C3×Q8—81no—
220C3×Q8#10412yes—
221C3×D4—21no—
222C3×D4—21no—
223C3×D4—42no—
224C3×D4—41no—
225C3×D4—41no—
226C3×D4—81no—
227C3×D4—81no—
228C3×D4—82no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—41no—
239C2×C3×C4—81no—
240C2×C3×C4—82no—
241C2×C3×C4C2×C22⋊C411yescharacteristic
242C2×C3×C4C4×D411yescharacteristic
243C2×C3×C4C4×D411yescharacteristic
244C2×C3×C4#12011yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12311yescharacteristic
247C2×Dic3—42no—
248C2×Dic3—81no—
249C2×Dic3—81no—
250C3×C8—41no—
251C3×C8—41no—
252C3⋊C8—22no—
253C3⋊C8—42no—
254C3⋊C8—41no—
255C3⋊C8—42no—
256#41—22no—
257#41—41no—
258#41—82no—
259C2×Q8—21no—
260C2×Q8#21711yescharacteristic
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—41no—
266C2×D4#21711yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—41no—
271C22×C4—121no—
272C22×C4D3×D411yescharacteristic
273C22×C4#23711yescharacteristic
274C22×C4C2×#6711yescharacteristic
275SD16—42no—
276D8—42no—
277M16—62no—
278M16—122no—
279M16—124no—
280M16—121no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—61no—
284C2×C8—61no—
285C2×C8—62no—
286C2×C8—121no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4#21711yescharacteristic
292C4⋊C4#21711yescharacteristic
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—62no—
302C42—121no—
303C42D3×D411yescharacteristic
304C42#23711yescharacteristic
305C42C22×Dic311yescharacteristic
306C42C2×#6711yescharacteristic
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—81no—
314C22×C3—81no—
315C22×C3—81no—
316C22×C3#38611yescharacteristic
317C3×C4—21no—
318C3×C4—22no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—41no—
324C3×C4—41no—
325C3×C4—81no—
326C3×C4C2×#10411yescharacteristic
327C3×C4#42111yescharacteristic
328Dic3—82no—
329C23—21no—
330C23—41no—
331C23#93911yescharacteristic
332Q8—21no—
333Q8—81no—
334Q8#83712yes—
335D4—21no—
336D4—21no—
337D4—42no—
338D4—41no—
339D4—41no—
340D4—81no—
341D4—81no—
342D4—82no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—81no—
354C2×C4—82no—
355C2×C4—122no—
356C2×C4—241no—
357C2×C4—241no—
358C2×C4D4×Dic311yescharacteristic
359C2×C4D4×Dic311yescharacteristic
360C2×C4#93511yescharacteristic
361C2×C4#93711yescharacteristic
362C2×C4#93911yescharacteristic
363C2×C4C2×#21711yescharacteristic
364C8—41no—
365C8—41no—
366C8—62no—
367C8—121no—
368C8—122no—
369C8—122no—
370C2×C3—41no—
371C2×C3—81no—
372C2×C3—161no—
373C2×C3#187111yescharacteristic
374C2×C3#190311yescharacteristic
375C2×C3#190611yescharacteristic
376C22—21no—
377C22—21no—
378C22—21no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22#522011yescharacteristic
386C4—21no—
387C4—22no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—81no—
395C4—242no—
396C4C2×#83711yescharacteristic
397C4#523611yescharacteristic
398C3#916211yescharacteristic, complemented, Sylow
399C2—41no—
400C2—81no—
401C2—161no—
402C2#7171311yescharacteristic
403C2#7209011yescharacteristic
404C2#7211411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3115C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3115D3
hypocenterC3#9162
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3115

Complements

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

NH ≅ G/Ntype
2. #82519C2semidirect
3. #71812C2semidirect
4. #71797C2semidirect
5. #71643C2semidirect
6. #67753C2semidirect
7. #67748C2semidirect
12. C3×#475C4semidirect
13. C3×#474C4semidirect
14. C3×#460C4semidirect
15. C3×#459C4semidirect
93. #475Dic3semidirect
94. #474Dic3semidirect
95. #460Dic3semidirect
96. #459Dic3semidirect
398. C3#9162semidirect

Maximal subgroups

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

subgroupindex
C3×#31152
#825192
#718122
#717972
#716432
#677532
#677482
#91623