Finite Group Databasebeta

← #10713864

Subgroups — #10713864

Order 768.

492 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 99 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#82386C211yesmaximal, characteristic
2C3×#1799C211yesmaximal, characteristic
3C4×#834C211yesmaximal, characteristic
4#67695C212yesmaximal
5#67695C212yesmaximal
6#13300—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#870C411yescharacteristic, complemented
9C2×C3×#99C2212yes—
10C3×#380—21no—
11#5221C2211yescharacteristic
12#5221C2211yescharacteristic
13C2×#834C2212yes—
14#5210—21no—
15#5210—21no—
16C4×#217—21no—
17#4977C411yescharacteristic, complemented
18#4975C411yescharacteristic, complemented
19#4948—21no—
20#4948—21no—
21C4×C4⋊Dic3C411yescharacteristic, complemented
22C3×#342—21no—
23#4526—21no—
24#2566—31no—
25#1799D311yescharacteristic
26C4×#99—31no—
27#1393—32no—
28#1393—32no—
29C22×C3×D4C2311yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C3×C4×D4—41no—
33C3×C4×D4—41no—
34C2×C3×C4⋊C4—41no—
35C2×C3×C4⋊C4C2×C411yescharacteristic
36C2×C3×C22⋊C4—22no—
37C2×C3×C22⋊C4D411yescharacteristic
38C2×C3×C22⋊C4D411yescharacteristic
39C2×C3×C42C2×C411yescharacteristic
40C2×#217—22no—
41C2×#217—21no—
42C2×#217—21no—
43C2×#217—21no—
44C2×#217—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3C2×C411yescharacteristic
47C2×C4⋊Dic3C2×C411yescharacteristic
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×C4×Dic3—41no—
53C2×C4×Dic3C2×C411yescharacteristic
54C2×C4×Dic3C2×C411yescharacteristic
55#870—41no—
56#870—41no—
57C3×#99—24no—
58C3×#95—41no—
59#834—22no—
60#834Q814yes—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—41no—
65C2×#125—31no—
66C2×C4×D4C2×D311yescharacteristic
67C2×#99—32no—
68C2×#99C2×D312yes—
69#396—61no—
70#396—61no—
71#390—31no—
72#390—31no—
73#385—61no—
74#385—61no—
75#384—31no—
76#382—31no—
77#380—21no—
78C4×C4⋊C4—31no—
79C4×C22⋊C4—61no—
80#342—21no—
81#342—61no—
82C24×C3C2×D411yescharacteristic
83C24×C3C2×D411yescharacteristic
84C2×C3×D4—41no—
85C2×C3×D4—41no—
86C2×C3×D4C2×Q812yes—
87C2×C3×D4#4112yes—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4—42no—
93C22×C3×C4—81no—
94C22×C3×C4—81no—
95C22×C3×C4C22⋊C411yescharacteristic
96C22×C3×C4C22⋊C411yescharacteristic
97C22×C3×C4C22⋊C411yescharacteristic
98C22×C3×C4C22⋊C411yescharacteristic
99C22×C3×C4C22×C411yescharacteristic
100C22×Dic3—21no—
101C22×Dic3—21no—
102C22×Dic3—21no—
103C22×Dic3—21no—
104C22×Dic3—42no—
105C3×C4⋊C4—41no—
106C3×C4⋊C4—82no—
107C3×C22⋊C4—41no—
108C3×C22⋊C4—41no—
109C3×C22⋊C4—42no—
110C3×C22⋊C4—42no—
111C3×C42—21no—
112C3×C42—21no—
113#217—24no—
114#217—42no—
115#217—42no—
116#217—42no—
117#217—81no—
118#217—81no—
119C4⋊Dic3—42no—
120C4⋊Dic3—41no—
121C4⋊Dic3—41no—
122Dic3⋊C4—42no—
123Dic3⋊C4—41no—
124Dic3⋊C4—41no—
125Dic3⋊C4—81no—
126Dic3⋊C4—81no—
127C4×Dic3—41no—
128C4×Dic3—41no—
129C4×Dic3—82no—
130C22×D4C22×D311yescharacteristic
131C23×C4D1211yescharacteristic
132C23×C4#6711yescharacteristic
133#125—121no—
134#125—121no—
135C4×D4—41no—
136C4×D4—41no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—61no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4C4×D311yescharacteristic
146C2×C22⋊C4—22no—
147C2×C22⋊C4—62no—
148C2×C22⋊C4—61no—
149C2×C22⋊C4—61no—
150C2×C22⋊C4—61no—
151C2×C22⋊C4—61no—
152C2×C22⋊C4D1211yescharacteristic
153C2×C22⋊C4#6711yescharacteristic
154C2×C42—31no—
155C2×C42—31no—
156C2×C42—121no—
157C2×C42C4×D311yescharacteristic
158#99—24no—
159#99—34no—
160#99—62no—
161#95—41no—
162#95—61no—
163#95—61no—
164#95—61no—
165#95—121no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—41no—
171C23×C3—41no—
172C23×C3—41no—
173C23×C3—41no—
