Finite Group Databasebeta

← #10708239

Subgroups — #10708239

Order 768.

378 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 57 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#78088C211yesmaximal, characteristic
2#69716C211yesmaximal, characteristic
3#68958C211yesmaximal, characteristic
4#68117C211yesmaximal, characteristic
5C3×#1352C211yesmaximal, characteristic
6#67492C211yesmaximal, characteristic
7#67447C211yesmaximal, characteristic
8#12255—31nomaximal, Sylow
9#5483C2211yescharacteristic
10C3×#399C2211yescharacteristic
11#4985—21no—
12#4984—21no—
13#4981—21no—
14#4970—21no—
15#4950—21no—
16#4950—21no—
17#4949C2211yescharacteristic
18#4946C2211yescharacteristic
19#4945—21no—
20#4722—21no—
21#4647—21no—
22C3×#342C2211yescharacteristic
23C3×#323C2211yescharacteristic
24#4527—21no—
25#4525C2211yescharacteristic
26#4467—21no—
27#4452—21no—
28#2807—31no—
29#2062—31no—
30#1614—31no—
31#1614—31no—
32#1352—31no—
33#1352D311yescharacteristic
34#1326—31no—
35C23×Dic3D411yescharacteristic
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4D411yescharacteristic
38C2×C3×C22⋊C4—21no—
39C2×C3×C22⋊C4—21no—
40C2×C3×C22⋊C4C2311yescharacteristic
41C2×#217—21no—
42C2×#217—21no—
43C2×#217D411yescharacteristic
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—41no—
47C2×C4⋊Dic3D411yescharacteristic
48#924—21no—
49#924—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×C4×Dic3—21no—
54#878—41no—
55#878—41no—
56C3×C22⋊C8—21no—
57C3×#95D411yescharacteristic
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—41no—
64C3⋊#95D411yescharacteristic
65#830—21no—
66#818—41no—
67#816—41no—
68#810—41no—
69#807—41no—
70#543—31no—
71#484—61no—
72#484—61no—
73#400—61no—
74#399—31no—
75#399C2×D311yescharacteristic
76#398—61no—
77#389—61no—
78#389—61no—
79#388—61no—
80#385—31no—
81#385—61no—
82#385—61no—
83#342—61no—
84#342C2×D311yescharacteristic
85#328—61no—
86#328—61no—
87#323—31no—
88#323C2×D311yescharacteristic
89C24×C3C2×D411yescharacteristic
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4—41no—
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×Dic3—21no—
96C22×Dic3—21no—
97C22×Dic3—21no—
98C22×Dic3—21no—
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—41no—
102C22×Dic3—81no—
103C2×Dic6—21no—
104C2×Dic6—21no—
105C2×C3×C8—41no—
106C3×C4⋊C4—41no—
107C3×C4⋊C4—42no—
108C3×C22⋊C4—22no—
109C3×C22⋊C4—41no—
110C3×C22⋊C4—42no—
111C3×C22⋊C4—41no—
112#217—21no—
113#217—41no—
114#217—41no—
115#217—81no—
116#217Q1612yes—
117C4⋊Dic3—21no—
118C4⋊Dic3—21no—
119C4⋊Dic3—41no—
120C4⋊Dic3—81no—
121C4⋊Dic3—81no—
122C4⋊Dic3—82no—
123Dic3⋊C4—41no—
124Dic3⋊C4—41no—
125Dic3⋊C4—42no—
126Dic3⋊C4—81no—
127Dic3⋊C4—81no—
128C4×Dic3—81no—
129C2×C3⋊C8—41no—
130C23×C4—31no—
131C22⋊Q8—61no—
132C22⋊Q8—61no—
133C22⋊Q8—121no—
134C22⋊Q8—121no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—31no—
137C2×C4⋊C4—61no—
138C2×C4⋊C4—61no—
139C2×C4⋊C4—61no—
140C2×C4⋊C4—61no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—121no—
143C2×C4⋊C4D1211yescharacteristic
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—31no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4—61no—
149C2×C22⋊C4C22×D311yescharacteristic
150C2×C42—61no—
151#107—121no—
152#107—121no—
153#103—121no—
154#103—121no—
155C22⋊C8—21no—
156C22⋊C8—61no—
157#95—31no—
158#95—61no—
159#95—61no—
160#95—61no—
161#95—61no—
