Finite Group Databasebeta

← #10691113

Subgroups — #10691113

Order 768.

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

normal subgroups only

This group has 446 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#82463C211yesmaximal, characteristic
2#82431C211yesmaximal, characteristic
3C3×#1479C211yesmaximal, characteristic
4#70793C211yesmaximal, characteristic
5#70792C211yesmaximal, characteristic
6#67746C211yesmaximal, characteristic
7#67738C211yesmaximal, characteristic
8#9178—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#431C2211yescharacteristic
13#5229—21no—
14#5225—21no—
15#5022—22no—
16#5021C412yescomplemented
17#5020C412yescomplemented
18#4980—22no—
19Dic3×C4⋊C4—21no—
20#4964—21no—
21#4937—21no—
22#4936C2211yescharacteristic
23C4×C4⋊Dic3C2211yescharacteristic
24#4932—21no—
25#4924C2211yescharacteristic
26C3×Q8⋊C8—22no—
27#4467—21no—
28#4467—21no—
29#4467—22no—
30#3078—31no—
31#2356—31no—
32#1518—31no—
33#1503—31no—
34#1479D311yescharacteristic
35#1303—31no—
36#1286—31no—
37C22×C3×Q8D411yescharacteristic
38C3×C4×Q8—21no—
39C3×C4×Q8—21no—
40C3×C4×Q8C2×C412yes—
41C3×C4×Q8D412yes—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C42—21no—
45C2×C3×C42C2311yescharacteristic
46#926—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49#924—21no—
50#924—21no—
51C2×C4×Dic3—22no—
52C2×C4×Dic3—21no—
53C2×#208—21no—
54#870—21no—
55#869C2×C411yescharacteristic
56#869C2×C411yescharacteristic
57C4×Dic6—21no—
58C4×Dic6—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C22⋊C8—21no—
62C3×C22⋊C8—21no—
63C3×C4×C8—21no—
64C3×C4×C8—21no—
65#835—22no—
66#835—22no—
67C3⋊#95—22no—
68C3⋊#95—22no—
69C3⋊#95—22no—
70C3⋊#95—21no—
71C3⋊#95D411yescharacteristic
72C3⋊#95D411yescharacteristic
73C3⋊#95D412yes—
74#808—22no—
75#807—22no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)C2×C412yes—
78#533—31no—
79C2×C4×Q8C2×D311yescharacteristic
80#479—62no—
81#477—32no—
82#475—32no—
83#431C2×D311yescharacteristic
84#431C2×D311yescharacteristic
85#428—61no—
86#423—31no—
87#419—61no—
88#391—61no—
89#387—62no—
90#383—61no—
91#382—31no—
92C4×C4⋊C4—31no—
93C4×C4⋊C4—61no—
94#376—61no—
95#328—61no—
96#328—61no—
97#328—62no—
98Q8⋊C8—22no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C2×C3×Q8—21no—
103C2×C3×Q8—21no—
104C2×C3×Q8C22⋊C412yes—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×Dic3—21no—
111C22×Dic3—22no—
112C22×Dic3—21no—
113C2×Dic6—21no—
114C2×Dic6—21no—
115C2×C3×C8—41no—
116C2×C3×C8—41no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4C22⋊C412yes—
121C3×C42—22no—
122C3×C42—22no—
123C3×C42C22×C411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42#4111yescharacteristic
126C3×C42#4111yescharacteristic
127#217—41no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132C4⋊Dic3—42no—
133C4⋊Dic3—41no—
134C4⋊Dic3—41no—
135Dic3⋊C4—41no—
136Dic3⋊C4—41no—
137C4×Dic3—44no—
138C4×Dic3—41no—
139C4×Dic3—41no—
140C4×Dic3—41no—
141#208—41no—
142#208—41no—
143C2×C3⋊C8—22no—
144C22×Q8D1211yescharacteristic
145C2×M16—61no—
146C4⋊Q8—31no—
147C4⋊Q8—31no—
148#125—61no—
149C22⋊Q8—61no—
150C22⋊Q8—61no—
151C4×Q8—21no—
152C4×Q8—21no—
153C4×Q8—61no—
154C4×Q8—61no—
155C4×Q8C4×D312yes—
156C4×Q8#6712yes—
157#117—61no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—61no—
160C2×C4⋊C4—61no—
161C2×C4⋊C4D1211yescharacteristic
162C2×C42—21no—
163C2×C42—62no—
164C2×C42—61no—
165C2×C42C22×D311yescharacteristic
166#107—62no—
167#106—62no—
168C4⋊C8—21no—
169C4⋊C8—21no—
170C4⋊C8—32no—
171C4⋊C8—61no—
172#103—62no—
173#103—62no—
174C22⋊C8—21no—
175C22⋊C8—21no—
176C4×C8—21no—
177C4×C8—21no—
178#95—31no—
179#95—31no—
180#95—32no—
181#95—62no—
182#95—62no—
183#95—62no—
184#95—61no—
185C23×C3#12011yescharacteristic
186C3×Q8—22no—
187C3×Q8—22no—
188C3×Q8—41no—
189C3×Q8—41no—
