Finite Group Databasebeta

← #10691118

Subgroups — #10691118

Order 768.

446 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 105 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#83367C211yesmaximal, characteristic
2#73643C211yesmaximal, characteristic
3C3×#1503C211yesmaximal, characteristic
4#71838C211yesmaximal, characteristic
5#70766C211yesmaximal, characteristic
6#67738C211yesmaximal, characteristic
7#67620C211yesmaximal, characteristic
8#9178—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×C4×C4⋊C4C2211yescharacteristic
13#5094C412yescomplemented
14#5091C412yescomplemented
15#5063—22no—
16#5004—21no—
17#4995—21no—
18#4978—22no—
19#4936C2211yescharacteristic
20C4×C4⋊Dic3—21no—
21#4929—21no—
22#4925C2211yescharacteristic
23#4924C2211yescharacteristic
24#4651—21no—
25#4648—21no—
26C3×#328—21no—
27C3×#328—21no—
28#4538—22no—
29#4484—22no—
30#3078—31no—
31#2356—31no—
32#1518—31no—
33#1503D311yescharacteristic
34#1479—31no—
35#1303—31no—
36#1286—31no—
37C22×Dic6D411yescharacteristic
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×C4⋊Q8C2×C411yescharacteristic
40C3×#125—21no—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8—21no—
43C3×C4×Q8—21no—
44C3×C4×Q8—21no—
45C3×#117—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C42—21no—
48C2×C3×C42C2311yescharacteristic
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3D411yescharacteristic
51C2×Dic3⋊C4—21no—
52C2×C4×Dic3—21no—
53C2×C4×Dic3—22no—
54C2×#208—21no—
55C4×Dic6—21no—
56C4×Dic6—21no—
57C4×Dic6C2×C412yes—
58C4×Dic6D412yes—
59C3×C4⋊C8—21no—
60C3×C22⋊C8—21no—
61C3×C4×C8—21no—
62C3×#95D411yescharacteristic
63C3×#95D411yescharacteristic
64C3⋊#95—22no—
65C3⋊#95—22no—
66C3⋊#95—22no—
67C3⋊#95—21no—
68C3⋊#95D412yes—
69#830—21no—
70#810—22no—
71#810—22no—
72#808—22no—
73#807—22no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)C2×C412yes—
77C4×C3⋊C8—21no—
78#533C2×D311yescharacteristic
79C2×C4×Q8—31no—
80#479—62no—
81#477—32no—
82#475—32no—
83#431—31no—
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—61no—
93C4×C4⋊C4C2×D311yescharacteristic
94#376—61no—
95#328—21no—
96#328—21no—
97#328—62no—
98Q8⋊C8—62no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—22no—
109C2×Dic6—21no—
110C2×Dic6—21no—
111C2×Dic6—21no—
112C2×Dic6—21no—
113C2×Dic6—21no—
114C2×Dic6C22⋊C412yes—
115C2×C3×C8—41no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—42no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C22⋊C4—41no—
124C3×C42—41no—
125C3×C42—41no—
126C3×C42—41no—
127C3×C42C22×C411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130C3×C42#4111yescharacteristic
131C4⋊Dic3—21no—
132C4⋊Dic3—41no—
133C4⋊Dic3—41no—
134C4⋊Dic3C22⋊C412yes—
135Dic3⋊C4—22no—
136Dic3⋊C4—22no—
137C4×Dic3—22no—
138C4×Dic3—22no—
139C4×Dic3—44no—
140#208—41no—
141#208—41no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—41no—
144C22×Q8—31no—
145C2×M16—61no—
146C4⋊Q8C4×D311yescharacteristic
147C4⋊Q8C4×D311yescharacteristic
148#125—21no—
149C22⋊Q8—21no—
150C22⋊Q8—21no—
151C4×Q8—21no—
152C4×Q8—21no—
153C4×Q8—32no—
154C4×Q8—32no—
155C4×Q8—61no—
156C4×Q8—61no—
157#117—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—61no—
161C2×C4⋊C4—61no—
162C2×C42—21no—
163C2×C42—61no—
164C2×C42—62no—
165C2×C42C22×D311yescharacteristic
166#107—62no—
167#106—62no—
168C4⋊C8—21no—
169C4⋊C8—32no—
170C4⋊C8—61no—
171C4⋊C8—61no—
172#103—62no—
173#103—62no—
174C22⋊C8—21no—
175C22⋊C8—61no—
176C4×C8—21no—
177C4×C8—61no—
178#95—32no—
179#95—62no—
180#95—62no—
181#95—62no—
182#95—61no—
183#95#6711yescharacteristic
