Finite Group Databasebeta

← #10708264

Subgroups — #10708264

Order 768.

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

normal subgroups only

This group has 342 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#77247C211yesmaximal, characteristic
2C3×#1605C211yesmaximal, characteristic
3#71904C211yesmaximal, characteristic
4#71901C211yesmaximal, characteristic
5#67648C211yesmaximal, characteristic
6#67457C211yesmaximal, characteristic
7#67452C211yesmaximal, characteristic
8#12260—31nomaximal, Sylow
9C3×#544C2211yescharacteristic
10C3×#451—21no—
11C3×#395C2211yescharacteristic
12#5230—21no—
13#5050—21no—
14#4995—21no—
15#4985—21no—
16#4979—21no—
17#4979C2211yescharacteristic
18#4977C2211yescharacteristic
19#4660—21no—
20#4658—21no—
21#4650—21no—
22C3×#328—21no—
23C3×#324C2211yescharacteristic
24#4538—21no—
25#4532—21no—
26#4530C2211yescharacteristic
27#4528C2211yescharacteristic
28#2844—31no—
29#2061—31no—
30#1605—31no—
31#1605D311yescharacteristic
32#1354—31no—
33#1354—31no—
34#1324—31no—
35C22×C3×Q8D411yescharacteristic
36C22×Dic6D411yescharacteristic
37C2×C3×Q16—41no—
38C3×C4⋊Q8—41no—
39C3×#125—41no—
40C3×C22⋊Q8—21no—
41C3×C22⋊Q8—21no—
42C3×#117—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4C2311yescharacteristic
45C2×#216—41no—
46C2×C4⋊Dic3D411yescharacteristic
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×C4×Dic3D411yescharacteristic
52C2×#208—21no—
53C3×#103—41no—
54C3×C22⋊C8—21no—
55C3×#95—21no—
56C3×#95D411yescharacteristic
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95D411yescharacteristic
66#830—21no—
67#810—41no—
68#808—41no—
69#807—41no—
70#544C2×D311yescharacteristic
71#451—21no—
72#451—61no—
73#428—61no—
74#400—61no—
75#400—61no—
76#400—61no—
77#395—31no—
78#395—61no—
79#395—61no—
80#395C2×D311yescharacteristic
81#382—31no—
82#382—61no—
83#343—31no—
84#339—61no—
85#328—21no—
86#328—61no—
87#324—31no—
88#324C2×D311yescharacteristic
89C2×C3×Q8—21no—
90C2×C3×Q8—21no—
91C2×C3×Q8—41no—
92C2×C3×Q8—81no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×Dic3—21no—
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C2×Dic6—41no—
103C2×Dic6—81no—
104C3×Q16—82no—
105C2×C3×C8—41no—
106C3×C4⋊C4—21no—
107C3×C4⋊C4—21no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113C3×C4⋊C4—81no—
114C3×C22⋊C4—41no—
115C3×C42—41no—
116#216—82no—
117C4⋊Dic3—41no—
118Dic3⋊C4—81no—
119Dic3⋊C4—81no—
120Dic3⋊C4—81no—
121Dic3⋊C4—81no—
122C4×Dic3—21no—
123C4×Dic3Q1612yescomplemented
124#208—22no—
125#208—41no—
126C2×C3⋊C8—41no—
127C2×C3⋊C8—41no—
128C22×Q8—31no—
129C22×Q8#6711yescharacteristic
130C2×Q16—41no—
131C2×Q16—121no—
132C2×M16—61no—
133C4⋊Q8—41no—
134#125—41no—
135C22⋊Q8—21no—
136C22⋊Q8—21no—
137#117—21no—
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⋊C4C22×D311yescharacteristic
145C2×C42—31no—
146#107—121no—
147#106—121no—
148#103—41no—
149#103—121no—
150C22⋊C8—21no—
151C22⋊C8—61no—
152#95—21no—
153#95—31no—
154#95—61no—
155#95—61no—
156#95—61no—
157#95—61no—
158#95—61no—
159#95—61no—
160#95—61no—
161#95—61no—
162#95#6711yescharacteristic
163C23×C3#12011yescharacteristic
164C3×Q8—42no—
165C3×Q8—81no—
166C3×Q8—81no—
167C3×Q8—81no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4C2×Q1611yescharacteristic
183C2×C3×C4#13711yescharacteristic
184C2×Dic3—22no—
185C2×Dic3—41no—
186C2×Dic3—41no—
187C2×Dic3—41no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—42no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195Dic6—42no—
196Dic6—81no—
197C3×C8—81no—
198C3⋊C8—42no—
199C3⋊C8—81no—
200C2×Q8—21no—
201C2×Q8—21no—
202C2×Q8—41no—
203C2×Q8—81no—
204C2×Q8—121no—
205C2×Q8—241no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—61no—
209C22×C4—61no—
210C22×C4—61no—
211C22×C4—61no—
212C22×C4D3×D411yescharacteristic
213C22×C4D3×D411yescharacteristic
214C22×C4C2×#6711yescharacteristic
215Q16—82no—
216Q16—242no—
217M16—62no—
218M16—121no—
219C2×C8—41no—
220C2×C8—121no—
221C2×C8—121no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—81no—
230C4⋊C4—121no—
231C4⋊C4—241no—
232C4⋊C4—241no—
233C4⋊C4—241no—
234C4⋊C4—241no—
235C22⋊C4—41no—
236C42—32no—
237C42—41no—
238C42—61no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3#45111yescharacteristic
242C22×C3#45611yescharacteristic
243C22×C3B(4,𝔽2)11yescharacteristic
244C3×C4—21no—
245C3×C4—21no—
246C3×C4—41no—
247C3×C4—41no—
248C3×C4—81no—
249C3×C4—81no—
250C3×C4—81no—
251C3×C4—81no—
252Dic3—42no—
253Dic3—81no—
254Dic3—81no—
255Dic3—81no—
256C23#93711yescharacteristic
257Q8—42no—
258Q8—81no—
259Q8—81no—
260Q8—81no—
261Q8—122no—
262Q8—241no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—62no—
278C2×C4—121no—
279C2×C4—121no—
280C2×C4—121no—
281C2×C4—121no—
282C2×C4—121no—
283C2×C4—121no—
284C2×C4—122no—
285C2×C4—121no—
286C2×C4—121no—
287C2×C4—121no—
288C2×C4D3×Q1611yescharacteristic
289C2×C4#91811yescharacteristic
290C8—81no—
291C8—122no—
292C8—241no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3#160511yescharacteristic
296C2×C3#219911yescharacteristic
297C2×C3#220311yescharacteristic
298C22—21no—
299C22—21no—
300C22#503211yescharacteristic
301C22#518611yescharacteristic
302C22#520511yescharacteristic
303C4—21no—
304C4—21no—
305C4—41no—
306C4—41no—
307C4—81no—
308C4—81no—
309C4—81no—
310C4—81no—
311C4—122no—
312C4—241no—
313C4—241no—
314C4—241no—
315C3#1226011yescharacteristic, complemented, Sylow
316C2—21no—
317C2—21no—
318C2#7098311yescharacteristic
319C2#7099311yescharacteristic
320C2#7099711yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1605C2
socleC22×C3B(4,𝔽2)
prime radicalC23×C3#120
hypercenter#1605D3
hypocenterC3#12260
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1605

Complements

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

NH ≅ G/Ntype
123. C4×Dic3Q16semidirect
315. C3#12260semidirect

Maximal subgroups

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

subgroupindex
#772472
C3×#16052
#719042
#719012
#676482
#674572
#674522
#122603