Finite Group Databasebeta

← #10690068

Subgroups — #10690068

Order 768.

430 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
1#79040C211yesmaximal, characteristic
2#79025C211yesmaximal, characteristic
3#70768C211yesmaximal, characteristic, complemented
4#70754C211yesmaximal, characteristic
5#70501C211yesmaximal, characteristic, complemented
6C3×#1301C211yesmaximal, characteristic, complemented
7#67703C211yesmaximal, characteristic, complemented
8#8994—31nomaximal, Sylow
9C4×#235C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14#5129—21no—
15#5122—21no—
16#5113—21no—
17#5106—21no—
18C2×C4⋊(C3⋊C8)C2211yescharacteristic
19C2×C4×C3⋊C8C2211yescharacteristic
20#4851C411yescharacteristic
21#4850C411yescharacteristic
22#4847—21no—
23#4846C411yescharacteristic
24#4842C411yescharacteristic
25#4840—22no—
26#4835—21no—
27#4833—21no—
28#4829—21no—
29#4484—21no—
30#4484—22no—
31#4483—21no—
32#4456—21no—
33#4456—21no—
34#3076—31no—
35#2925—31no—
36#1493—31no—
37#1481—31no—
38#1476—31no—
39#1301D311yescharacteristic, complemented
40#1277—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
49#924—21no—
50#924—21no—
51C22×C3⋊C8—21no—
52C22×C3⋊C8—21no—
53C4×D12C2×C411yescharacteristic
54C4×D12D411yescharacteristic
55#872—21no—
56C42×D3—21no—
57#870—21no—
58#869C2×C411yescharacteristic
59#869C2×C411yescharacteristic
60C4×Dic6—21no—
61C4×Dic6—21no—
62C4×Dic6C2×C411yescharacteristic
63C4×Dic6D411yescharacteristic
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8C2×C411yescharacteristic
68C3×C4⋊C8C2×C411yescharacteristic
69C3×C4⋊C8D412yes—
70C3×C22⋊C8—21no—
71C3×C4×C8—21no—
72#821—21no—
73#821—21no—
74(C2×D3)⋊C8—22no—
75(C2×D3)⋊C8—22no—
76#818—21no—
77#818—21no—
78#817—21no—
79#817—21no—
80#816—21no—
81#816—21no—
82#815—22no—
83C8×Dic3—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)D411yescharacteristic
86C4⋊(C3⋊C8)D411yescharacteristic
87C4×C3⋊C8—21no—
88C4×C3⋊C8—21no—
89#533—31no—
90C4×#41—31no—
91#479—61no—
92#478—61no—
93#477—31no—
94#476—31no—
95#475—31no—
96#474—31no—
97#433—62no—
98#432—61no—
99#431—61no—
100#431C2×D311yescharacteristic
101#427—61no—
102#423C2×D311yescharacteristic
103C2×C4⋊C8—31no—
104C2×C4⋊C8C2×D311yescharacteristic
105#416—61no—
106#408—61no—
107#405—61no—
108C2×C4×C8—31no—
109Q8⋊C8—61no—
110Q8⋊C8—61no—
111Q8⋊C8—61no—
112Q8⋊C8—62no—
113D4⋊C8—61no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4#4111yescharacteristic
116C22×C3×C4#4111yescharacteristic
117C2×#67—21no—
118#235—21no—
119#235—21no—
120#235—21no—
121#235—21no—
122C2×D12C22⋊C411yescharacteristic
123C2×C4×D3—21no—
124C2×C4×D3—21no—
125C2×C4×D3—21no—
126C2×Dic6—21no—
127C2×Dic6—21no—
128C2×Dic6C22⋊C411yescharacteristic
129C3×M16—41no—
130C3×M16—41no—
131C2×C3×C8—21no—
132C2×C3×C8—21no—
133C2×C3×C8—22no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C2×C3×C8—41no—
137C3×C42C22×C411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141#217—21no—
142#217—41no—
143(C2×D3)⋊C4—21no—
144(C2×D3)⋊C4—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3C22⋊C411yescharacteristic
150C4⋊Dic3C22⋊C411yescharacteristic
151Dic3⋊C4—21no—
152Dic3⋊C4—21no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155C4×Dic3—21no—
156C4×Dic3—21no—
157C4×Dic3—21no—
158C4×Dic3—41no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—21no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—41no—
164C2×C3⋊C8—41no—
165C2×C3⋊C8—41no—
166C2×#41—31no—
167C2×M16—21no—
168C22×C8—21no—
169C22×C8—61no—
170C22×C8—61no—
171C4⋊Q8—31no—
172C4⋊Q8—31no—
173#125—61no—
174C22⋊Q8—61no—
175C22⋊Q8—61no—
176C4×Q8—31no—
177C4×Q8—31no—
178C4×Q8—61no—
179C4×Q8—61no—
180C4×D4—31no—
181C4×D4—31no—
182C4×D4—61no—
183C4×D4—61no—
184#117—31no—
185#117—61no—
186#117—61no—
187C2×C42—61no—
188C2×C42C22×D311yescharacteristic
189#107—61no—
190#107—61no—
191#106—61no—
192#106—61no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—31no—
197C4⋊C8—31no—
198C4⋊C8—61no—
199C4⋊C8C4×D311yescharacteristic
200C4⋊C8C4×D311yescharacteristic
201C4⋊C8D1212yes—
202#103—61no—
203#103—61no—
204#102—61no—
205#102—61no—
206C22⋊C8—21no—
207C22⋊C8—62no—
208C22⋊C8—62no—
209#97—62no—
