Finite Group Databasebeta

← #10691178

Subgroups — #10691178

Order 768.

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

normal subgroups only

This group has 502 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#82391C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic
3C3×C4×#100C211yesmaximal, characteristic
4#69863C212yesmaximal, complemented
5#67666C212yesmaximal, complemented
6#9189—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#870C2211yescharacteristic
9C2×C3×#100—21no—
10C3×C4×M16C2212yes—
11#5222—21no—
12#5210—21no—
13#5210—21no—
14#5207—21no—
15C2×#830—21no—
16#5011—24no—
17#4983—21no—
18#4953C411yescharacteristic
19#4953C411yescharacteristic
20#4949C411yescharacteristic
21#4949C411yescharacteristic
22#4937C2211yescharacteristic
23#4926C2212yes—
24C3×#343—21no—
25#4465—24no—
26#2973—31no—
27#2633—31no—
28C4×#100D311yescharacteristic
29#1528—32no—
30#1288—32no—
31C22×C3×D4D411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C2×C3×M16—22no—
35C3×C4×D4—22no—
36C3×C4×D4—22no—
37C2×C3×C4⋊C4D411yescharacteristic
38C2×C3×C22⋊C4C2×C411yescharacteristic
39C2×C3×C22⋊C4C2×C411yescharacteristic
40C2×C3×C42C2311yescharacteristic
41C2×#217—21no—
42C2×#217—21no—
43C2×#217—21no—
44C2×#217—21no—
45C2×C4⋊Dic3C2×C411yescharacteristic
46C2×C4⋊Dic3C2×C411yescharacteristic
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×#208—22no—
52C22×C3⋊C8—22no—
53#870—22no—
54#870—22no—
55C3×#100—24no—
56C3×#97—22no—
57C3×C4×C8—22no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95C2×C411yescharacteristic
63C3⋊#95C2×C411yescharacteristic
64#830—22no—
65#830—22no—
66#830—22no—
67C4⋊(C3⋊C8)—22no—
68#803D414yes—
69C2×#125—31no—
70C2×C4×D4C2×D311yescharacteristic
71#435—64no—
72#432—32no—
73C2×#100—21no—
74#407—61no—
75C2×C22⋊C8—61no—
76C4×M16C2×D312yes—
77#399—31no—
78#399—31no—
79#398—61no—
80#397—31no—
81#397—31no—
82#394—61no—
83#385—61no—
84#385—61no—
85#383—31no—
86#343—21no—
87#330—64no—
88C24×C3C22⋊C411yescharacteristic
89C24×C3C22⋊C411yescharacteristic
90C2×C3×D4—22no—
91C2×C3×D4—22no—
92C2×C3×D4—41no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—81no—
98C22×C3×C4—81no—
99C22×C3×C4C22⋊C411yescharacteristic
100C22×C3×C4C22⋊C411yescharacteristic
101C22×C3×C4C22×C411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×Dic3—21no—
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C3×M16—24no—
109C3×M16—42no—
110C2×C3×C8—42no—
111C3×C4⋊C4—22no—
112C3×C22⋊C4—22no—
113C3×C22⋊C4—22no—
114C3×C42C2×D412yes—
115C3×C42#4112yes—
116#217—42no—
117#217—42no—
118#217—81no—
119#217—81no—
120C4⋊Dic3—22no—
121C4⋊Dic3—22no—
122Dic3⋊C4—41no—
123Dic3⋊C4—42no—
124Dic3⋊C4—41no—
125Dic3⋊C4—81no—
126Dic3⋊C4—81no—
127#208—82no—
128C2×C3⋊C8—24no—
129C2×C3⋊C8—82no—
130C22×D4#6711yescharacteristic
131C23×C4D1211yescharacteristic
132C23×C4D1211yescharacteristic
133C2×M16—22no—
134C2×M16—62no—
135C22×C8—62no—
136#125—62no—
137#125—62no—
138C4×D4—22no—
139C4×D4—22no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4—31no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4—61no—
146C2×C4⋊C4#6711yescharacteristic
147C2×C22⋊C4—61no—
148C2×C22⋊C4—61no—
149C2×C22⋊C4—61no—
150C2×C22⋊C4—61no—
151C2×C22⋊C4C4×D311yescharacteristic
152C2×C22⋊C4C4×D311yescharacteristic
153C2×C42C22×D311yescharacteristic
154C4⋊C8—62no—
