Finite Group Databasebeta

← #10692474

Subgroups — #10692474

Order 768.

492 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C3×#2337C211yesmaximal, characteristic
2#82949C211yesmaximal, characteristic
3#71718C211yesmaximal, characteristic
4#69566C211yesmaximal, characteristic, complemented
5#69493C211yesmaximal, characteristic, complemented
6#67633C211yesmaximal, characteristic, complemented
7#67621C211yesmaximal, characteristic, complemented
8#9379—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#391—21no—
12C3×#388—21no—
13C3×#388—21no—
14C3×#386—21no—
15C3×#385—21no—
16C3×#383C2211yescharacteristic
17C3×C4×C4⋊C4C2211yescharacteristic
18C3×C4×C22⋊C4—21no—
19#5228—21no—
20#5218—21no—
21#5060C412yescomplemented
22#5055C412yescomplemented
23#5054—22no—
24#4926C2211yescharacteristic
25#4926C2211yescharacteristic
26C4×#208C2211yescharacteristic
27#4540—22no—
28#4523—21no—
29#4523—21no—
30#4452—21no—
31#4452—21no—
32#3215—31no—
33#2337D311yescharacteristic
34#1986—31no—
35#1527—31no—
36#1515—31no—
37#1288—31no—
38#1285—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C3×#124—21no—
43C3×#124—21no—
44C3×#124C2×C412yes—
45C3×#124D412yes—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53C2×C3×C42—21no—
54C2×C3×C42D411yescharacteristic
55C2×C3×C42D411yescharacteristic
56C2×C3×C42C2311yescharacteristic
57C2×#216—22no—
58C2×#214—22no—
59#926—21no—
60#924—21no—
61#924—21no—
62C2×#208—21no—
63#870—21no—
64#869C2×C411yescharacteristic, complemented
65#869C2×C411yescharacteristic, complemented
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×#95—21no—
69C3×#95—21no—
70C3×#95—21no—
71C3×#95—21no—
72C3×#95—21no—
73C3×#95D411yescharacteristic
74C3×#95D411yescharacteristic
75#835—22no—
76#832—22no—
77#830—21no—
78#830—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81#803—21no—
82#803—21no—
83#803C2×C412yescomplemented
84C4×C3⋊C8—21no—
85#533—31no—
86C2×#124C2×D311yescharacteristic
87#497—32no—
88#489—32no—
89#464—62no—
90#451—61no—
91#450—61no—
92#432—31no—
93#432—31no—
94C4×M16—31no—
95#391—21no—
96#388—21no—
97#388—21no—
98#386—21no—
99#385—21no—
100#383C2×D311yescharacteristic
101C4×C4⋊C4C2×D311yescharacteristic
102C4×C22⋊C4—21no—
103#339—61no—
104#339—61no—
105#329—62no—
106#328—61no—
107#328—61no—
108C24×C3—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—81no—
111C2×C3×Q8C22⋊C412yes—
112C2×C3×D4—21no—
113C2×C3×D4—81no—
114C2×C3×D4C22⋊C412yes—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—42no—
121C22×C3×C4—81no—
122C22×C3×C4—81no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C2×Dic6—21no—
127C2×Dic6—21no—
128C3×C4⋊C4—42no—
129C3×C4⋊C4—81no—
130C3×C22⋊C4—22no—
131C3×C22⋊C4—22no—
132C3×C22⋊C4—22no—
133C3×C22⋊C4—22no—
134C3×C22⋊C4—42no—
135C3×C22⋊C4—81no—
136C3×C22⋊C4—81no—
137C3×C42—41no—
138C3×C42—41no—
139C3×C42—81no—
140C3×C42C22×C411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143C3×C42#4111yescharacteristic
144#217—41no—
145#216—82no—
146#214—82no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153C4×Dic3—41no—
154#208—22no—
155#208—41no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C22×Q8#6711yescharacteristic
160C22×D4#6711yescharacteristic
161C23×C4—21no—
162C2×Q16—62no—
163C2×SD16—62no—
164C2×M16—61no—
165C4⋊Q8—31no—
166C4⋊Q8—31no—
167#125—61no—
168#124—21no—
169#124—21no—
170#124C4×D312yes—
171#124D1212yes—
172C22⋊Q8—61no—
173C22⋊Q8—61no—
174C4×Q8—61no—
175C4×Q8—61no—
176#117—61no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C42—21no—
185C2×C42#6711yescharacteristic
186C2×C42#6711yescharacteristic
187C2×C42C22×D311yescharacteristic
188C4⋊C8—61no—
189C4⋊C8—61no—
190#103—62no—
191#102—62no—
192C22⋊C8—61no—
193C22⋊C8—61no—
