Finite Group Databasebeta

← #10691271

Subgroups — #10691271

Order 768.

478 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79276C211yesmaximal, characteristic
2#79025C211yesmaximal, characteristic
3#70792C211yesmaximal, characteristic, complemented
4#70529C211yesmaximal, characteristic
5#70501C211yesmaximal, characteristic, complemented
6C3×#1301C211yesmaximal, characteristic, complemented
7#67704C211yesmaximal, characteristic, complemented
8#9207—31nomaximal, Sylow
9C4×#235C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×#825—21no—
15#5113—21no—
16#5106—21no—
17C4×C4⋊Dic3C2211yescharacteristic
18C2×C4×C3⋊C8C2211yescharacteristic
19#4851C411yescharacteristic, complemented
20#4850C411yescharacteristic, complemented
21#4847—21no—
22#4846C411yescharacteristic, complemented
23#4842C411yescharacteristic, complemented
24#4834—21no—
25#4830—21no—
26#4829—21no—
27#67⋊C8—22no—
28#4685—21no—
29C4×D3×C8—21no—
30#4555—21no—
31#4484—21no—
32#4484—21no—
33#4467—22no—
34#3076—31no—
35C8×#41—31no—
36#1806—31no—
37#1503—31no—
38#1493—31no—
39#1301D311yescharacteristic, complemented
40#1279—31no—
41C2×#235D411yescharacteristic
42C2×C3×M16—21no—
43C22×C3×C8—21no—
44C2×C3×C42C2311yescharacteristic
45C4×#67—21no—
46C4×#67—21no—
47#926—21no—
48#926D411yescharacteristic
49C2×C4⋊Dic3—21no—
50#924—21no—
51#924—21no—
52C2×C4×Dic3—21no—
53C22×C3⋊C8—21no—
54C2×D3×C8—22no—
55C4×D12C2×C411yescharacteristic
56C4×D12D411yescharacteristic
57#872—21no—
58C42×D3—21no—
59#870—21no—
60#869C2×C411yescharacteristic
61#869C2×C411yescharacteristic
62C4×Dic6—21no—
63C4×Dic6—21no—
64C4×Dic6C2×C411yescharacteristic
65C4×Dic6D411yescharacteristic
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8C2×C411yescharacteristic
71C3×C4⋊C8C2×C411yescharacteristic
72C3×C22⋊C8D412yes—
73C3×C4×C8—21no—
74C3⋊#95—21no—
75#825—42no—
76#821—21no—
77#821—21no—
78(C2×D3)⋊C8—22no—
79#818—21no—
80#818—21no—
81#817—21no—
82#817—21no—
83#816—21no—
84#816—21no—
85Dic3⋊C8—22no—
86C8×Dic3—22no—
87C4×C3⋊C8—21no—
88C4×C3⋊C8D411yescharacteristic
89C4×C3⋊C8D411yescharacteristic
90#533—31no—
91C4×#41—31no—
92#479—61no—
93#478—61no—
94#477—31no—
95#476—31no—
96#475—31no—
97#474—31no—
98C8×Q8—61no—
99C8×D4—61no—
100C8×D4—62no—
101#431—61no—
102#431C2×D311yescharacteristic
103#427—61no—
104#423C2×D311yescharacteristic
105C2×C4⋊C8C2×D311yescharacteristic
106C2×#104—61no—
107#416—61no—
108C2×C4×C8—31no—
109C2×C4×C8—61no—
110C4×C4⋊C4—31no—
111#339—61no—
112#328—62no—
113Q8⋊C8—61no—
114Q8⋊C8—61no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C2×#67—21no—
119C22×Dic3—41no—
120#235—21no—
121#235—21no—
122#235SD1612yes—
123#235Q1612yes—
124C2×D12C22⋊C411yescharacteristic
125C2×C4×D3—21no—
126C2×C4×D3—21no—
127C2×C4×D3—21no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×Dic6C22⋊C411yescharacteristic
131C3×M16—41no—
132C3×M16—41no—
133C2×C3×C8—21no—
134C2×C3×C8—21no—
135C2×C3×C8—22no—
136C2×C3×C8—41no—