162#95—61no—
163#95—121no—
164#95D1211yescharacteristic
165C23×C3—21no—
166C23×C3—21no—
167C23×C3—41no—
168C23×C3—41no—
169C23×C3#12011yescharacteristic
170C23×C3C2×Q1611yescharacteristic
171C23×C3#13711yescharacteristic
172C2×C3×C4—22no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—42no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—81no—
184C2×C3×C4—81no—
185C2×Dic3—22no—
186C2×Dic3—41no—
187C2×Dic3—41no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—42no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—81no—
201C2×Dic3—81no—
202C2×Dic3—81no—
203Dic6—81no—
204Dic6—81no—
205C3×C8—81no—
206C3⋊C8—81no—
207C24D3×D411yescharacteristic
208C2×Q8—61no—
209C2×Q8—61no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—41no—
213C22×C4—61no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4—61no—
219C22×C4—121no—
220C22×C4—241no—
221C22×C4C2×D1211yescharacteristic
222C22×C4D3×D411yescharacteristic
223C2×C8—41no—
224C2×C8—121no—
225C4⋊C4—41no—
226C4⋊C4—42no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—121no—
230C4⋊C4—122no—
231C4⋊C4—121no—
232C4⋊C4—121no—
233C4⋊C4—241no—
234C4⋊C4—241no—
235C4⋊C4—241no—
236C4⋊C4—241no—
237C4⋊C4—242no—
238C22⋊C4—22no—
239C22⋊C4—32no—
240C22⋊C4—41no—
241C22⋊C4—42no—
242C22⋊C4—41no—
243C22⋊C4—61no—
244C22⋊C4—121no—
245C22⋊C4—121no—
246C22⋊C4—241no—
247C42—241no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3—81no—
257C22×C3#45111yescharacteristic
258C22×C3#45611yescharacteristic
259C22×C3B(4,𝔽2)11yescharacteristic
260C3×C4—42no—
261C3×C4—41no—
262C3×C4—41no—
263C3×C4—82no—
264C3×C4—81no—
265Dic3—42no—
266Dic3—81no—
267Dic3—81no—
268Dic3—81no—
269Dic3—81no—
270Dic3—81no—
271C23—21no—
272C23—21no—
273C23—41no—
274C23—41no—
275C23#88211yescharacteristic
276C23#91511yescharacteristic
277C23D3×Q1611yescharacteristic
278Q8—241no—
279Q8—241no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—62no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4—121no—
299C2×C4—121no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4—122no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—241no—
309C2×C4—241no—
310C2×C4—241no—
311C8—81no—
312C8—241no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—41no—
316C2×C3—41no—
317C2×C3#160511yescharacteristic
318C2×C3#219811yescharacteristic
319C2×C3#220011yescharacteristic
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—81no—
329C22#474111yescharacteristic
330C22#480911yescharacteristic
331C22#482611yescharacteristic
332C4—42no—
333C4—41no—
334C4—41no—
335C4—82no—
336C4—81no—
337C4—122no—
338C4—241no—
339C4—241no—
340C4—241no—
341C4—241no—
342C4—241no—
343C3#1225511yescharacteristic, complemented, Sylow
344C2—21no—
345C2—21no—
346C2—41no—
347C2—41no—
348C2#6832711yescharacteristic
349C2#6913911yescharacteristic
350C2#6914811yescharacteristic
subgroupG/H
centerC22#4741
commutator, cosocleC2×C3×C22⋊C4C23
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1352C2
socleC22×C3B(4,𝔽2)
prime radicalC24×C3C2×D4
hypercenter#1352D3
hypocenterC3#12255
LCSG ▷ C2×C3×C22⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1352

Complements

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

NH ≅ G/Ntype
343. C3#12255semidirect

Maximal subgroups

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

subgroupindex
#780882
#697162
#689582
#681172
C3×#13522
#674922
#674472
#122553