194#97—32no—
195#97—61no—
196#97—61no—
197C4×C8—61no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95—21no—
203#95#6711yescharacteristic
204#95#6711yescharacteristic
205C23×C3—22no—
206C23×C3—41no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3#12011yescharacteristic
213C3×Q8—42no—
214C3×Q8—81no—
215C3×D4—42no—
216C3×D4—81no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—84no—
237C2×C3×C4C2×C22⋊C411yescharacteristic
238C2×C3×C4C4×D411yescharacteristic
239C2×C3×C4C4×D411yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12311yescharacteristic
243C2×Dic3—41no—
244C2×Dic3—41no—
245Dic6—42no—
246Dic6—81no—
247C3⋊C8—42no—
248C3⋊C8—42no—
249C3⋊C8—81no—
250C24—21no—
251C2×Q8—21no—
252C2×Q8—61no—
253C2×Q8—61no—
254C2×Q8—81no—
255C2×Q8(C2×D3)⋊C412yes—
256C2×D4—21no—
257C2×D4—81no—
258C2×D4(C2×D3)⋊C412yes—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—42no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4C2×#6711yescharacteristic
268C22×C4C2×#6711yescharacteristic
269C22×C4C2×#6711yescharacteristic
270Q16—242no—
271SD16—242no—
272M16—62no—
273M16—121no—
274C2×C8—62no—
275C2×C8—121no—
276C2×C8—121no—
277C4⋊C4—42no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—81no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—42no—
290C22⋊C4—81no—
291C22⋊C4—81no—
292C22⋊C4—121no—
293C42—41no—
294C42—41no—
295C42—81no—
296C42—121no—
297C42C2×C4×D311yescharacteristic
298C42C2×D1211yescharacteristic
299C42#23511yescharacteristic
300C42#23511yescharacteristic
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—42no—
304C22×C3—42no—
305C22×C3—42no—
306C22×C3—81no—
307C22×C3—81no—
308C22×C3—81no—
309C22×C3—81no—
310C22×C3#38611yescharacteristic
311C22×C3#45511yescharacteristic
312C22×C3#45611yescharacteristic
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—42no—
320C3×C4—42no—
321C3×C4—81no—
322C3×C4—81no—
323C3×C4—81no—
324Dic3—81no—
325Dic3—81no—
326C23—22no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23#95311yescharacteristic
334Q8—42no—
335Q8—81no—
336Q8—122no—
337Q8—241no—
338D4—42no—
339D4—81no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—84no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4C2×(C2×D3)⋊C411yescharacteristic
363C2×C4C4×#6711yescharacteristic
364C2×C4C4×#6711yescharacteristic
365C2×C4#92911yescharacteristic
366C2×C4#93011yescharacteristic
367C2×C4#93011yescharacteristic
368C8—122no—
369C8—122no—
370C8—241no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—82no—
374C2×C3#188211yescharacteristic
375C2×C3#190611yescharacteristic
376C2×C3#191111yescharacteristic
377C22—21no—
378C22—21no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#521311yescharacteristic
387C22#524211yescharacteristic
388C22#524611yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—42no—
396C4—42no—
397C4—81no—
398C4—81no—
399C4—81no—
400C4—241no—
401C4—241no—
402C3#937911yescharacteristic, complemented, Sylow
403C2—21no—
404C2—21no—
405C2—82no—
406C2#7164511yescharacteristic
407C2#7165211yescharacteristic
408C2#7166611yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2337C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#2337D3
hypocenterC3#9379
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2337

Complements

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

NH ≅ G/Ntype
4. #69566C2semidirect
5. #69493C2semidirect
6. #67633C2semidirect
7. #67621C2semidirect
21. #5060C4semidirect
22. #5055C4semidirect
64. #869C2×C4semidirect
65. #869C2×C4semidirect
83. #803C2×C4semidirect
402. C3#9379semidirect

Maximal subgroups

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

subgroupindex
C3×#23372
#829492
#717182
#695662
#694932
#676332
#676212
#93793