Finite Group Databasebeta

← #10694350

Subgroups — #10694350

Order 768.

422 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83344C211yesmaximal, characteristic
2C3×#1503C211yesmaximal, characteristic
3#71841C211yesmaximal, characteristic
4#70792C211yesmaximal, characteristic
5#70741C211yesmaximal, characteristic
6#69493C211yesmaximal, characteristic
7C4×#808C211yesmaximal, characteristic
8#9720—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×C4×C4⋊C4C2211yescharacteristic
13#5093C411yescharacteristic, complemented
14#5092C411yescharacteristic, complemented
15#5079C411yescharacteristic, complemented
16#5078C411yescharacteristic, complemented
17#5051—21no—
18#5051—21no—
19C4×#216—21no—
20C4×#216—21no—
21C2×#808—21no—
22C4×C4⋊Dic3C2211yescharacteristic
23C4×#208C2211yescharacteristic
24C2×C4×C3⋊C8C2211yescharacteristic
25C3×#328—22no—
26#4532—22no—
27#4529—21no—
28#4523—22no—
29#4467—22no—
30#4455—21no—
31#4454—21no—
32#3170—31no—
33#1986—31no—
34#1986—31no—
35C4×#106—31no—
36#1503—31no—
37#1503D311yescharacteristic
38#1451—31no—
39C3×C4⋊Q8C2×C411yescharacteristic
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×#125—21no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C3×#117—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C42—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#216—21no—
51C2×#216—21no—
52#926—21no—
53C2×C4⋊Dic3—21no—
54#924—21no—
55#924—21no—
56C2×C4×Dic3—21no—
57C2×#208D412yes—
58C22×C3⋊C8D411yescharacteristic
59C22×C3⋊C8D411yescharacteristic
60#870—21no—
61#869C2×C411yescharacteristic
62#869C2×C411yescharacteristic
63C4×Dic6—21no—
64C4×Dic6—21no—
65C3×C4⋊C8—21no—
66C3×C22⋊C8D412yes—
67C3×C4×C8—21no—
68C3×#95—21no—
69#835—21no—
70#835—21no—
71C3⋊#95—21no—
72#810—21no—
73#810—21no—
74#808—24no—
75#807—21no—
76#807—21no—
77#803—21no—
78C4×C3⋊C8—21no—
79C4×C3⋊C8C2×C411yescharacteristic
80C4×C3⋊C8C2×C411yescharacteristic
81C4×C3⋊C8D411yescharacteristic
82C4×C3⋊C8D411yescharacteristic
83#533—31no—
84#533C2×D311yescharacteristic
85#500—31no—
86#494—31no—
87#487—31no—
88#487—31no—
89#467—61no—
90#467—61no—
91C4×Q16—61no—
92C4×Q16—61no—
93#431C2×D311yescharacteristic
94C2×#106—61no—
95C4×M16—31no—
96C2×C4×C8—31no—
97C4×C4⋊C4—31no—
98C4×C4⋊C4C2×D311yescharacteristic
99#340—61no—
100#339—62no—
101#339—62no—
102#328—22no—
103#328—62no—
104#322—61no—
105#322—61no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C22×C3×C4—41no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×Dic3—41no—
113C2×Dic6—21no—
114C2×Dic6—21no—
115C2×C3×C8—22no—
116C3×C4⋊C4—22no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C22⋊C4—41no—
125C3×C42—41no—
126C3×C42—42no—
127C3×C42C22×C411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130C3×C42#4111yescharacteristic
131#217—41no—
132#216—42no—
133#216—42no—
134C4⋊Dic3—22no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—41no—
140Dic3⋊C4—41no—
141Dic3⋊C4—41no—
142C4×Dic3—41no—
143C4×Dic3—42no—
144#208—22no—
145#208—22no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—41no—
149C2×C3⋊C8C22⋊C411yescharacteristic
150C2×C3⋊C8C22⋊C411yescharacteristic
151C2×C3⋊C8C22⋊C411yescharacteristic
152C2×C3⋊C8C22⋊C411yescharacteristic
153C2×Q16—61no—
154C2×Q16—61no—
155C2×M16—32no—
156C22×C8—31no—
157C22×C8—31no—
158C4⋊Q8—31no—
159C4⋊Q8—31no—
160C4⋊Q8C4×D311yescharacteristic
161C4⋊Q8C4×D311yescharacteristic
162#125—21no—
163#125—61no—
164C22⋊Q8—21no—
165C22⋊Q8—21no—
166C22⋊Q8—61no—
167C22⋊Q8—61no—
168C4×Q8—21no—
169C4×Q8—21no—
170C4×Q8—61no—
171C4×Q8—61no—
172#117—21no—
173#117—61no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—61no—
176C2×C42—21no—
177C2×C42—61no—
178C2×C42C22×D311yescharacteristic
179#107—61no—
180#107—61no—