210C4×C8—21no—
211C4×C8—62no—
212C4×C8—61no—
213C4×C8—61no—
214C23×C3#12311yescharacteristic
215C22×D3—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—41no—
219C2×C3×C4C2×C22⋊C411yescharacteristic
220C2×C3×C4C4×D411yescharacteristic
221C2×C3×C4C4×D411yescharacteristic
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#13511yescharacteristic
226C2×C3×C4#13511yescharacteristic
227C2×C3×C4#13711yescharacteristic
228C2×C3×C4#13711yescharacteristic
229#67—41no—
230#67—41no—
231C2×Dic3—21no—
232C2×Dic3—21no—
233C2×Dic3—21no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236D12—21no—
237D12—21no—
238C4×D3—41no—
239C4×D3—41no—
240C4×D3—41no—
241C4×D3—41no—
242C4×D3—81no—
243Dic6—21no—
244Dic6—21no—
245Dic6—42no—
246Dic6—41no—
247Dic6—41no—
248C3×C8—41no—
249C3×C8—41no—
250C3×C8—42no—
251C3×C8—81no—
252C3⋊C8—42no—
253C3⋊C8—41no—
254C3⋊C8—41no—
255#41—61no—
256#41—61no—
257#41—61no—
258#41—61no—
259C2×Q8—31no—
260C2×Q8—61no—
261C2×Q8—61no—
262C2×D4—31no—
263C2×D4—61no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4#23511yescharacteristic
268C22×C4D3×D411yescharacteristic
269C22×C4#23911yescharacteristic
270M16—41no—
271M16—41no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—22no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—62no—
279C2×C8—61no—
280C2×C8—61no—
281C2×C8—121no—
282C2×C8—121no—
283C2×C8—121no—
284C2×C8—121no—
285C4⋊C4—31no—
286C4⋊C4—31no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—121no—
294C4⋊C4—121no—
295C22⋊C4—61no—
296C22⋊C4—61no—
297C22⋊C4—61no—
298C22⋊C4—121no—
299C42—61no—
300C42—61no—
301C42—61no—
302C42—121no—
303C42C2×C4×D311yescharacteristic
304C42C2×D1211yescharacteristic
305C42D3×D411yescharacteristic
306C42D3×D411yescharacteristic
307C22×C3—21no—
308C22×C3—21no—
309C22×C3#38611yescharacteristic
310C22×C3#48511yescharacteristic
311C22×C3#48511yescharacteristic
312C2×D3—41no—
313C2×D3—41no—
314C2×D3—41no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4#41611yescharacteristic
319C3×C4#41911yescharacteristic
320C3×C4#44311yescharacteristic
321C3×C4#44411yescharacteristic
322C3×C4#45211yescharacteristic
323C3×C4#45211yescharacteristic
324Dic3—41no—
325Dic3—41no—
326Dic3—41no—
327Dic3—81no—
328Dic3—81no—
329C23—61no—
330C23#89411yescharacteristic
331Q8—61no—
332Q8—61no—
333Q8—122no—
334Q8—121no—
335Q8—121no—
336D4—61no—
337D4—61no—
338D4—121no—
339D4—121no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—241no—
353C2×C4C4×D1211yescharacteristic
354C2×C4D3×C22⋊C411yescharacteristic
355C2×C4#88211yescharacteristic
356C2×C4#88411yescharacteristic
357C2×C4#89311yescharacteristic
358C2×C4#89511yescharacteristic
359C2×C4#91211yescharacteristic
360C2×C4#91511yescharacteristic
361C2×C4#91611yescharacteristic
362C2×C4#91811yescharacteristic
363C8—41no—
364C8—41no—
365C8—42no—
366C8—81no—
367C8—122no—
368C8—121no—
369C8—121no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3#187311yescharacteristic
373C2×C3#187611yescharacteristic
374C2×C3#188111yescharacteristic
375D3—81no—
376C22—21no—
377C22—21no—
378C22—121no—
379C22—121no—
380C22—121no—
381C22#466511yescharacteristic
382C22#486811yescharacteristic
383C22#488811yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—121no—
388C4—121no—
389C4—121no—
390C4—241no—
391C4—241no—
392C4#477211yescharacteristic
393C4#478311yescharacteristic
394C4#480211yescharacteristic
395C4#480811yescharacteristic
396C4#489411yescharacteristic
397C4#489511yescharacteristic
398C3#899411yescharacteristic, complemented, Sylow
399C2—21no—
400C2—21no—
401C2—241no—
402C2#6887911yescharacteristic
403C2#6894611yescharacteristic
404C2#6903511yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1301C2
socleC22×C3#386
prime radicalC2×#235D4
cosocleC2×C3×C42C23
hypercenter#1301D3
hypocenterC3#8994
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. #70768C2semidirect
5. #70501C2semidirect
6. C3×#1301C2semidirect
7. #67703C2semidirect
39. #1301D3semidirect
398. C3#8994semidirect

Maximal subgroups

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

subgroupindex
#790402
#790252
#707682
#707542
#705012
C3×#13012
#677032
#89943