Finite Group Databasebeta

← #10705758

Subgroups — #10705758

Order 768.

652 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 652 conjugacy classes of subgroups — too many to draw. Only its 87 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#82378C211yesmaximal, characteristic
2C3×#1465C211yesmaximal, characteristic, complemented
3C4×#834C211yesmaximal, characteristic, complemented
4#67448C214yesmaximal, complemented
5#11776—31nomaximal, Sylow
6C2×C3×C4×D4C2211yescharacteristic
7C2×#940—21no—
8C2×#930—21no—
9C2×#930—21no—
10C2×#874C411yescharacteristic, complemented
11C3×#407—21no—
12C2×#834—21no—
13C4×#217—21no—
14#4988—21no—
15#4960C412yescomplemented
16#4960C2212yes—
17C4×C4⋊Dic3C411yescharacteristic, complemented
18C3×#324—21no—
19C3×#323C2212yes—
20#4526C2212yes—
21#2541—31no—
22C4×#99—31no—
23#1465D311yescharacteristic, complemented
24#1343—34no—
25C22×C3×D4D411yescharacteristic
26C23×C3×C4D411yescharacteristic
27C23×C3×C4D411yescharacteristic
28C22×#67—22no—
29C22×#67—22no—
30C22×D12—21no—
31C22×D12—21no—
32C22×D12C2×C412yes—
33C2×C3×M16—41no—
34C3×C4×D4—41no—
35C3×C4×D4—41no—
36C2×C3×C4⋊C4D411yescharacteristic
37C2×C3×C22⋊C4C2×C411yescharacteristic
38C2×C3×C22⋊C4C2311yescharacteristic
39C2×C3×C42C2×C411yescharacteristic
40C2×#217—22no—
41#940—44no—
42#930—44no—
43#930—44no—
44C2×(C2×D3)⋊C4—22no—
45C2×(C2×D3)⋊C4—22no—
46C2×C4⋊Dic3—21no—
47C2×C4×Dic3—21no—
48C2×C4×Dic3—41no—
49#874—41no—
50#874—41no—
51C3×C22⋊C8—22no—
52C3×C22⋊C8—41no—
53#834—24no—
54C3⋊#95—41no—
55C3⋊#95C2×C412yes—
56C2×#127—31no—
57C2×#127—61no—
58C2×#121—61no—
59C2×#121—61no—
60C2×C4×D4C2×D311yescharacteristic
61C2×#99—61no—
62#407—21no—
63#392—32no—
64#392—32no—
65#390—61no—
66C4×C4⋊C4—31no—
67C4×C22⋊C4—61no—
68#342—32no—
69#324—21no—
70#323C2×D312yes—
71C24×C3C22⋊C411yescharacteristic
72C24×C3C2×D411yescharacteristic
73C23×D3—22no—
74C2×C3×D4—22no—
75C2×C3×D4—22no—
76C2×C3×D4—41no—
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C22×C3×C4—42no—
82C22×C3×C4—42no—
83C22×C3×C4C22⋊C411yescharacteristic
84C22×C3×C4C22⋊C411yescharacteristic
85C22×C3×C4C22⋊C411yescharacteristic
86C22×C3×C4C22×C411yescharacteristic
87C22×C3×C4C2×D411yescharacteristic
88C2×#67—44no—
89C2×#67—44no—
90C2×#67—44no—
91C2×#67—44no—
92C2×#67—82no—
93C2×#67—82no—
94C22×Dic3—22no—
95C22×Dic3—42no—
96C2×D12—42no—
97C2×D12—42no—
98C2×D12—42no—
99C2×D12—42no—
100C2×D12—42no—
101C2×D12—41no—
102C2×D12—41no—
103C2×D12—42no—
104C2×D12—82no—
105C3×M16—82no—
106C2×C3×C8—42no—
107C3×C4⋊C4—41no—
108C3×C22⋊C4—41no—
109C3×C22⋊C4—41no—
110C3×C42—21no—
111C3×C42—21no—
112#217—24no—
113#217—42no—
114(C2×D3)⋊C4—44no—
115(C2×D3)⋊C4—44no—
116C4⋊Dic3—24no—
117C4×Dic3—42no—
118C4×Dic3—82no—
119C22×D4—32no—
120C22×D4—62no—
121C22×D4—62no—
122C22×D4—61no—
123C22×D4—61no—
124C22×D4D1211yescharacteristic
125C23×C4D1211yescharacteristic
126C23×C4#6711yescharacteristic
127C2×M16—41no—
128#127—121no—
129#127—121no—
130#127—124no—
131#121—124no—
132#121—124no—
133C4×D4—41no—
134C4×D4—41no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4#6711yescharacteristic
137C2×C22⋊C4—62no—
138C2×C22⋊C4—62no—
139C2×C22⋊C4—62no—
140C2×C22⋊C4C4×D311yescharacteristic
141C2×C22⋊C4C22×D311yescharacteristic
142C2×C42—61no—
143C2×C42—121no—
144C2×C42C4×D311yescharacteristic
145#99—64no—
146C22⋊C8—22no—
147C22⋊C8—41no—
148#95—32no—
149#95—121no—
150C23×C3—21no—
151C23×C3—21no—
152C23×C3—21no—
153C23×C3—21no—
154C23×C3—21no—
155C23×C3—41no—
156C23×C3—41no—
157C23×C3—41no—
158C23×C3—41no—
159C23×C3C2×C22⋊C411yescharacteristic
160C23×C3AGL(1,ℤ/8)12yes—
161C22×D3—42no—
162C22×D3—42no—
163C22×D3—42no—
164C22×D3—42no—
165C22×D3—42no—
166C22×D3—42no—
167C22×D3—42no—
168C3×D4—44no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—22no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—42no—
