Finite Group Databasebeta

← #10691470

Subgroups — #10691470

Order 768.

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

normal subgroups only

This group has 474 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#83368C211yesmaximal, characteristic
2#73618C211yesmaximal, characteristic
3C3×#1503C211yesmaximal, characteristic
4#71839C211yesmaximal, characteristic
5#70770C211yesmaximal, characteristic
6#67736C211yesmaximal, characteristic
7#67620C211yesmaximal, characteristic
8#9236—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×C4×C4⋊C4C2211yescharacteristic
13#5091C412yescomplemented
14#5090C412yescomplemented
15#5050—21no—
16#5044—21no—
17C4×#216—22no—
18#4973—21no—
19#4972—21no—
20#4932—21no—
21C4×Dic3⋊C4—21no—
22C42×Dic3C2211yescharacteristic
23#4925C2211yescharacteristic
24#4924C2211yescharacteristic
25#4652—22no—
26C3×#328—22no—
27#4532—21no—
28#4532—21no—
29#4456—22no—
30#3104—31no—
31#2295—31no—
32#1994—31no—
33#1503D311yescharacteristic
34#1487—31no—
35#1303—31no—
36#1276—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×#216—22no—
50C2×C4⋊Dic3D411yescharacteristic
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—22no—
53C2×C4×Dic3—21no—
54C2×C4×Dic3—21no—
55C2×C4×Dic3—21no—
56C2×C4×Dic3D411yescharacteristic
57C2×C4×Dic3D411yescharacteristic
58C2×#208—21no—
59C4×Dic6—21no—
60C4×Dic6—21no—
61C4×Dic6C2×C412yes—
62C4×Dic6D412yes—
63C3×C4⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×C4×C8—21no—
66C3×#95D412yes—
67#835—22no—
68C3⋊#95—22no—
69C3⋊#95—22no—
70C3⋊#95—22no—
71#830—21no—
72#810—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#484—61no—
81#477—32no—
82#462—32no—
83#451—61no—
84C4×Q16—62no—
85#431—31no—
86#431C2×D311yescharacteristic
87#423—31no—
88#391—61no—
89#389—61no—
90#387—62no—
91C4×C4⋊C4—61no—
92C4×C4⋊C4C2×D311yescharacteristic
93#376—61no—
94C43—31no—
95#339—61no—
96#339—61no—
97#328—22no—
98Q8⋊C8—62no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C22×C3×C4—22no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—21no—
109C2×Dic6—21no—
110C2×Dic6—42no—
111C2×Dic6C22⋊C412yes—
112C2×C3×C8—41no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—42no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C22⋊C4—41no—
121C3×C42—41no—
122C3×C42—41no—
123C3×C42—41no—
124C3×C42C22×C411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42#4111yescharacteristic
127C3×C42#4111yescharacteristic
128#216—44no—
129C4⋊Dic3—21no—
130C4⋊Dic3C22⋊C412yes—
131Dic3⋊C4—22no—
132Dic3⋊C4—22no—
133Dic3⋊C4—44no—
134C4×Dic3—22no—
135C4×Dic3—22no—
136C4×Dic3—21no—
137C4×Dic3—21no—
138C4×Dic3—21no—
139C4×Dic3—42no—
140C4×Dic3—42no—
141C4×Dic3Q1612yescomplemented
142#208—41no—
143#208—41no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—41no—
146C22×Q8—31no—
147C2×Q16—62no—
148C2×M16—61no—
149C4⋊Q8C4×D311yescharacteristic
150C4⋊Q8C4×D311yescharacteristic
151#125—21no—
152C22⋊Q8—21no—
153C22⋊Q8—21no—
154C4×Q8—21no—
155C4×Q8—21no—
156C4×Q8—32no—
157C4×Q8—32no—
158C4×Q8—61no—
159C4×Q8—61no—
160#117—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—31no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4—62no—
165C2×C42—21no—
166C2×C42—31no—
167C2×C42—31no—
168C2×C42—61no—
169C2×C42—61no—
170C2×C42—61no—
171C2×C42C22×D311yescharacteristic
172#107—62no—
173C4⋊C8—21no—
174C4⋊C8—32no—
175C4⋊C8—61no—
176C4⋊C8—61no—
177#103—62no—
178#103—62no—
179C22⋊C8—21no—
180C22⋊C8—61no—
181C4×C8—21no—
182C4×C8—61no—
183#95—62no—
184#95—62no—
185#95—62no—
186#95#6712yes—
187C23×C3#12011yescharacteristic
188C3×Q8—42no—
189C3×Q8—81no—
190C2×C3×C4—22no—