181#106—64no—
182C4⋊C8—21no—
183#103—61no—
184#103—61no—
185#103—61no—
186#103—61no—
187C22⋊C8#6712yes—
188#97—61no—
189C4×C8—21no—
190C4×C8—31no—
191C4×C8—31no—
192C4×C8—31no—
193C4×C8—31no—
194C4×C8—61no—
195#95—21no—
196#95—61no—
197C23×C3#12711yescharacteristic
198C3×Q8—41no—
199C3×Q8—41no—
200C3×Q8—81no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—41no—
204C2×C3×C4—42no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—81no—
208C2×C3×C4—81no—
209C2×C3×C4C2×C22⋊C411yescharacteristic
210C2×C3×C4C4×D411yescharacteristic
211C2×C3×C4C4×D411yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12411yescharacteristic
215C2×C3×C4#13712yes—
216C2×C3×C4#13712yes—
217C2×Dic3—42no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—81no—
221C2×Dic3—81no—
222Dic6—41no—
223Dic6—41no—
224Dic6—81no—
225C3×C8—42no—
226C3⋊C8—21no—
227C3⋊C8—21no—
228C3⋊C8—21no—
229C3⋊C8—42no—
230C3⋊C8#10412yes—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—61no—
234C2×Q8—61no—
235C22×C4—41no—
236C22×C4—121no—
237C22×C4D3×D411yescharacteristic
238C22×C4D3×D411yescharacteristic
239C22×C4C2×#6711yescharacteristic
240Q16—122no—
241Q16—122no—
242M16—62no—
243M16—62no—
244C2×C8—22no—
245C2×C8—31no—
246C2×C8—31no—
247C2×C8—31no—
248C2×C8—31no—
249C2×C8—62no—
250C2×C8—62no—
251C2×C8—121no—
252C4⋊C4—22no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—62no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—121no—
266C4⋊C4—121no—
267C4⋊C4—121no—
268C22⋊C4—41no—
269C22⋊C4—121no—
270C42—41no—
271C42—42no—
272C42—121no—
273C42—122no—
274C42C2×C4×D311yescharacteristic
275C42#23511yescharacteristic
276C42D3×D411yescharacteristic
277C42#23711yescharacteristic
278C22×C3—21no—
279C22×C3—21no—
280C22×C3#39011yescharacteristic
281C22×C3#49711yescharacteristic
282C22×C3#49711yescharacteristic
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—82no—
287C3×C4—81no—
288C3×C4—81no—
289C3×C4C2×#10411yescharacteristic
290C3×C4#42111yescharacteristic
291C3×C4#44112yes—
292C3×C4#47012yes—
293Dic3—82no—
294Dic3—81no—
295Dic3—81no—
296C23#94011yescharacteristic
297Q8—41no—
298Q8—41no—
299Q8—81no—
300Q8—121no—
301Q8—121no—
302Q8—241no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—122no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—241no—
315C2×C4—241no—
316C2×C4D3×C22⋊C411yescharacteristic
317C2×C4#67⋊C411yescharacteristic
318C2×C4#88411yescharacteristic
319C2×C4#88511yescharacteristic
320C2×C4#91512yes—
321C2×C4#91812yes—
322C2×C4C4×#6711yescharacteristic
323C2×C4#93911yescharacteristic
324C8—32no—
325C8—42no—
326C8—61no—
327C8—61no—
328C8—61no—
329C8—122no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3#197111yescharacteristic
333C2×C3#197611yescharacteristic
334C2×C3#198011yescharacteristic
335C22—21no—
336C22—21no—
337C22#495711yescharacteristic
338C22#516711yescharacteristic
339C22#519011yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—82no—
344C4—81no—
345C4—81no—
346C4—242no—
347C4—241no—
348C4—241no—
349C4#478912yes—
350C4#481612yes—
351C4D3×#10411yescharacteristic
352C4#482111yescharacteristic
353C3#972011yescharacteristic, complemented, Sylow
354C2—21no—
355C2—21no—
356C2#7004211yescharacteristic
357C2#7025911yescharacteristic
358C2#7035811yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1503C2
socleC22×C3#390
prime radicalC23×C3#127
cosocleC2×C3×C42C23
hypercenter#1503D3
hypocenterC3#9720
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
13. #5093C4semidirect
14. #5092C4semidirect
15. #5079C4semidirect
16. #5078C4semidirect
353. C3#9720semidirect

Maximal subgroups

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

subgroupindex
#833442
C3×#15032
#718412
#707922
#707412
#694932
C4×#8082
#97203