190C2×C3×C4—22no—
191C2×C3×C4—22no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—42no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4C2×C22⋊C411yescharacteristic
198C2×C3×C4C4×D411yescharacteristic
199C2×C3×C4C4×D411yescharacteristic
200C2×C3×C4#12111yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×C3×C4#12311yescharacteristic
203C2×C3×C4#13511yescharacteristic
204C2×C3×C4#13511yescharacteristic
205C2×C3×C4#13711yescharacteristic
206C2×C3×C4#13711yescharacteristic
207C2×Dic3—22no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—42no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—42no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—42no—
218Dic6—81no—
219Dic6—81no—
220C3×C8—81no—
221C3×C8—81no—
222C3⋊C8—42no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—61no—
229C2×Q8—61no—
230C2×Q8(C2×D3)⋊C412yes—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—61no—
234C22×C4—62no—
235C22×C4—61no—
236C22×C4C2×D1211yescharacteristic
237C22×C4D3×D411yescharacteristic
238C22×C4D3×D411yescharacteristic
239M16—121no—
240M16—121no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—62no—
244C4⋊C4—22no—
245C4⋊C4—21no—
246C4⋊C4—22no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—122no—
252C4⋊C4—121no—
253C4⋊C4—121no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C4⋊C4(C2×D3)⋊C412yes—
257C22⋊C4—121no—
258C42—22no—
259C42—22no—
260C42—124no—
261C42—121no—
262C42—121no—
263C42—121no—
264C42C2×C4×D311yescharacteristic
265C42#23711yescharacteristic
266C42#23711yescharacteristic
267C42C2×#6711yescharacteristic
268C22×C3—21no—
269C22×C3—21no—
270C22×C3#38611yescharacteristic
271C22×C3#45211yescharacteristic
272C22×C3#45211yescharacteristic
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—42no—
278C3×C4—42no—
279C3×C4#41611yescharacteristic
280C3×C4#41911yescharacteristic
281C3×C4#48511yescharacteristic
282C3×C4#48511yescharacteristic
283Dic3—42no—
284Dic3—81no—
285Dic3—82no—
286Dic3—81no—
287Dic3—81no—
288C23#88211yescharacteristic
289Q8—22no—
290Q8—22no—
291Q8—41no—
292Q8—41no—
293Q8—241no—
294Q8—241no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—62no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—122no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—122no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—122no—
313C2×C4#67⋊C411yescharacteristic
314C2×C4#67⋊C411yescharacteristic
315C2×C4#88611yescharacteristic
316C2×C4#90411yescharacteristic
317C2×C4#90411yescharacteristic
318C2×C4#90911yescharacteristic
319C2×C4#90911yescharacteristic
320C2×C4C2×(C2×D3)⋊C411yescharacteristic
321C2×C4#93911yescharacteristic
322C2×C4#93911yescharacteristic
323C8—81no—
324C8—81no—
325C8—122no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3#187311yescharacteristic
329C2×C3#190511yescharacteristic
330C2×C3#190611yescharacteristic
331C22—21no—
332C22—21no—
333C22#473311yescharacteristic
334C22#473311yescharacteristic
335C22#495811yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—42no—
341C4—42no—
342C4—122no—
343C4—241no—
344C4—242no—
345C4—241no—
346C4—241no—
347C4#472311yescharacteristic
348C4#472311yescharacteristic
349C4#511311yescharacteristic
350C4#512011yescharacteristic
351C3#917811yescharacteristic, complemented, Sylow
352C2—21no—
353C2—21no—
354C2#6993111yescharacteristic
355C2#7033411yescharacteristic
356C2#7039511yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1479C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#1479D3
hypocenterC3#9178
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
16. #5021C4semidirect
17. #5020C4semidirect
351. C3#9178semidirect

Maximal subgroups

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

subgroupindex
#824632
#824312
C3×#14792
#707932
#707922
#677462
#677382
#91783