Finite Group Databasebeta

← #10690810

Subgroups — #10690810

Order 768.

530 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 119 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#82074C211yesmaximal, characteristic
2C2×#4926C211yesmaximal, characteristic
3C3×#1749C211yesmaximal, characteristic
4C2×#4465C212yesmaximal, complemented
5#67659C212yesmaximal, complemented
6#9125—31nomaximal, Sylow
7C22×C3×C42C2211yescharacteristic
8C2×#870C2211yescharacteristic
9C2×C3×C22⋊C8—21no—
10C2×C3×#97C2212yes—
11#5212C411yescharacteristic
12#5212C411yescharacteristic
13#5210—21no—
14#5210—21no—
15C2×#830—21no—
16#4936C2211yescharacteristic
17#4933—21no—
18#4933—21no—
19#4928C411yescharacteristic
20#4928C411yescharacteristic
21#4926—24no—
22C2×C4⋊(C3⋊C8)—21no—
23C2×#803C2212yes—
24C3×#336—21no—
25#4465—24no—
26C2×#432—31no—
27#2851—31no—
28#1749D311yescharacteristic
29C2×#330—32no—
30#1295—32no—
31C23×C3×C4C2×C411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C22×C3×C8—22no—
35C2×C3×C42—22no—
36C2×C3×C42—22no—
37C2×C3×C42C2×C411yescharacteristic
38C2×C3×C42D411yescharacteristic
39C2×C3×C42D411yescharacteristic
40C2×C3×C42C2311yescharacteristic
41C2×#217—21no—
42C2×#217—21no—
43C2×C4⋊Dic3C2×C411yescharacteristic
44C2×C4⋊Dic3C2×C411yescharacteristic
45C2×C4⋊Dic3C2×C411yescharacteristic
46C2×C4⋊Dic3C2×C411yescharacteristic
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C22×C3⋊C8—22no—
52#870—22no—
53#870—22no—
54C3×C22⋊C8—24no—
55C3×#97—22no—
56C3×#97—22no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63#830—24no—
64C4⋊(C3⋊C8)—24no—
65#803—22no—
66#803D414yes—
67C2×#125—31no—
68C22×C42C2×D311yescharacteristic
69#432—64no—
70C2×C4⋊C8—61no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—61no—
73C2×#97—32no—
74C2×#97C2×D312yes—
75#389—31no—
76#389—31no—
77#387—61no—
78#387—61no—
79#385—61no—
80#385—61no—
81#382—31no—
82#380—31no—
83#380—31no—
84#336—21no—
85#330—64no—
86C24×C3C22⋊C411yescharacteristic
87C22×C3×C4—22no—
88C22×C3×C4—22no—
89C22×C3×C4—22no—
90C22×C3×C4—21no—
91C22×C3×C4—22no—
92C22×C3×C4—21no—
93C22×C3×C4—41no—
94C22×C3×C4—41no—
95C22×C3×C4—41no—
96C22×C3×C4—41no—
97C22×C3×C4C22⋊C411yescharacteristic
98C22×C3×C4C22⋊C411yescharacteristic
99C22×C3×C4C22⋊C411yescharacteristic
100C22×C3×C4C22×C411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×Dic3—21no—
104C22×Dic3—21no—
105C22×Dic3—21no—
106C22×Dic3—21no—
107C2×C3×C8—24no—
108C2×C3×C8—42no—
109C2×C3×C8—42no—
110C3×C42—22no—
111C3×C42—21no—
112C3×C42—22no—
113C3×C42—21no—
114C3×C42C2×D412yes—
115C3×C42#4112yes—
116#217—42no—
117#217—42no—
118C4⋊Dic3—22no—
119C4⋊Dic3—22no—
120C4⋊Dic3—41no—
121C4⋊Dic3—41no—
122Dic3⋊C4—42no—
123Dic3⋊C4—42no—
124Dic3⋊C4—41no—
125Dic3⋊C4—42no—
126Dic3⋊C4—41no—
127C2×C3⋊C8—24no—
128C2×C3⋊C8—42no—
129C2×C3⋊C8—42no—
130C23×C4C4×D311yescharacteristic
131C23×C4D1211yescharacteristic
132C23×C4#6711yescharacteristic
133C22×C8—22no—
134C22×C8—62no—
135#125—62no—
136#125—62no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4—61no—
145C2×C22⋊C4—61no—
146C2×C22⋊C4—61no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42C4×D311yescharacteristic
150C2×C42D1211yescharacteristic
151C2×C42#6711yescharacteristic
152C2×C42C22×D311yescharacteristic
153C4⋊C8—64no—
154C22⋊C8—24no—
155C22⋊C8—64no—
156#97—22no—
157#97—22no—
158#97—34no—
159#97—62no—
160#95—61no—
161#95—61no—
162#95—61no—