191C2×C3×C4—41no—
192C2×C3×C4—42no—
193C2×C3×C4—42no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—42no—
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×C4C2×Q1611yescharacteristic
204C2×C3×C4#13511yescharacteristic
205C2×C3×C4#13711yescharacteristic
206C2×C3×C4#13711yescharacteristic
207C2×Dic3—22no—
208C2×Dic3—22no—
209C2×Dic3—22no—
210C2×Dic3—22no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—42no—
215C2×Dic3—42no—
216C2×Dic3—41no—
217Dic6—22no—
218Dic6—22no—
219Dic6—42no—
220C3×C8—81no—
221C3⋊C8—42no—
222C3⋊C8—81no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—32no—
226C2×Q8—61no—
227C2×Q8—122no—
228C22×C4—22no—
229C22×C4—61no—
230C22×C4—61no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4D3×D411yescharacteristic
234C22×C4D3×D411yescharacteristic
235C22×C4C2×#6711yescharacteristic
236Q16—124no—
237M16—121no—
238M16—121no—
239C2×C8—41no—
240C2×C8—62no—
241C2×C8—121no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—32no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—61no—
251C4⋊C4—62no—
252C4⋊C4—62no—
253C4⋊C4—124no—
254C22⋊C4—41no—
255C42—32no—
256C42—41no—
257C42—41no—
258C42—41no—
259C42—62no—
260C42—62no—
261C42—61no—
262C42—61no—
263C42—61no—
264C42—122no—
265C42—122no—
266C42C2×C4×D311yescharacteristic
267C42#23511yescharacteristic
268C42D3×D411yescharacteristic
269C42#23711yescharacteristic
270C22×C3—21no—
271C22×C3—21no—
272C22×C3#38611yescharacteristic
273C22×C3#45111yescharacteristic
274C22×C3#45211yescharacteristic
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—82no—
279C3×C4—81no—
280C3×C4—81no—
281C3×C4#42112yes—
282C3×C4C4×Q1611yescharacteristic
283C3×C4#44111yescharacteristic
284C3×C4#46211yescharacteristic
285C3×C4#46411yescharacteristic
286Dic3—42no—
287Dic3—42no—
288Dic3—42no—
289Dic3—42no—
290C23#93711yescharacteristic
291Q8—42no—
292Q8—62no—
293Q8—62no—
294Q8—81no—
295Q8—122no—
296C2×C4—22no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—62no—
304C2×C4—62no—
305C2×C4—62no—
306C2×C4—62no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—122no—
311C2×C4—122no—
312C2×C4—121no—
313C2×C4D3×C22⋊C411yescharacteristic
314C2×C4#67⋊C411yescharacteristic
315C2×C4#88311yescharacteristic
316C2×C4#88411yescharacteristic
317C2×C4#91511yescharacteristic
318C2×C4#91611yescharacteristic
319C2×C4D3×Q1611yescharacteristic
320C2×C4#91811yescharacteristic
321C2×C4C4×#6711yescharacteristic
322C2×C4#93811yescharacteristic
323C8—81no—
324C8—122no—
325C8—241no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3#187811yescharacteristic
329C2×C3#189711yescharacteristic
330C2×C3#190611yescharacteristic
331C22—21no—
332C22—21no—
333C22#495611yescharacteristic
334C22#517311yescharacteristic
335C22#518611yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—82no—
340C4—81no—
341C4—81no—
342C4—122no—
343C4—122no—
344C4—122no—
345C4—122no—
346C4#478911yescharacteristic
347C4#479011yescharacteristic
348C4#479511yescharacteristic
349C4#479811yescharacteristic
350C4#482112yes—
351C3#923611yescharacteristic, complemented, Sylow
352C2—21no—
353C2—21no—
354C2#7005511yescharacteristic
355C2#7028111yescharacteristic
356C2#7038311yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1503C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#1503D3
hypocenterC3#9236
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. #5091C4semidirect
14. #5090C4semidirect
141. C4×Dic3Q16semidirect
351. C3#9236semidirect

Maximal subgroups

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

subgroupindex
#833682
#736182
C3×#15032
#718392
#707702
#677362
#676202
#92363