Finite Group Databasebeta

← #10691473

Subgroups — #10691473

Order 768.

474 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2295C211yesmaximal, characteristic
2#82447C211yesmaximal, characteristic
3#71622C211yesmaximal, characteristic
4#69471C211yesmaximal, characteristic
5#69396C211yesmaximal, characteristic
6#67620C211yesmaximal, characteristic
7#67611C211yesmaximal, characteristic
8#9236—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#391—21no—
12C3×#389—21no—
13C3×#387—22no—
14C3×C4×C4⋊C4—21no—
15C3×C4×C4⋊C4C2211yescharacteristic
16C3×#376—21no—
17C3×C43C2211yescharacteristic
18#5228—21no—
19#5031C412yescomplemented
20C4×#216—22no—
21#5021C412yescomplemented
22#4970—21no—
23#4925C2211yescharacteristic
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26#4535—22no—
27#4523—21no—
28#4523—21no—
29#4452—22no—
30#3104—31no—
31#2295D311yescharacteristic
32#1994—31no—
33#1503—31no—
34#1487—31no—
35#1303—31no—
36#1276—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⋊C4—22no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4D411yescharacteristic
46C2×C3×C42—21no—
47C2×C3×C42—21no—
48C2×C3×C42—21no—
49C2×C3×C42—21no—
50C2×C3×C42D411yescharacteristic
51C2×C3×C42D411yescharacteristic
52C2×C3×C42C2311yescharacteristic
53C2×#216—22no—
54#926—21no—
55#924—21no—
56#924—21no—
57C2×#208—21no—
58#870—21no—
59#869C2×C411yescharacteristic
60#869C2×C411yescharacteristic
61C4×Dic6—21no—
62C4×Dic6—21no—
63C3×#95—22no—
64C3×#95—22no—
65C3×#95—22no—
66C3×#95D412yes—
67#835—22no—
68#830—21no—
69#830—21no—
70#810—22no—
71#807—22no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)C2×C412yes—
76C4×C3⋊C8—21no—
77C4×C3⋊C8—21no—
78#533—31no—
79C2×C4×Q8C2×D311yescharacteristic
80#484—61no—
81#477—32no—
82#462—32no—
83#451—61no—
84C4×Q16—62no—
85#431—31no—
86#431—31no—
87#423—31no—
88#391—21no—
89#389—21no—
90#387—22no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4C2×D311yescharacteristic
93#376—21no—
94C43C2×D311yescharacteristic
95#339—61no—
96#339—61no—
97#328—62no—
98Q8⋊C8—62no—
99C2×C3×Q8—21no—
100C2×C3×Q8—42no—
101C2×C3×Q8C22⋊C412yes—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4—22no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C2×Dic6—21no—
111C2×Dic6—21no—
112C3×C4⋊C4—22no—
113C3×C4⋊C4—22no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—42no—
116C3×C4⋊C4—44no—
117C3×C4⋊C4C22⋊C412yes—
118C3×C42—22no—
119C3×C42—22no—
120C3×C42—21no—
121C3×C42—21no—
122C3×C42—21no—
123C3×C42—42no—
124C3×C42—42no—
125C3×C42—41no—
126C3×C42—41no—
127C3×C42Q1612yescomplemented
128C3×C42C22×C411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42#4111yescharacteristic
131C3×C42#4111yescharacteristic
132#217—41no—
133#216—44no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138Dic3⋊C4—41no—
139Dic3⋊C4—41no—
140C4×Dic3—41no—
141#208—41no—
142#208—41no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—41no—
145C2×C3⋊C8—41no—
146C22×Q8#6711yescharacteristic
147C2×Q16—62no—
148C2×M16—61no—
149C4⋊Q8—31no—
150C4⋊Q8—31no—
151#125—61no—
152C22⋊Q8—61no—
153C22⋊Q8—61no—
154C4×Q8—21no—
155C4×Q8—21no—
156C4×Q8—61no—
157C4×Q8—61no—
158C4×Q8C4×D312yes—
159C4×Q8D1212yes—
160#117—61no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4#6711yescharacteristic
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42#6711yescharacteristic
170C2×C42#6711yescharacteristic
171C2×C42C22×D311yescharacteristic
172#107—62no—
173C4⋊C8—32no—
174C4⋊C8—61no—
175C4⋊C8—61no—
176C4⋊C8—61no—
177#103—62no—
178#103—62no—
179C22⋊C8—61no—
180C22⋊C8—61no—
181C4×C8—61no—
