Finite Group Databasebeta

← #10695225

Subgroups — #10695225

Order 768.

680 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4929C211yesmaximal, characteristic
2C3×#1797C211yesmaximal, characteristic
3C4×#830C211yesmaximal, characteristic
4#67674C214yesmaximal
5#9878—31nomaximal, Sylow
6C23×Dic6C411yescharacteristic
7C22×C3×C42C2211yescharacteristic
8C22×C4⋊Dic3C411yescharacteristic
9C2×C3×C4⋊C8—21no—
10C2×C3×C22⋊C8C2212yes—
11C2×C3⋊#95C412yes—
12C2×C3⋊#95C2212yes—
13C2×#830C2212yes—
14#4929—21no—
15#4929—21no—
16#4929—21no—
17#4929—21no—
18C2×C4×C3⋊C8—21no—
19C3×#336—21no—
20#4524—21no—
21C2×#391—31no—
22#1797D311yescharacteristic
23C4×C22⋊C8—31no—
24#1280—34no—
25C23×C3×C4C2×C411yescharacteristic
26C23×C3×C4C2×C411yescharacteristic
27C23×C3×C4C2311yescharacteristic
28C23×Dic3C2×C412yes—
29C23×Dic3C2×C412yes—
30C22×Dic6—21no—
31C22×Dic6—21no—
32C22×Dic6—22no—
33C22×Dic6—42no—
34C22×Dic6—42no—
35C22×Dic6C814yescomplemented
36C22×C3×C8—22no—
37C2×C3×C42—42no—
38C2×C3×C42D411yescharacteristic
39C2×C3×C42D411yescharacteristic
40C2×C3×C42D411yescharacteristic
41C2×C3×C42D411yescharacteristic
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—42no—
45C22×C3⋊C8—22no—
46C3×C4⋊C8—42no—
47C3×C22⋊C8—24no—
48C3⋊#95—22no—
49C3⋊#95—22no—
50C3⋊#95—22no—
51C3⋊#95—22no—
52#830—22no—
53#830Q814yescomplemented
54C4×C3⋊C8—42no—
55C23×Q8—31no—
56C22×C4⋊C4—31no—
57C22×C42C2×D311yescharacteristic
58C2×C4⋊C8—21no—
59C2×C22⋊C8—32no—
60C2×C22⋊C8C2×D312yes—
61C2×C4×C8—61no—
62#391—61no—
63#391—61no—
64#391—61no—
65#391—61no—
66C2×#95—32no—
67C2×#95—32no—
68#336—21no—
69#336—61no—
70C24×C3C22×C411yescharacteristic
71C22×C3×C4—22no—
72C22×C3×C4—22no—
73C22×C3×C4—22no—
74C22×C3×C4—22no—
75C22×C3×C4—41no—
76C22×C3×C4—41no—
77C22×C3×C4—41no—
78C22×C3×C4C22⋊C411yescharacteristic
79C22×C3×C4C22⋊C411yescharacteristic
80C22×C3×C4C22⋊C411yescharacteristic
81C22×C3×C4C22⋊C411yescharacteristic
82C22×C3×C4C2×C812yes—
83C22×C3×C4M1612yes—
84C22×C3×C4C2×D411yescharacteristic
85C22×C3×C4C2×D411yescharacteristic
86C22×C3×C4C2×Q812yes—
87C22×C3×C4#4112yes—
88C22×Dic3—22no—
89C22×Dic3—22no—
90C22×Dic3—22no—
91C22×Dic3—24no—
92C22×Dic3—42no—
93C22×Dic3—42no—
94C22×Dic3—42no—
95C22×Dic3—42no—
96C22×Dic3C2×C814yes—
97C2×Dic6—24no—
98C2×Dic6—24no—
99C2×Dic6—42no—
100C2×Dic6—42no—
101C2×Dic6—48no—
102C2×Dic6—42no—
103C2×Dic6—42no—
104C2×Dic6—48no—
105C2×C3×C8—42no—
106C2×C3×C8—42no—
107C2×C3×C8—42no—
108C3×C42—22no—
109C3×C42—22no—
110C3×C42—41no—
111C3×C42—41no—
112C4⋊Dic3—42no—
113C4⋊Dic3—42no—
114C2×C3⋊C8—24no—
115C2×C3⋊C8—42no—
116C2×C3⋊C8—42no—
117C22×Q8—34no—
118C22×Q8—62no—
119C22×Q8—61no—
120C22×Q8—61no—
121C22×Q8—122no—
122C22×Q8—122no—
123C23×C4—32no—
124C23×C4—32no—
125C23×C4C4×D311yescharacteristic
126C23×C4C4×D311yescharacteristic
127C23×C4C22×D311yescharacteristic
128C22×C8—22no—
129C22×C8—62no—
130C2×C4⋊C4—61no—
131C2×C4⋊C4—61no—
132C2×C4⋊C4—122no—
133C2×C42—42no—
134C2×C42D1211yescharacteristic
135C2×C42D1211yescharacteristic
136C2×C42#6711yescharacteristic
137C2×C42#6711yescharacteristic
138C4⋊C8—42no—
139C22⋊C8—24no—
140C22⋊C8—34no—
141C22⋊C8—62no—
142C4×C8—122no—
143#95—62no—
144#95—62no—
145#95—62no—
146#95—62no—
147C23×C3—21no—
148C23×C3—21no—
149C23×C3—21no—
150C23×C3—21no—
151C23×C3C2×C22⋊C411yescharacteristic
152C23×C3C4×Q812yes—
153C23×C3C22×C811yescharacteristic
154C23×C3C2×M1611yescharacteristic
155C23×C3#13112yes—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—24no—
165C2×C3×C4—24no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—42no—
170C2×C3×C4—42no—
171C2×C3×C4—44no—
172C2×C3×C4C22⋊C812yes—
173C2×C3×C4C22⋊C812yes—
174C2×C3×C4#9912yes—
175C2×C3×C4#10012yes—
176C2×C3×C4#10112yes—
177C2×C3×C4C22⋊Q812yes—
178C2×C3×C4#12411yescharacteristic
179C2×C3×C4C4⋊Q811yescharacteristic