155#100—24no—
156C22⋊C8—62no—
157C22⋊C8—62no—
158C22⋊C8—62no—
159#97—22no—
160#97—34no—
161C4×C8—22no—
162#95—31no—
163#95—31no—
164#95—61no—
165#95—61no—
166#95—61no—
167#95—61no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—41no—
174C23×C3—41no—
175C23×C3—41no—
176C23×C3—41no—
177C23×C3#10412yes—
178C23×C3C2×C22⋊C411yescharacteristic
179C3×D4—44no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—81no—
191C2×C3×C4—81no—
192C2×C3×C4C4×D412yes—
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12112yes—
195C2×C3×C4#12311yescharacteristic
196C2×Dic3—41no—
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—
208C3×C8—44no—
209C3⋊C8—44no—
210C24(C2×D3)⋊C411yescharacteristic
211C24(C2×D3)⋊C411yescharacteristic
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—41no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—61no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4(C2×D3)⋊C411yescharacteristic
226C22×C4(C2×D3)⋊C411yescharacteristic
227C22×C4C2×C4×D311yescharacteristic
228C22×C4C2×D1211yescharacteristic
229C22×C4C2×#6711yescharacteristic
230M16—24no—
231M16—42no—
232M16—242no—
233C2×C8—42no—
234C2×C8—64no—
235C2×C8—242no—
236C4⋊C4—22no—
237C4⋊C4—62no—
238C4⋊C4—62no—
239C4⋊C4—121no—
240C4⋊C4—121no—
241C4⋊C4—122no—
242C4⋊C4—241no—
243C4⋊C4—241no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—122no—
247C22⋊C4—122no—
248C22⋊C4—241no—
249C22⋊C4—241no—
250C42D3×D412yes—
251C42#23712yes—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—22no—
255C22×C3—42no—
256C22×C3—42no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—81no—
263C22×C3—81no—
264C22×C3#38611yescharacteristic
265C22×C3C2×#10411yescharacteristic
266C22×C3#42111yescharacteristic
267C3×C4—22no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—22no—
271C3×C4—41no—
272C3×C4—41no—
273Dic3—81no—
274Dic3—81no—
275Dic3—81no—
276Dic3—81no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23#80512yes—
287C23C2×(C2×D3)⋊C411yescharacteristic
288D4—44no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4#67⋊C412yes—
314C2×C4#88211yescharacteristic
315C2×C4#88611yescharacteristic
316C2×C4#93912yes—
317C8—44no—
318C8—124no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—42no—
322C2×C3—81no—
323C2×C3#187111yescharacteristic
324C2×C3#191111yescharacteristic
325C2×C3#191511yescharacteristic
326C22—21no—
327C22—21no—
328C22—22no—
329C22—42no—
330C22—42no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—81no—
337C22—81no—
338C22C2×#80511yescharacteristic
339C22#495811yescharacteristic
340C22#500511yescharacteristic
341C4—22no—
342C4—21no—
343C4—21no—
344C4—22no—
345C4—41no—
346C4—41no—
347C4—241no—
348C4—241no—
349C4—241no—
350C4—241no—
351C3#918911yescharacteristic, complemented, Sylow
352C2—21no—
353C2—21no—
354C2—42no—
355C2—81no—
356C2#6945311yescharacteristic
357C2#7007111yescharacteristic
358C2#7013811yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×C4×#100C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenterC4×#100D3
hypocenterC3#9189
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
4. #69863C2semidirect
5. #67666C2semidirect
351. C3#9189semidirect

Maximal subgroups

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

subgroupindex
#823912
#823022
C3×C4×#1002
#698632
#676662
#91893