182C4×C8—61no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95#6712yes—
187C23×C3#12011yescharacteristic
188C3×Q8—22no—
189C3×Q8—22no—
190C3×Q8—42no—
191C2×C3×C4—22no—
192C2×C3×C4—22no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4C2×C22⋊C411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4C4×D411yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12311yescharacteristic
212C2×C3×C4C2×Q1611yescharacteristic
213C2×C3×C4#13511yescharacteristic
214C2×C3×C4#13711yescharacteristic
215C2×C3×C4#13711yescharacteristic
216C2×Dic3—41no—
217C2×Dic3—41no—
218Dic6—42no—
219Dic6—81no—
220C3⋊C8—42no—
221C3⋊C8—81no—
222C3⋊C8—81no—
223C2×Q8—21no—
224C2×Q8—42no—
225C2×Q8—61no—
226C2×Q8—61no—
227C2×Q8(C2×D3)⋊C412yes—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4C2×#6711yescharacteristic
234C22×C4C2×#6711yescharacteristic
235C22×C4C2×#6711yescharacteristic
236Q16—124no—
237M16—121no—
238M16—121no—
239C2×C8—62no—
240C2×C8—121no—
241C2×C8—121no—
242C4⋊C4—22no—
243C4⋊C4—21no—
244C4⋊C4—22no—
245C4⋊C4—42no—
246C4⋊C4—44no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—121no—
252C4⋊C4—121no—
253C4⋊C4(C2×D3)⋊C412yes—
254C22⋊C4—121no—
255C42—22no—
256C42—22no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—42no—
261C42—42no—
262C42—41no—
263C42—41no—
264C42—121no—
265C42#21612yescomplemented
266C42C2×C4×D311yescharacteristic
267C42C2×D1211yescharacteristic
268C42#23511yescharacteristic
269C42#23511yescharacteristic
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—42no—
279C3×C4—42no—
280C3×C4—42no—
281C3×C4—42no—
282C3×C4—82no—
283C3×C4#42112yes—
284C3×C4C4×Q1611yescharacteristic
285C3×C4#44111yescharacteristic
286C3×C4#46211yescharacteristic
287C3×C4#46411yescharacteristic
288Dic3—81no—
289Dic3—81no—
290C23#95311yescharacteristic
291Q8—22no—
292Q8—22no—
293Q8—42no—
294Q8—122no—
295Q8—241no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4C2×(C2×D3)⋊C411yescharacteristic
314C2×C4C4×#6711yescharacteristic
315C2×C4C4×#6711yescharacteristic
316C2×C4#92911yescharacteristic
317C2×C4#93011yescharacteristic
318C2×C4#93011yescharacteristic
319C2×C4#94211yescharacteristic
320C2×C4C2×#21611yescharacteristic
321C2×C4#95011yescharacteristic
322C2×C4#95111yescharacteristic
323C8—122no—
324C8—241no—
325C8—241no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3#187811yescharacteristic
329C2×C3#189711yescharacteristic
330C2×C3#190611yescharacteristic
331C22—21no—
332C22—21no—
333C22#521311yescharacteristic
334C22#522811yescharacteristic
335C22#523911yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—42no—
340C4—42no—
341C4—42no—
342C4—42no—
343C4—82no—
344C4—241no—
345C4—241no—
346C4#500512yes—
347C4#502811yescharacteristic
348C4C4×#21611yescharacteristic
349C4#503011yescharacteristic
350C4#503111yescharacteristic
351C3#923611yescharacteristic, complemented, Sylow
352C2—21no—
353C2—21no—
354C2#7159111yescharacteristic
355C2#7163211yescharacteristic
356C2#7164511yescharacteristic
subgroupG/H
centerC2×C4#930
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2295C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#2295D3
hypocenterC3#9236
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2295

Complements

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

NH ≅ G/Ntype
19. #5031C4semidirect
21. #5021C4semidirect
127. C3×C42Q16semidirect
265. C42#216semidirect
351. C3#9236semidirect

Maximal subgroups

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

subgroupindex
C3×#22952
#824472
#716222
#694712
#693962
#676202
#676112
#92363