163#95—61no—
164#95—61no—
165#95—61no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3#10412yes—
173C23×C3C2×C22⋊C411yescharacteristic
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—44no—
189C2×C3×C4—44no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4#10112yes—
193C2×C3×C4C4×D412yes—
194C2×C3×C4#12011yescharacteristic
195C2×C3×C4#12112yes—
196C2×C3×C4#12311yescharacteristic
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C3×C8—44no—
210C3⋊C8—44no—
211C24(C2×D3)⋊C411yescharacteristic
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4—61no—
225C22×C4—61no—
226C22×C4(C2×D3)⋊C411yescharacteristic
227C22×C4(C2×D3)⋊C411yescharacteristic
228C22×C4(C2×D3)⋊C411yescharacteristic
229C22×C4C2×C4×D311yescharacteristic
230C22×C4C2×D1211yescharacteristic
231C22×C4C2×#6711yescharacteristic
232C2×C8—24no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—64no—
236C2×C8—122no—
237C2×C8—122no—
238C4⋊C4—62no—
239C4⋊C4—62no—
240C4⋊C4—121no—
241C4⋊C4—121no—
242C4⋊C4—122no—
243C4⋊C4—122no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C4⋊C4—122no—
247C22⋊C4—122no—
248C22⋊C4—122no—
249C42—22no—
250C42—21no—
251C42—21no—
252C42—22no—
253C42D3×D412yes—
254C42#23712yes—
255C22×C3—22no—
256C22×C3—22no—
257C22×C3—22no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3#38611yescharacteristic
263C22×C3C2×#10111yescharacteristic
264C22×C3#41311yescharacteristic
265C22×C3C2×#10411yescharacteristic
266C22×C3#42111yescharacteristic
267C22×C3#42111yescharacteristic
268C22×C3#42111yescharacteristic
269C3×C4—22no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—22no—
273C3×C4—22no—
274C3×C4—41no—
275C3×C4—41no—
276Dic3—81no—
277Dic3—81no—
278Dic3—81no—
279Dic3—81no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23#80512yes—
287C23C2×(C2×D3)⋊C411yescharacteristic
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4#82412yes—
319C2×C4#67⋊C412yes—
320C2×C4#88211yescharacteristic
321C2×C4#88611yescharacteristic
322C2×C4#93912yes—
323C8—44no—
324C8—124no—
325C2×C3—42no—
326C2×C3#152811yescharacteristic
327C2×C3#153211yescharacteristic
328C2×C3#153511yescharacteristic
329C2×C3#153711yescharacteristic
330C2×C3#187111yescharacteristic
331C2×C3#189711yescharacteristic
332C2×C3#190111yescharacteristic
333C22—22no—
334C22—22no—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22C2×#80511yescharacteristic
341C22#495811yescharacteristic
342C22#500511yescharacteristic
343C22#513211yescharacteristic
344C22C2×#82411yescharacteristic
345C22#513911yescharacteristic
346C22#513911yescharacteristic
347C4—22no—
348C4—21no—
349C4—21no—
350C4—22no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—241no—
355C4—241no—
356C4—241no—
357C4—241no—
358C3#912511yescharacteristic, complemented, Sylow
359C2—42no—
360C2#6945311yescharacteristic
361C2#6979911yescharacteristic
362C2#6986311yescharacteristic
363C2#6996611yescharacteristic
364C2#6998611yescharacteristic
365C2#7080911yescharacteristic
366C2#7081511yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1749C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1749D3
hypocenterC3#9125
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
4. C2×#4465C2semidirect
5. #67659C2semidirect
358. C3#9125semidirect

Maximal subgroups

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

subgroupindex
#820742
C2×#49262
C3×#17492
C2×#44652
#676592
#91253