Finite Group Databasebeta

← #10708231

Subgroups — #10708231

Order 768.

388 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 59 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#74544C211yesmaximal, characteristic
2#69594C211yesmaximal, characteristic
3#68117C211yesmaximal, characteristic
4#68115C211yesmaximal, characteristic
5C3×#1326C211yesmaximal, characteristic
6#67451C211yesmaximal, characteristic
7#67449C211yesmaximal, characteristic
8#12254—31nomaximal, Sylow
9#5483C2211yescharacteristic
10C3×#385C2211yescharacteristic
11#5212—21no—
12#5210—21no—
13#4986—21no—
14#4984—21no—
15#4951—21no—
16#4949C2211yescharacteristic
17#4947C2211yescharacteristic
18#4739—21no—
19#4722—21no—
20#4656—21no—
21#4654—21no—
22C3×#323C2211yescharacteristic
23C3×#323C2211yescharacteristic
24#4527—21no—
25#4526C2211yescharacteristic
26#4467—21no—
27#4467—21no—
28#2742—31no—
29#2062—31no—
30#1614—31no—
31#1602—31no—
32#1352—31no—
33#1351—31no—
34#1326D311yescharacteristic
35C23×C3×C4D411yescharacteristic
36C22×Dic6D411yescharacteristic
37C2×C3×C4⋊C4—21no—
38C2×C3×C22⋊C4D411yescharacteristic
39C2×C3×C22⋊C4C2311yescharacteristic
40C2×#217—21no—
41C2×#217—21no—
42C2×#217—21no—
43C2×#217—21no—
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3D411yescharacteristic
48#924—21no—
49#924—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—41no—
52C2×C4×Dic3—21no—
53C2×Dic12—41no—
54#878—41no—
55#878—41no—
56C3×C22⋊C8—21no—
57C3×C22⋊C8—21no—
58C3×#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—41no—
65C3⋊#95D411yescharacteristic
66C3⋊#95D411yescharacteristic
67#818—41no—
68#816—41no—
69#816—41no—
70#543—31no—
71#484—61no—
72#451—61no—
73#400—61no—
74#399—31no—
75#395—61no—
76#393—31no—
77#389—61no—
78#388—61no—
79#387—61no—
80#385—61no—
81#385C2×D311yescharacteristic
82#380—61no—
83#342—31no—
84#342—61no—
85#328—61no—
86#328—61no—
87#323C2×D311yescharacteristic
88#323C2×D311yescharacteristic
89C24×C3C2×D411yescharacteristic
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4—42no—
93C22×C3×C4—41no—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×Dic3—21no—
97C22×Dic3—21no—
98C22×Dic3—21no—
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C22×Dic3—41no—
103C2×Dic6—21no—
104C2×Dic6—21no—
105C2×Dic6—41no—
106C2×Dic6—81no—
107C2×C3×C8—41no—
108C2×C3×C8—41no—
109C3×C4⋊C4—42no—
110C3×C22⋊C4—21no—
111C3×C22⋊C4—41no—
112C3×C22⋊C4Q1612yescomplemented
113#217—22no—
114#217—41no—
115#217—41no—
116#217—41no—
117#217—81no—
118#217—81no—
119C4⋊Dic3—21no—
120C4⋊Dic3—21no—
121C4⋊Dic3—42no—
122C4⋊Dic3—41no—
123C4⋊Dic3—81no—
124C4⋊Dic3—81no—
125Dic3⋊C4—41no—
126Dic3⋊C4—41no—
127Dic3⋊C4—81no—
128Dic3⋊C4—82no—
129C4×Dic3—81no—
130Dic12—82no—
131C22×Q8—31no—
132C23×C4D1211yescharacteristic
133C2×Q16—121no—
134C22⋊Q8—61no—
135C22⋊Q8—61no—
136C22⋊Q8—121no—
137C22⋊Q8—121no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—61no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4—121no—
145C2×C22⋊C4—61no—
146C2×C22⋊C4—61no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4—61no—
149C2×C22⋊C4D1211yescharacteristic
150C2×C22⋊C4C22×D311yescharacteristic
151C2×C42—61no—
152#107—121no—
153#103—121no—
154#103—121no—
155C22⋊C8—21no—
156C22⋊C8—21no—
157#95—21no—
158#95—31no—
159#95—31no—
160#95—61no—
161#95—61no—
162#95—61no—
163#95—61no—
164#95—61no—
165#95—121no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—41no—
