Finite Group Databasebeta

← #10708024

Subgroups — #10708024

Order 768.

394 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C3×#2784C211yesmaximal, characteristic
2#69719C211yesmaximal, characteristic, complemented
3#69707C211yesmaximal, characteristic, complemented
4#69643C211yesmaximal, characteristic
5#67652C211yesmaximal, characteristic, complemented
6#67648C211yesmaximal, characteristic, complemented
7#67640C211yesmaximal, characteristic
8#12223—31nomaximal, Sylow
9#5500C2211yescharacteristic
10C3×#400C2211yescharacteristic
11C3×#396—21no—
12C3×#395C2211yescharacteristic
13C3×#393—21no—
14C3×#390C2211yescharacteristic
15C3×#387—21no—
16C3×#386—21no—
17C3×#385—21no—
18C3×#380—21no—
19#5228—21no—
20#5215—21no—
21#4970—21no—
22#4530C2211yescharacteristic
23#4528C2211yescharacteristic
24#4528C2211yescharacteristic
25#4523—21no—
26#4452—21no—
27#4452—21no—
28#2784D311yescharacteristic
29#2027—31no—
30#1617—31no—
31#1605—31no—
32#1354—31no—
33#1353—31no—
34#1318—31no—
35C22×C3×Q8D411yescharacteristic
36C22×C3×D4D411yescharacteristic
37C23×C3×C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C4⋊C4C2311yescharacteristic
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C42D411yescharacteristic
49C2×#216—41no—
50#926—21no—
51#924—21no—
52#924—21no—
53C2×#208—21no—
54#890—41no—
55#888—41no—
56C3×#95—21no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95—21no—
60C3×#95—21no—
61C3×#95D411yescharacteristic
62C3×#95D411yescharacteristic
63#835—41no—
64#832—41no—
65#832—41no—
66#830—21no—
67#830—21no—
68#810—41no—
69#807—41no—
70#544—31no—
71#484—61no—
72#451—61no—
73#418—61no—
74#400C2×D311yescharacteristic
75#396—21no—
76#395C2×D311yescharacteristic
77#393—21no—
78#390C2×D311yescharacteristic
79#387—21no—
80#386—21no—
81#385—21no—
82#380—21no—
83#343—31no—
84#339—61no—
85#328—61no—
86#328—61no—
87#324—31no—
88#324—31no—
89C24×C3—21no—
90C2×C3×Q8—41no—
91C2×C3×Q8—81no—
92C2×C3×D4—21no—
93C2×C3×D4—41no—
94C2×C3×D4—41no—
95C2×C3×D4Q1612yescomplemented
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4—42no—
102C22×C3×C4—81no—
103C22×C3×C4—81no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C2×Dic6—21no—
108C2×Dic6—21no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—42no—
112C3×C4⋊C4—81no—
113C3×C4⋊C4—81no—
114C3×C22⋊C4—42no—
115C3×C22⋊C4—81no—
116C3×C22⋊C4—81no—
117C3×C22⋊C4—81no—
118C3×C22⋊C4—81no—
119C3×C42—41no—
120#217—41no—
121#216—82no—
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126Dic3⋊C4—41no—
127Dic3⋊C4—41no—
128C4×Dic3—41no—
129#208—41no—
130#208—41no—
131C2×C3⋊C8—41no—
132C2×C3⋊C8—41no—
133C2×C3⋊C8—41no—
134C22×Q8#6711yescharacteristic
135C22×D4#6711yescharacteristic
136C23×C4—21no—
137C2×Q16—121no—
138C2×M16—61no—
139C4⋊Q8—121no—
140#125—121no—
141C22⋊Q8—61no—
142C22⋊Q8—61no—
143#117—61no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4#6711yescharacteristic
148C2×C4⋊C4C22×D311yescharacteristic
149C2×C22⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C42#6711yescharacteristic
155#107—121no—
156#103—121no—
157#103—121no—
158#102—121no—
159#102—121no—
160C22⋊C8—61no—
161C22⋊C8—61no—
162#95—21no—
163#95—21no—
164#95—21no—
165#95—21no—
166#95—21no—
167#95#6711yescharacteristic
168#95#6711yescharacteristic
169C23×C3—22no—
170C23×C3—41no—
171C23×C3—41no—
172C23×C3—41no—
173C23×C3—41no—