174C23×C3C2×C22⋊C411yescharacteristic
175C23×C3C22⋊Q812yes—
176C23×C3#12411yescharacteristic
177C23×C3C4⋊Q811yescharacteristic
178C3×D4—42no—
179C3×D4—42no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—41no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—81no—
192C2×C3×C4—81no—
193C2×C3×C4—82no—
194C2×C3×C4—82no—
195C2×C3×C4—82no—
196C2×C3×C4#9911yescharacteristic
197C2×C3×C4#9911yescharacteristic
198C2×C3×C4C4×Q812yes—
199C2×Dic3—22no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—44no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—82no—
213C2×Dic3—82no—
214C24C2×D1211yescharacteristic
215C24C2×#6711yescharacteristic
216C2×D4—41no—
217C2×D4—41no—
218C2×D4#23712yes—
219C2×D4D3×Q812yes—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—42no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—81no—
230C22×C4—81no—
231C22×C4—122no—
232C22×C4(C2×D3)⋊C411yescharacteristic
233C22×C4(C2×D3)⋊C411yescharacteristic
234C22×C4(C2×D3)⋊C411yescharacteristic
235C22×C4(C2×D3)⋊C411yescharacteristic
236C22×C4C2×C4×D311yescharacteristic
237C4⋊C4—41no—
238C4⋊C4—82no—
239C4⋊C4—121no—
240C4⋊C4—121no—
241C4⋊C4—122no—
242C4⋊C4—122no—
243C4⋊C4—121no—
244C4⋊C4—121no—
245C4⋊C4—241no—
246C4⋊C4—241no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—64no—
252C22⋊C4—122no—
253C22⋊C4—122no—
254C22⋊C4—122no—
255C22⋊C4—241no—
256C22⋊C4—241no—
257C42—21no—
258C42—21no—
259C42—121no—
260C42—121no—
261C42—242no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—22no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—42no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—81no—
275C22×C3—81no—
276C22×C3#39111yescharacteristic
277C22×C3C2×#9911yescharacteristic
278C22×C3#41011yescharacteristic
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—22no—
282C3×C4—41no—
283C3×C4—41no—
284C3×C4—41no—
285C3×C4—162no—
286Dic3—42no—
287Dic3—81no—
288Dic3—84no—
289Dic3—81no—
290Dic3—81no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23#89712yes—
300C23C2×(C2×D3)⋊C411yescharacteristic
301C23#93611yescharacteristic
302C23#94411yescharacteristic
303D4—42no—
304D4—42no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—62no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—124no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—242no—
335C2×C4—242no—
336C2×C4#80611yescharacteristic
337C2×C4#80611yescharacteristic
338C2×C4#88712yes—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—42no—
342C2×C3—41no—
343C2×C3—41no—
344C2×C3#199611yescharacteristic
345C2×C3#212611yescharacteristic
346C2×C3#212711yescharacteristic
347C22—21no—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—21no—
352C22—42no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—81no—
360C22—81no—
361C22C2×#80611yescharacteristic
362C22#497311yescharacteristic
363C22#500611yescharacteristic
364C4—21no—
365C4—21no—
366C4—22no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—122no—
371C4—162no—
372C4—241no—
373C4—244no—
374C4—241no—
375C4—241no—
376C3#1330011yescharacteristic, complemented, Sylow
377C2—21no—
378C2—21no—
379C2—41no—
380C2—42no—
381C2—41no—
382C2#6952911yescharacteristic
383C2#7001911yescharacteristic
384C2#7006011yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC22×D4C22×D3
FittingC3×#1799C2
socleC22×C3C2×#99
prime radical, cosocleC22×C3×D4C23
hypercenter#1799D3
hypocenterC3#13300
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
8. C2×#870C4semidirect
17. #4977C4semidirect
18. #4975C4semidirect
21. C4×C4⋊Dic3C4semidirect
376. C3#13300semidirect

Maximal subgroups

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

subgroupindex
#823862
C3×#17992
C4×#8342
#676952
#133003