169C23×C3—41no—
170C23×C3#12011yescharacteristic
171C23×C3C2×Q1611yescharacteristic
172C23×C3#13711yescharacteristic
173C2×C3×C4—22no—
174C2×C3×C4—22no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—42no—
184C2×C3×C4—81no—
185C2×Dic3—41no—
186C2×Dic3—41no—
187C2×Dic3—41no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—42no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—81no—
201C2×Dic3—81no—
202C2×Dic3—81no—
203C2×Dic3—81no—
204Dic6—42no—
205Dic6—81no—
206Dic6—81no—
207Dic6—81no—
208C3×C8—81no—
209C3×C8—81no—
210C24C2×D1211yescharacteristic
211C2×Q8—61no—
212C2×Q8—61no—
213C2×Q8—121no—
214C2×Q8—241no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—41no—
219C22×C4—61no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4—61no—
225C22×C4—121no—
226C22×C4D3×D411yescharacteristic
227C22×C4D3×D411yescharacteristic
228Q16—242no—
229C2×C8—41no—
230C2×C8—41no—
231C4⋊C4—42no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—121no—
235C4⋊C4—122no—
236C4⋊C4—121no—
237C4⋊C4—121no—
238C4⋊C4—241no—
239C4⋊C4—241no—
240C4⋊C4—241no—
241C4⋊C4—242no—
242C22⋊C4—21no—
243C22⋊C4—41no—
244C22⋊C4—62no—
245C22⋊C4—121no—
246C22⋊C4—121no—
247C22⋊C4—121no—
248C22⋊C4—241no—
249C22⋊C4—241no—
250C22⋊C4Dic1212yescomplemented
251C42—241no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—41no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—81no—
261C22×C3#45111yescharacteristic
262C22×C3#45611yescharacteristic
263C22×C3B(4,𝔽2)11yescharacteristic
264C3×C4—42no—
265C3×C4—41no—
266C3×C4—42no—
267C3×C4—41no—
268Dic3—81no—
269Dic3—81no—
270Dic3—82no—
271Dic3—81no—
272Dic3—81no—
273Dic3—81no—
274Dic3—81no—
275C23—21no—
276C23—21no—
277C23—41no—
278C23—41no—
279C23#88211yescharacteristic
280C23C2×Dic1211yescharacteristic
281C23#90911yescharacteristic
282Q8—122no—
283Q8—241no—
284Q8—241no—
285Q8—241no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—81no—
298C2×C4—121no—
299C2×C4—121no—
300C2×C4—122no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—241no—
314C2×C4—241no—
315C2×C4—241no—
316C2×C4—241no—
317C8—81no—
318C8—81no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—41no—
322C2×C3—41no—
323C2×C3#160511yescharacteristic
324C2×C3#219711yescharacteristic
325C2×C3#220011yescharacteristic
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—81no—
335C22#473911yescharacteristic
336C22#474111yescharacteristic
337C22#475211yescharacteristic
338C4—42no—
339C4—41no—
340C4—42no—
341C4—41no—
342C4—241no—
343C4—241no—
344C4—242no—
345C4—241no—
346C4—241no—
347C4—241no—
348C4—241no—
349C3#1225411yescharacteristic, complemented, Sylow
350C2—21no—
351C2—21no—
352C2—41no—
353C2—41no—
354C2#6813911yescharacteristic
355C2#6913211yescharacteristic
356C2#6913911yescharacteristic
subgroupG/H
centerC22#4741
commutator, cosocleC2×C3×C22⋊C4C23
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1326C2
socleC22×C3B(4,𝔽2)
prime radicalC24×C3C2×D4
hypercenter#1326D3
hypocenterC3#12254
LCSG ▷ C2×C3×C22⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1326

Complements

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

NH ≅ G/Ntype
112. C3×C22⋊C4Q16semidirect
250. C22⋊C4Dic12semidirect
349. C3#12254semidirect

Maximal subgroups

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

subgroupindex
#745442
#695942
#681172
#681152
C3×#13262
#674512
#674492
#122543