174C23×C3—41no—
175C23×C3—41no—
176C23×C3#12011yescharacteristic
177C3×Q8—42no—
178C3×Q8—81no—
179C3×D4—42no—
180C3×D4—81no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—22no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
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—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—84no—
203C2×C3×C4C2×Q1611yescharacteristic
204C2×C3×C4#13711yescharacteristic
205C2×Dic3—41no—
206C2×Dic3—41no—
207Dic6—81no—
208Dic6—81no—
209C3⋊C8—81no—
210C3⋊C8—81no—
211C3⋊C8—81no—
212C24—21no—
213C2×Q8—41no—
214C2×Q8—61no—
215C2×Q8—61no—
216C2×Q8—81no—
217C2×D4—21no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4#21612yescomplemented
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—42no—
227C22×C4—81no—
228C22×C4—81no—
229C22×C4C2×#6711yescharacteristic
230C22×C4C2×#6711yescharacteristic
231C22×C4C2×#6711yescharacteristic
232Q16—242no—
233M16—121no—
234M16—121no—
235C2×C8—121no—
236C2×C8—121no—
237C2×C8—121no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—81no—
246C4⋊C4—81no—
247C4⋊C4—121no—
248C4⋊C4—121no—
249C22⋊C4—42no—
250C22⋊C4—81no—
251C22⋊C4—81no—
252C22⋊C4—81no—
253C22⋊C4—81no—
254C22⋊C4—121no—
255C42—41no—
256C42—121no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—42no—
260C22×C3—42no—
261C22×C3—42no—
262C22×C3—81no—
263C22×C3—81no—
264C22×C3—81no—
265C22×C3—81no—
266C22×C3#45111yescharacteristic
267C22×C3#45511yescharacteristic
268C22×C3B(4,𝔽2)11yescharacteristic
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—42no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—81no—
275C3×C4—81no—
276C3×C4—81no—
277C3×C4—81no—
278Dic3—81no—
279Dic3—81no—
280C23—22no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23#95311yescharacteristic
288Q8—42no—
289Q8—81no—
290Q8—241no—
291Q8—241no—
292D4—42no—
293D4—81no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—84no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4#94211yescharacteristic
319C2×C4C2×#21611yescharacteristic
320C8—241no—
321C8—241no—
322C8—241no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—82no—
326C2×C3#160211yescharacteristic
327C2×C3#220011yescharacteristic
328C2×C3#220411yescharacteristic
329C22—21no—
330C22—21no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22#522811yescharacteristic
339C22#524211yescharacteristic
340C22#524411yescharacteristic
341C4—21no—
342C4—21no—
343C4—42no—
344C4—41no—
345C4—41no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4—81no—
350C4—241no—
351C4—241no—
352C3#1222311yescharacteristic, complemented, Sylow
353C2—21no—
354C2—21no—
355C2—82no—
356C2#7189311yescharacteristic
357C2#7193011yescharacteristic
358C2#7194711yescharacteristic
subgroupG/H
centerC22#5244
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2784C2
socleC22×C3B(4,𝔽2)
prime radicalC22×C3×D4D4
hypercenter#2784D3
hypocenterC3#12223
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2784

Complements

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

NH ≅ G/Ntype
2. #69719C2semidirect
3. #69707C2semidirect
5. #67652C2semidirect
6. #67648C2semidirect
95. C2×C3×D4Q16semidirect
220. C2×D4#216semidirect
352. C3#12223semidirect

Maximal subgroups

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

subgroupindex
C3×#27842
#697192
#697072
#696432
#676522
#676482
#676402
#122233