184#95#6711yescharacteristic
185C23×C3#12011yescharacteristic
186C3×Q8—81no—
187C3×Q8—81no—
188C2×C3×C4—22no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4C2×C22⋊C411yescharacteristic
199C2×C3×C4C4×D411yescharacteristic
200C2×C3×C4C4×D411yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12311yescharacteristic
204C2×C3×C4#13511yescharacteristic
205C2×C3×C4#13511yescharacteristic
206C2×C3×C4#13711yescharacteristic
207C2×C3×C4#13711yescharacteristic
208C2×Dic3—22no—
209C2×Dic3—22no—
210C2×Dic3—42no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—42no—
214C2×Dic3—42no—
215C2×Dic3—42no—
216Dic6—22no—
217Dic6—22no—
218Dic6—41no—
219Dic6—41no—
220C3×C8—81no—
221C3⋊C8—42no—
222C3⋊C8—81no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—32no—
226C2×Q8—61no—
227C2×Q8—61no—
228C2×Q8—61no—
229C2×Q8—61no—
230C2×Q8—61no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—62no—
236C22×C4D3×D411yescharacteristic
237C22×C4D3×D411yescharacteristic
238C22×C4C2×#6711yescharacteristic
239M16—121no—
240M16—121no—
241C2×C8—41no—
242C2×C8—62no—
243C2×C8—121no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—32no—
249C4⋊C4—42no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—62no—
253C4⋊C4—61no—
254C4⋊C4—62no—
255C4⋊C4—121no—
256C4⋊C4—121no—
257C22⋊C4—41no—
258C42—41no—
259C42—41no—
260C42—41no—
261C42—62no—
262C42—62no—
263C42—124no—
264C42C2×C4×D311yescharacteristic
265C42#23511yescharacteristic
266C42D3×D411yescharacteristic
267C42#23711yescharacteristic
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—81no—
279C3×C4—81no—
280C3×C4—81no—
281C3×C4#41611yescharacteristic
282C3×C4#41911yescharacteristic
283C3×C4#48511yescharacteristic
284C3×C4#48511yescharacteristic
285Dic3—42no—
286Dic3—42no—
287Dic3—82no—
288C23#93711yescharacteristic
289Q8—62no—
290Q8—62no—
291Q8—81no—
292Q8—81no—
293Q8—121no—
294Q8—121no—
295C2×C4—22no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—62no—
306C2×C4—62no—
307C2×C4—122no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—122no—
311C2×C4—122no—
312C2×C4—122no—
313C2×C4D3×C22⋊C411yescharacteristic
314C2×C4#67⋊C411yescharacteristic
315C2×C4#88311yescharacteristic
316C2×C4#88411yescharacteristic
317C2×C4#91511yescharacteristic
318C2×C4#91611yescharacteristic
319C2×C4#91811yescharacteristic
320C2×C4#91911yescharacteristic
321C2×C4C4×#6711yescharacteristic
322C2×C4#93811yescharacteristic
323C8—81no—
324C8—122no—
325C8—241no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3#187311yescharacteristic
329C2×C3#190511yescharacteristic
330C2×C3#190611yescharacteristic
331C22—21no—
332C22—21no—
333C22#495611yescharacteristic
334C22#517311yescharacteristic
335C22#518711yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—42no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—122no—
345C4—122no—
346C4—242no—
347C4#480211yescharacteristic
348C4#480411yescharacteristic
349C4#481411yescharacteristic
350C4#481511yescharacteristic
351C3#917811yescharacteristic, complemented, Sylow
352C2—21no—
353C2—21no—
354C2#7005211yescharacteristic
355C2#7026911yescharacteristic
356C2#7039111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1503C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#1503D3
hypocenterC3#9178
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
13. #5094C4semidirect
14. #5091C4semidirect
351. C3#9178semidirect

Maximal subgroups

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

subgroupindex
#833672
#736432
C3×#15032
#718382
#707662
#677382
#676202
#91783