137C2×C3×C8—41no—
138C2×C3×C8—41no—
139C3×C42C22×C411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143#217—21no—
144#217—41no—
145(C2×D3)⋊C4—21no—
146(C2×D3)⋊C4—21no—
147C4⋊Dic3—22no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—41no—
153C4⋊Dic3C22⋊C411yescharacteristic
154C4⋊Dic3C22⋊C411yescharacteristic
155Dic3⋊C4—21no—
156Dic3⋊C4—21no—
157Dic3⋊C4—41no—
158Dic3⋊C4—41no—
159C4×Dic3—21no—
160C4×Dic3—21no—
161C4×Dic3—21no—
162C4×Dic3—41no—
163C4×Dic3—41no—
164C4×Dic3—41no—
165C2×C3⋊C8—21no—
166C2×C3⋊C8—21no—
167C2×C3⋊C8—42no—
168D3×C8—44no—
169C2×#41—31no—
170C2×M16—21no—
171C22×C8—21no—
172C22×C8—62no—
173C22×C8—61no—
174C4⋊Q8—31no—
175C4⋊Q8—31no—
176#125—61no—
177C22⋊Q8—61no—
178C22⋊Q8—61no—
179C4×Q8—31no—
180C4×Q8—31no—
181C4×Q8—61no—
182C4×Q8—61no—
183C4×D4—31no—
184C4×D4—31no—
185C4×D4—61no—
186C4×D4—61no—
187#117—31no—
188#117—61no—
189#117—61no—
190C2×C4⋊C4—61no—
191C2×C42—61no—
192C2×C42—61no—
193C2×C42C22×D311yescharacteristic
194#107—61no—
195#107—61no—
196#106—61no—
197#106—61no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—62no—
203C4⋊C8C4×D311yescharacteristic
204C4⋊C8C4×D311yescharacteristic
205#104—122no—
206#103—61no—
207#103—61no—
208#102—61no—
209#102—61no—
210C22⋊C8—62no—
211C22⋊C8D1212yes—
212C4×C8—21no—
213C4×C8—31no—
214C4×C8—31no—
215C4×C8—62no—
216C4×C8—61no—
217#95—61no—
218C23×C3#12111yescharacteristic
219C22×D3—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—41no—
223C2×C3×C4C2×C22⋊C411yescharacteristic
224C2×C3×C4C4×D411yescharacteristic
225C2×C3×C4C4×D411yescharacteristic
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12311yescharacteristic
229C2×C3×C4C2×SD1611yescharacteristic
230C2×C3×C4C2×Q1611yescharacteristic
231C2×C3×C4#13511yescharacteristic
232C2×C3×C4#13511yescharacteristic
233#67—22no—
234#67—22no—
235C2×Dic3—21no—
236C2×Dic3—21no—
237C2×Dic3—21no—
238C2×Dic3—42no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—81no—
242C2×Dic3—81no—
243D12—21no—
244D12#10312yes—
245C4×D3—22no—
246C4×D3—22no—
247C4×D3—42no—
248C4×D3—41no—
249C4×D3—41no—
250Dic6—21no—
251Dic6—41no—
252Dic6—41no—
253Dic6—81no—
254Dic6#10312yes—
255C3×C8—42no—
256C3×C8—41no—
257C3×C8—41no—
258C3×C8—81no—
259C3⋊C8—22no—
260C3⋊C8—41no—
261#41—32no—
262#41—32no—
263#41—61no—
264#41—61no—
265C2×Q8—31no—
266C2×Q8—61no—
267C2×Q8—61no—
268C2×D4—31no—
269C2×D4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—121no—
274C22×C4C2×D1211yescharacteristic
275C22×C4D3×D411yescharacteristic
276C22×C4#23911yescharacteristic
277M16—41no—
278M16—41no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—22no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—61no—
286C2×C8—61no—
287C2×C8—124no—
288C2×C8—122no—
289C4⋊C4—31no—
290C4⋊C4—31no—
291C4⋊C4—62no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C4⋊C4—121no—
299C4⋊C4—121no—
300C4⋊C4—121no—
301C22⋊C4—61no—
302C22⋊C4—61no—
303C22⋊C4—61no—
304C22⋊C4—121no—