176C2×C3×C4—42no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—81no—
181C2×C3×C4—81no—
182C2×C3×C4#9911yescharacteristic
183C2×C3×C4#9911yescharacteristic
184C2×C3×C4#10412yes—
185#67—88no—
186#67—88no—
187C2×Dic3—24no—
188C2×Dic3—42no—
189C2×Dic3—42no—
190C2×Dic3—44no—
191C2×Dic3—82no—
192C2×Dic3—82no—
193D12—82no—
194D12—84no—
195D12—84no—
196D12—82no—
197C3×C8—82no—
198C24—62no—
199C24(C2×D3)⋊C411yescharacteristic
200C24C2×D1211yescharacteristic
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—41no—
204C2×D4—122no—
205C2×D4—122no—
206C2×D4—122no—
207C2×D4—122no—
208C2×D4—122no—
209C2×D4—124no—
210C2×D4—124no—
211C2×D4—124no—
212C2×D4—124no—
213C2×D4—121no—
214C2×D4—121no—
215C2×D4—122no—
216C2×D4—242no—
217C2×D4—242no—
218C2×D4—242no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—62no—
226C22×C4—122no—
227C22×C4(C2×D3)⋊C411yescharacteristic
228C22×C4(C2×D3)⋊C411yescharacteristic
229C22×C4(C2×D3)⋊C411yescharacteristic
230C22×C4C2×C4×D311yescharacteristic
231C22×C4C2×#6711yescharacteristic
232M16—82no—
233C2×C8—42no—
234C4⋊C4—41no—
235C4⋊C4—64no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—64no—
239C22⋊C4—124no—
240C22⋊C4—124no—
241C22⋊C4—122no—
242C42—21no—
243C42—21no—
244C42—122no—
245C42—242no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—22no—
249C22×C3—42no—
250C22×C3—42no—
251C22×C3—41no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3—81no—
257C22×C3—81no—
258C22×C3C2×#9911yescharacteristic
259C22×C3#41811yescharacteristic
260C22×C3C2×#10411yescharacteristic
261C2×D3—82no—
262C2×D3—82no—
263C2×D3—82no—
264C2×D3—82no—
265C2×D3—82no—
266C2×D3—82no—
267C2×D3—82no—
268C3×C4—21no—
269C3×C4—22no—
270C3×C4—41no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4#35312yes—
274Dic3—44no—
275Dic3—84no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—122no—
286C23—122no—
287C23—122no—
288C23—122no—
289C23—122no—
290C23—122no—
291C23—122no—
292C23#90812yes—
293C23C2×(C2×D3)⋊C411yescharacteristic
294D4—44no—
295D4—244no—
296D4—242no—
297D4—244no—
298D4—248no—
299D4—248no—
300D4—242no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—64no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—122no—
316C2×C4—122no—
317C2×C4—124no—
318C2×C4—242no—
319C2×C4—242no—
320C2×C4#80611yescharacteristic
321C2×C4#80611yescharacteristic
322C2×C4#82512yes—
323C8—82no—
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—42no—
327C2×C3—81no—
328C2×C3#150711yescharacteristic
329C2×C3#212211yescharacteristic
330C2×C3C2×#35311yescharacteristic
331D3—162no—
332C22—21no—
333C22—21no—
334C22—22no—
335C22—42no—
336C22—42no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—81no—
343C22—81no—
344C22—242no—
345C22—242no—
346C22—242no—
347C22—242no—
348C22—242no—
349C22—242no—
350C22—242no—
351C22C2×#80611yescharacteristic
352C22#513111yescharacteristic
353C22C2×#82511yescharacteristic
354C4—22no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—124no—
360C4—244no—
361C4#447412yes—
362C3#1177611yescharacteristic, complemented, Sylow
363C2—21no—
364C2—21no—
365C2—42no—
366C2—81no—
367C2—482no—
368C2#6997711yescharacteristic
369C2C2×#447411yescharacteristic
370C2#7014611yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1465C2
socleC22×C3C2×#99
prime radical#82378C2
cosocleC2×C3×C22⋊C4C23
hypercenter#1465D3
hypocenterC3#11776
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
2. C3×#1465C2semidirect
3. C4×#834C2semidirect
4. #67448C2semidirect
10. C2×#874C4semidirect
15. #4960C4semidirect
17. C4×C4⋊Dic3C4semidirect
23. #1465D3semidirect
362. C3#11776semidirect

Maximal subgroups

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

subgroupindex
#823782
C3×#14652
C4×#8342
#674482
#117763