Finite Group Databasebeta

← #190333

Subgroups — #190333

Order 512.

847 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34187C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic
3#17365C211yesmaximal, characteristic
4#8774C218yesmaximal, complemented
5#8070C212yesmaximal
6#7976C212yesmaximal
7C22×C4×Q8C2211yescharacteristic
8#2920C2212yes—
9#2920C2212yes—
10#2356C414yes—
11#2356C2214yes—
12#2355—22no—
13#2355—22no—
14#2313—21no—
15#2313—21no—
16#2310—21no—
17#2285C412yes—
18#2283C412yes—
19#2283C2211yescharacteristic
20#2283C2211yescharacteristic
21#2274—21no—
22#1821—21no—
23#1801—22no—
24#1784—21no—
25C4×#101C2214yes—
26#1467C2214yes—
27#1463—21no—
28#1463—21no—
29#1461—22no—
30#1300C2218yes—
31#1288C2218yes—
32C23×Q8D411yescharacteristic
33C2×C4×Q8—22no—
34C2×C4×Q8—22no—
35C2×C4×Q8—22no—
36C2×C4×Q8—22no—
37C2×C4×Q8C2×C412yes—
38C2×C4×Q8D412yes—
39C2×C4×Q8D412yes—
40C2×C4×Q8C2312yes—
41C2×#117—22no—
42C22×C4⋊C4D411yescharacteristic
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C42C2×C411yescharacteristic
46C22×C42C2×C411yescharacteristic
47C22×C42C2311yescharacteristic
48#432—22no—
49#432—22no—
50#432—22no—
51#432C2314yes—
52#431—22no—
53#431—22no—
54#431—22no—
55#431—22no—
56#423—22no—
57#423—22no—
58C2×#101—24no—
59#407—22no—
60#407—22no—
61C4×M16—22no—
62C4×M16C2314yes—
63#391—24no—
64#391—24no—
65#387—24no—
66#387—24no—
67#387—24no—
68#387—24no—
69#384—22no—
70#383C2×C412yes—
71#383C2×C414yes—
72#383C2312yes—
73#382—22no—
74#382—21no—
75#382—21no—
76#382C2×C414yes—
77#381—22no—
78#381—21no—
79#381—21no—
80#380—22no—
81#380C2×C412yes—
82#380C2×C412yes—
83C4×C4⋊C4—24no—
84C4×C4⋊C4—24no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4C2×C414yes—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—21no—
92C4×C22⋊C4—21no—
93#376C2×C412yes—
94#376C2×C414yes—
95#376C2312yes—
96#343—24no—
97#324—24no—
98#324—24no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8—44no—
102C22×Q8C22⋊C412yes—
103C22×Q8C2×D412yes—
104C23×C4C22⋊C411yescharacteristic
105C23×C4C22⋊C411yescharacteristic
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22×C411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C2×M16—24no—
112C2×M16—42no—
113C4×Q8—24no—
114C4×Q8—24no—
115C4×Q8—24no—
116C4×Q8—24no—
117C4×Q8—24no—
118C4×Q8—24no—
119C4×Q8—24no—
120C4×Q8—24no—
121#117—48no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4C22⋊C412yes—
133C2×C4⋊C4C22⋊C412yes—
134C2×C4⋊C4C22⋊C412yes—
135C2×C4⋊C4C2×D412yes—
136C2×C4⋊C4C2×D412yes—
137C2×C4⋊C4C2×D412yes—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—24no—
149C2×C42—24no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42C22⋊C412yes—
153C2×C42C22⋊C412yes—
154C2×C42C22×C412yes—
155C2×C42C22×C412yes—
156C2×C42C22×C411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42#4112yes—
160C2×C42#4112yes—
161C2×C42#4112yes—
162C2×C42#4112yes—
163C2×C42C2411yescharacteristic
164C4⋊C8—24no—
165C4⋊C8—42no—
166C4⋊C8—42no—
167C4⋊C8—42no—
168#101—48no—
169C22⋊C8—24no—
170C22⋊C8—42no—
171C22⋊C8—42no—
172C22⋊C8—42no—
173#97—24no—
174#97—24no—
175C4×C8—24no—
176C4×C8—42no—
177#95—24no—
178#95—22no—
179#95—24no—
180#95—22no—
181#95—24no—
182#95—24no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95C22×C414yes—
188#95C22×C414yes—
189C24C2×C22⋊C411yescharacteristic
190C2×Q8—24no—
191C2×Q8—24no—
192C2×Q8—24no—
193C2×Q8—24no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—48no—
197C2×Q8—48no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—24no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—81no—
213C22×C4—81no—
214C22×C4—81no—
215C22×C4C2×C22⋊C412yes—
216C22×C4C2×C22⋊C411yescharacteristic
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4C2×C22⋊C412yes—
219C22×C4C2×C22⋊C412yes—
220C22×C4C2×C22⋊C412yes—
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4#11711yescharacteristic
223C22×C4#11711yescharacteristic
224C22×C4#11711yescharacteristic
225C22×C4#11711yescharacteristic
226C22×C4#12312yes—
227C22×C4#12312yes—
228C22×C4#12312yes—
229C22×C4#12312yes—
230C22×C4C23×C411yescharacteristic
231C22×C4C22×D411yescharacteristic
232C22×C4C22×D411yescharacteristic
233M16—48no—
234M16—84no—
235C2×C8—44no—
236C2×C8—44no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—48no—
246C4⋊C4—84no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C42—22no—
250C42—22no—
251C42—22no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—22no—
257C42—48no—
258C42—82no—
259C42—82no—
260C42—84no—
261C42#11712yes—
262C42#11712yes—
263C42#12312yes—
264C42#12312yes—
265C42#12312yes—
266C42#12312yes—
267C42C2×#4112yes—
268C42C2×#4112yes—
269C23—22no—
270C23—41no—
271C23—41no—
272C23#38111yescharacteristic
273C23#38111yescharacteristic
274C23C22×C22⋊C411yescharacteristic
275Q8—48no—
276Q8—48no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4#38111yescharacteristic
315C2×C4#38112yes—
316C2×C4#38112yes—
317C2×C4#38111yescharacteristic
318C2×C4#42114yes—
319C2×C4C2×#11711yescharacteristic
320C2×C4C2×#11711yescharacteristic
321C2×C4C2×#12312yes—
322C2×C4C2×#12312yes—
323C8—84no—
324C8—84no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—44no—
334C22C2×#38111yescharacteristic
335C22C2×#42112yes—
336C4—22no—
337C4—24no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—42no—
345C4—42no—
346C4—42no—
347C4—84no—
348C4—84no—
349C4#183012yes—
350C2—22no—
351C2—41no—
352C2—41no—
353C2#2070511yescharacteristic
354C2#2070811yescharacteristic
355C2C2×#183011yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8774C2semidirect

Maximal subgroups

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

subgroupindex
#341872
#207272
#173652
#87742
#80702
#79762