180C2×Dic3—24no—
181C2×Dic3—44no—
182C2×Dic3—42no—
183C2×Dic3—42no—
184C2×Dic3—44no—
185C2×Dic3—42no—
186C2×Dic3—42no—
187C2×Dic3—42no—
188C2×Dic3—84no—
189Dic6—48no—
190Dic6—48no—
191C3×C8—82no—
192C3⋊C8—44no—
193C24C2×C4×D311yescharacteristic
194C2×Q8—64no—
195C2×Q8—64no—
196C2×Q8—128no—
197C2×Q8—122no—
198C2×Q8—122no—
199C2×Q8—128no—
200C2×Q8—122no—
201C2×Q8—122no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—34no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—62no—
211C22×C4—62no—
212C22×C4—62no—
213C22×C4—64no—
214C22×C4—122no—
215C22×C4—122no—
216C22×C4—122no—
217C22×C4—122no—
218C22×C4D3×C812yes—
219C22×C4ΓL(1,𝔽25)12yes—
220C22×C4(C2×D3)⋊C411yescharacteristic
221C22×C4(C2×D3)⋊C411yescharacteristic
222C22×C4(C2×D3)⋊C411yescharacteristic
223C22×C4(C2×D3)⋊C411yescharacteristic
224C22×C4C2×D1211yescharacteristic
225C22×C4#23712yes—
226C22×C4D3×Q812yes—
227C22×C4C2×#6711yescharacteristic
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—64no—
232C2×C8—122no—
233C2×C8—122no—
234C4⋊C4—122no—
235C4⋊C4—122no—
236C42—22no—
237C42—22no—
238C42—41no—
239C42—41no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—22no—
245C22×C3—22no—
246C22×C3—22no—
247C22×C3—22no—
248C22×C3#39111yescharacteristic
249C22×C3C2×C22⋊C811yescharacteristic
250C22×C3#40811yescharacteristic
251C22×C3C2×#9911yescharacteristic
252C22×C3#41011yescharacteristic
253C22×C3C2×#10011yescharacteristic
254C22×C3C2×#10111yescharacteristic
255C22×C3C8×Q812yes—
256C22×C3#44612yes—
257C3×C4—22no—
258C3×C4—22no—
259C3×C4—22no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—41no—
263C3×C4#32314yes—
264Dic3—44no—
265Dic3—82no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23#88712yes—
271C23C2×D3×C811yescharacteristic
272C23C2×ΓL(1,𝔽25)11yescharacteristic
273C23#90712yes—
274C23C2×(C2×D3)⋊C411yescharacteristic
275Q8—128no—
276Q8—128no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—64no—
294C2×C4—124no—
295C2×C4—122no—
296C2×C4—122no—
297C2×C4—124no—
298C2×C4—122no—
299C2×C4—122no—
300C2×C4—122no—
301C2×C4—244no—
302C2×C4#80612yes—
303C2×C4(C2×D3)⋊C812yes—
304C2×C4(C2×D3)⋊C812yes—
305C2×C4#82312yes—
306C2×C4#82412yes—
307C2×C4#89712yes—
308C2×C4#93611yescharacteristic
309C2×C4#94411yescharacteristic
310C8—82no—
311C8—124no—
312C2×C3—22no—
313C2×C3—22no—
314C2×C3C2×#32311yescharacteristic
315C2×C3#146111yescharacteristic
316C2×C3#146911yescharacteristic
317C2×C3#147211yescharacteristic
318C2×C3#198411yescharacteristic
319C2×C3#199611yescharacteristic
320C2×C3#199711yescharacteristic
321C22—21no—
322C22—21no—
323C22—22no—
324C22—22no—
325C22—21no—
326C22—21no—
327C22—22no—
328C22—22no—
329C22#483012yes—
330C22#483112yes—
331C22C2×#80611yescharacteristic
332C22#497311yescharacteristic
333C22#500611yescharacteristic
334C22C2×(C2×D3)⋊C811yescharacteristic
335C22#512911yescharacteristic
336C22C2×#82311yescharacteristic
337C22C2×#82411yescharacteristic
338C4—22no—
339C4—22no—
340C4—21no—
341C4—22no—
342C4—21no—
343C4—41no—
344C4—124no—
345C4—242no—
346C4#446814yes—
347C3#987811yescharacteristic, complemented, Sylow
348C2—22no—
349C2—22no—
350C2#6952911yescharacteristic
351C2C2×#446811yescharacteristic
352C2#6997211yescharacteristic
353C2#7047211yescharacteristic
354C2#7052411yescharacteristic
355C2#7078611yescharacteristic
356C2#7082811yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC23×C3C22×C8
FrattiniC23×C4C22×D3
FittingC3×#1797C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C22×C4
cosocleC23×C3×C4C23
hypercenter#1797D3
hypocenterC3#9878
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
35. C22×Dic6C8semidirect
53. #830Q8semidirect
347. C3#9878semidirect

Maximal subgroups

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

subgroupindex
C2×#49292
C3×#17972
C4×#8302
#676742
#98783