Finite Group Databasebeta

← #10708166

Subgroups — #10708166

Order 768.

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

normal subgroups only

This group has 398 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#77592C211yesmaximal, characteristic
2C3×#1608C211yesmaximal, characteristic
3#71895C211yesmaximal, characteristic
4#71890C211yesmaximal, characteristic
5#67649C211yesmaximal, characteristic
6#67450C211yesmaximal, characteristic
7#67447C211yesmaximal, characteristic
8#12244—31nomaximal, Sylow
9C3×#543C2211yescharacteristic
10C2×#879—21no—
11C3×#483—21no—
12C3×#397C2211yescharacteristic
13#5043—21no—
14#4982—21no—
15#4953C2211yescharacteristic
16#4946—21no—
17#4946C2211yescharacteristic
18#4942—21no—
19#4657—21no—
20#4652—21no—
21#4647—21no—
22C3×#328—21no—
23C3×#323C2211yescharacteristic
24#4538—21no—
25#4526—21no—
26#4526C2211yescharacteristic
27#4525C2211yescharacteristic
28#2722—31no—
29#2062—31no—
30#1608—31no—
31#1608D311yescharacteristic
32#1344—31no—
33#1344—31no—
34#1326—31no—
35C23×Dic3D411yescharacteristic
36C3×C22⋊Q8—21no—
37C3×C22⋊Q8—21no—
38C3×C22⋊Q8—41no—
39C3×C22⋊Q8—41no—
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4C2311yescharacteristic
44C2×#217—21no—
45C2×#217—21no—
46C2×#217D411yescharacteristic
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3D411yescharacteristic
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—41no—
55C2×C4×Dic3—21no—
56#879—42no—
57#879—42no—
58C3×#106—41no—
59C3×#103—41no—
60C3×C22⋊C8—21no—
61C3×#95—21no—
62C3×#95D411yescharacteristic
63#835—41no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—41no—
69C3⋊#95D411yescharacteristic
70#830—21no—
71#808—41no—
72#543C2×D311yescharacteristic
73C2×#126—61no—
74#483—21no—
75#483—61no—
76#398—61no—
77#398—61no—
78#397—31no—
79#397C2×D311yescharacteristic
80#389—61no—
81#387—61no—
82#385—31no—
83#385—61no—
84#380—61no—
85#342—31no—
86#342—61no—
87#328—21no—
88#328—61no—
89#323—31no—
90#323C2×D311yescharacteristic
91C24×C3C2×D411yescharacteristic
92C2×C3×Q8—21no—
93C2×C3×Q8—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×Dic3—21no—
101C22×Dic3—21no—
102C22×Dic3—21no—
103C22×Dic3—21no—
104C22×Dic3—42no—
105C22×Dic3—41no—
106C22×Dic3—41no—
107C2×C3×C8—41no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113C3×C22⋊C4—41no—
114C3×C22⋊C4—41no—
115C3×C22⋊C4—41no—
116C3×C22⋊C4—41no—
117C3×C22⋊C4—41no—
118#217—21no—
119#217—22no—
120#217—41no—
121#217—41no—
122#217—41no—
123#217SD1612yes—
124C4⋊Dic3—42no—
125C4⋊Dic3—41no—
126Dic3⋊C4—42no—
127Dic3⋊C4—42no—
128Dic3⋊C4—42no—
129Dic3⋊C4—42no—
130Dic3⋊C4—81no—
131Dic3⋊C4—82no—
132C4×Dic3—42no—
133C2×C3⋊C8—41no—
134C23×C4—31no—
135#126—122no—
136#126—122no—
137C22⋊Q8—21no—
138C22⋊Q8—21no—
139C22⋊Q8—41no—
140C22⋊Q8—41no—
141C2×C4⋊C4—31no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4—61no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—121no—
149C2×C4⋊C4#6711yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—31no—
153C2×C22⋊C4—61no—
154C2×C22⋊C4—61no—
155C2×C22⋊C4C22×D311yescharacteristic
156C2×C42—61no—
157#106—41no—
158#106—121no—
159#103—41no—
160#103—121no—
161C22⋊C8—21no—
162C22⋊C8—61no—
163#95—21no—
164#95—31no—
165#95—61no—
166#95—61no—
167#95—61no—
168#95—61no—
169#95—121no—
170#95#6711yescharacteristic
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—41no—
174C23×C3—41no—
175C23×C3#12011yescharacteristic
176C23×C3C2×SD1611yescharacteristic
177C23×C3AGL(1,ℤ/8)11yescharacteristic
178C3×Q8—81no—
179C3×Q8—81no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—81no—
193C2×C3×C4—81no—
194C2×Dic3—22no—
195C2×Dic3—22no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—42no—
201C2×Dic3—41no—
202C2×Dic3—42no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—81no—
209C2×Dic3—81no—
210C2×Dic3—81no—
211C3×C8—81no—
212C3⋊C8—81no—
213C24D3×D411yescharacteristic
214C2×Q8—21no—
215C2×Q8—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4—122no—
225C22×C4—121no—
226C22×C4—121no—
227C22×C4D3×D411yescharacteristic
228C22×C4C2×#6711yescharacteristic
229C2×C8—41no—
230C2×C8—121no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—121no—
237C4⋊C4—122no—
238C4⋊C4—122no—
239C4⋊C4—122no—
240C4⋊C4—122no—
241C4⋊C4—122no—
242C4⋊C4—241no—
243C4⋊C4—242no—
244C22⋊C4—32no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—61no—
251C22⋊C4—62no—
252C22⋊C4—121no—
253C22⋊C4—121no—
254C22⋊C4—121no—
255C42—122no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—81no—
265C22×C3#45011yescharacteristic
266C22×C3#45611yescharacteristic
267C22×C3B(4,𝔽2)11yescharacteristic
268C3×C4—41no—
269C3×C4—41no—
270C3×C4—81no—
271C3×C4—81no—
272C3×C4—81no—
273C3×C4—81no—
274Dic3—42no—
275Dic3—42no—
276Dic3—81no—
277Dic3—82no—
278Dic3—81no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23D3×SD1611yescharacteristic
284C23#91411yescharacteristic
285C23#93711yescharacteristic
286Q8—81no—
287Q8—81no—
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—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—62no—
301C2×C4—62no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—122no—
309C2×C4—122no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—241no—
317C2×C4—241no—
318C2×C4—241no—
319C8—81no—
320C8—241no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—41no—
324C2×C3—41no—
325C2×C3#160411yescharacteristic
326C2×C3#219711yescharacteristic
327C2×C3#220111yescharacteristic
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—81no—
337C22#503211yescharacteristic
338C22#516911yescharacteristic
339C22#520511yescharacteristic
340C4—41no—
341C4—41no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—122no—
347C4—122no—
348C4—241no—
349C4—242no—
350C4—241no—
351C3#1224411yescharacteristic, complemented, Sylow
352C2—21no—
353C2—21no—
354C2—41no—
355C2—41no—
356C2#7097511yescharacteristic
357C2#7099011yescharacteristic
358C2#7100311yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C22⋊C4C23
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1608C2
socleC22×C3B(4,𝔽2)
prime radicalC24×C3C2×D4
hypercenter#1608D3
hypocenterC3#12244
LCSG ▷ C2×C3×C22⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1608

Complements

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

NH ≅ G/Ntype
351. C3#12244semidirect

Maximal subgroups

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

subgroupindex
#775922
C3×#16082
#718952
#718902
#676492
#674502
#674472
#122443