Finite Group Databasebeta

← #10708323

Subgroups — #10708323

Order 768.

372 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 55 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#77613C211yesmaximal, characteristic
2C3×#1608C211yesmaximal, characteristic
3#71894C211yesmaximal, characteristic
4#71893C211yesmaximal, characteristic
5#67649C211yesmaximal, characteristic
6#67449C211yesmaximal, characteristic
7#67445C211yesmaximal, characteristic
8#12269—31nomaximal, Sylow
9C3×#543C2211yescharacteristic
10C3×#483—21no—
11C3×#397C2211yescharacteristic
12#5050—21no—
13#4982—21no—
14#4951—21no—
15#4947C2211yescharacteristic
16#4946—21no—
17#4943—21no—
18#4943C2211yescharacteristic
19#4659—21no—
20#4651—21no—
21#4646—21no—
22C3×#328—21no—
23C3×#323C2211yescharacteristic
24#4538—21no—
25#4526—21no—
26#4526C2211yescharacteristic
27#4525C2211yescharacteristic
28#2764—31no—
29#2030—31no—
30#1608D311yescharacteristic
31#1602—31no—
32#1351—31no—
33#1344—31no—
34#1317—31no—
35C23×Dic3D411yescharacteristic
36C22×Dic6D411yescharacteristic
37C3×C22⋊Q8—21no—
38C3×C22⋊Q8—21no—
39C3×C22⋊Q8—41no—
40C3×C22⋊Q8—41no—
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4C2311yescharacteristic
45C2×#217—21no—
46C2×#217—21no—
47C2×#217D411yescharacteristic
48C2×#216—41no—
49C2×C4⋊Dic3—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×C4×Dic3—21no—
53C3×#106—41no—
54C3×#103—41no—
55C3×C22⋊C8—21no—
56C3×#95—21no—
57C3×#95D411yescharacteristic
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—41no—
66C3⋊#95—41no—
67C3⋊#95D411yescharacteristic
68#830—21no—
69#810—41no—
70#543C2×D311yescharacteristic
71#483—21no—
72#451—61no—
73#400—61no—
74#398—61no—
75#397C2×D311yescharacteristic
76#393—31no—
77#391—61no—
78#388—61no—
79#385—61no—
80#383—61no—
81#381—31no—
82#381—61no—
83#342—31no—
84#342—61no—
85#328—21no—
86#328—61no—
87#323—31no—
88#323C2×D311yescharacteristic
89C24×C3C2×D411yescharacteristic
90C2×C3×Q8—21no—
91C2×C3×Q8—21no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×Dic3—21no—
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C22×Dic3—21no—
103C22×Dic3—21no—
104C22×Dic3—41no—
105C22×Dic3—41no—
106C22×Dic3—41no—
107C2×Dic6—41no—
108C2×Dic6—42no—
109C2×C3×C8—41no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—41no—
113C3×C4⋊C4—41no—
114C3×C4⋊C4—41no—
115C3×C22⋊C4—41no—
116C3×C22⋊C4—41no—
117C3×C22⋊C4—41no—
118C3×C22⋊C4—41no—
119C3×C22⋊C4—41no—
120#217—42no—
121#217—41no—
122#217—41no—
123#217—41no—
124#216—82no—
125C4⋊Dic3—81no—
126Dic3⋊C4—81no—
127Dic3⋊C4—81no—
128C4×Dic3—42no—
129C2×C3⋊C8—41no—
130C22×Q8—31no—
131C23×C4—31no—
132C2×Q16—121no—
133C22⋊Q8—21no—
134C22⋊Q8—21no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C2×C4⋊C4—61no—
138C2×C4⋊C4—61no—
139C2×C4⋊C4—61no—
140C2×C4⋊C4#6711yescharacteristic
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—31no—
144C2×C22⋊C4—61no—
145C2×C22⋊C4—61no—
146C2×C22⋊C4C22×D311yescharacteristic
147C2×C42—61no—
148#106—41no—
149#103—41no—
150#103—121no—
151C22⋊C8—21no—
152C22⋊C8—61no—
153#95—21no—
154#95—31no—
155#95—61no—
156#95—61no—
157#95—61no—
158#95—61no—
159#95—61no—
160#95—61no—
161#95—61no—
162#95—121no—
163#95—121no—
164#95#6711yescharacteristic
165C23×C3—21no—
166C23×C3—21no—
167C23×C3—41no—
168C23×C3—41no—
169C23×C3#12011yescharacteristic
170C23×C3#13511yescharacteristic
171C23×C3#13711yescharacteristic
172C3×Q8—81no—
173C3×Q8—81no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
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×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—81no—
187C2×C3×C4—81no—
188C2×Dic3—22no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—42no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—81no—
202C2×Dic3—81no—
203C2×Dic3—81no—
204C2×Dic3—81no—
205Dic6—42no—
206Dic6—81no—
207C3×C8—81no—
208C3⋊C8—81no—
209C24D3×D411yescharacteristic
210C2×Q8—21no—
211C2×Q8—21no—
212C2×Q8—121no—
213C2×Q8—122no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—61no—
219C22×C4—61no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4—121no—
225C22×C4—121no—
226C22×C4—121no—
227C22×C4D3×D411yescharacteristic
228C22×C4C2×#6711yescharacteristic
229Q16—242no—
230C2×C8—41no—
231C2×C8—121no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—241no—
238C4⋊C4—241no—
239C4⋊C4—241no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—122no—
246C22⋊C4—121no—
247C22⋊C4—121no—
248C22⋊C4—121no—
249C42—122no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3—41no—
257C22×C3—41no—
258C22×C3—81no—
259C22×C3#45211yescharacteristic
260C22×C3#45511yescharacteristic
261C22×C3B(4,𝔽2)11yescharacteristic
262C3×C4—41no—
263C3×C4—41no—
264C3×C4—81no—
265C3×C4—81no—
266C3×C4—81no—
267C3×C4—81no—
268Dic3—42no—
269Dic3—81no—
270Dic3—81no—
271Dic3—81no—
272Dic3—161no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23#91511yescharacteristic
278C23#91611yescharacteristic
279C23#93711yescharacteristic
280Q8—81no—
281Q8—81no—
282Q8—122no—
283Q8—241no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—62no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—121no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—122no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—241no—
312C2×C4—241no—
313C2×C4—241no—
314C2×C4—241no—
315C8—81no—
316C8—241no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—41no—
320C2×C3—41no—
321C2×C3#160611yescharacteristic
322C2×C3#219711yescharacteristic
323C2×C3#220111yescharacteristic
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22#503211yescharacteristic
334C22#517311yescharacteristic
335C22#520211yescharacteristic
336C4—41no—
337C4—41no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—122no—
343C4—241no—
344C4—241no—
345C4—241no—
346C4—481no—
347C3#1226911yescharacteristic, complemented, Sylow
348C2—21no—
349C2—21no—
350C2—41no—
351C2—41no—
352C2#7097311yescharacteristic
353C2#7099011yescharacteristic
354C2#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#12269
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
347. C3#12269semidirect

Maximal subgroups

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

subgroupindex
#776132
C3×#16082
#718942
#718932
#676492
#674492
#674452
#122693