305C42—61no—
306C42—61no—
307C42—61no—
308C42—121no—
309C42—121no—
310C42—121no—
311C42C2×C4×D311yescharacteristic
312C42#23511yescharacteristic
313C42D3×D411yescharacteristic
314C42D3×D411yescharacteristic
315C22×C3—21no—
316C22×C3—21no—
317C22×C3#38611yescharacteristic
318C22×C3#46511yescharacteristic
319C22×C3#46711yescharacteristic
320C2×D3—22no—
321C2×D3—41no—
322C2×D3—41no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4C2×#10311yescharacteristic
327C3×C4#41611yescharacteristic
328C3×C4#44311yescharacteristic
329C3×C4#44411yescharacteristic
330C3×C4#46511yescharacteristic
331C3×C4#46711yescharacteristic
332Dic3—22no—
333Dic3—41no—
334Dic3—41no—
335Dic3—82no—
336Dic3—81no—
337Dic3—81no—
338C23—61no—
339C23#89511yescharacteristic
340Q8—32no—
341Q8—61no—
342Q8—121no—
343Q8—121no—
344Q8—241no—
345D4—32no—
346D4—62no—
347D4—62no—
348D4—61no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—61no—
353C2×C4—62no—
354C2×C4—62no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—122no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—122no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—241no—
364C2×C4—241no—
365C2×C4C4×D1211yescharacteristic
366C2×C4D3×C22⋊C411yescharacteristic
367C2×C4#88211yescharacteristic
368C2×C4#88411yescharacteristic
369C2×C4#89311yescharacteristic
370C2×C4#89411yescharacteristic
371C2×C4#91211yescharacteristic
372C2×C4D3×SD1611yescharacteristic
373C2×C4#91611yescharacteristic
374C2×C4D3×Q1611yescharacteristic
375C8—42no—
376C8—41no—
377C8—41no—
378C8—62no—
379C8—81no—
380C8—121no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3#187611yescharacteristic
384C2×C3#189211yescharacteristic
385C2×C3#190611yescharacteristic
386D3—42no—
387C22—21no—
388C22—21no—
389C22—62no—
390C22—121no—
391C22—121no—
392C22#466511yescharacteristic
393C22#486411yescharacteristic
394C22#488711yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—62no—
399C4—121no—
400C4—121no—
401C4—242no—
402C4—241no—
403C4—241no—
404C4#477211yescharacteristic
405C4#477811yescharacteristic
406C4D3×#10311yescharacteristic
407C4#481311yescharacteristic
408C4#489411yescharacteristic
409C4#489511yescharacteristic
410C3#920711yescharacteristic, complemented, Sylow
411C2—21no—
412C2—21no—
413C2—122no—
414C2#6888211yescharacteristic
415C2#6894611yescharacteristic
416C2#6903911yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1301C2
socleC22×C3#386
prime radicalC2×#235D4
cosocleC2×C3×C42C23
hypercenter#1301D3
hypocenterC3#9207
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
3. #70792C2semidirect
5. #70501C2semidirect
6. C3×#1301C2semidirect
7. #67704C2semidirect
19. #4851C4semidirect
20. #4850C4semidirect
22. #4846C4semidirect
23. #4842C4semidirect
39. #1301D3semidirect
410. C3#9207semidirect

Maximal subgroups

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

subgroupindex
#792762
#790252
#707922
#705292
#705012
C3×#13012
#677042
#92073