Finite Group Databasebeta

← #190288

Subgroups — #190288

Order 512.

847 conjugacy classes of subgroups.

The subgroup lattice has 383 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#8769C214yesmaximal, complemented
5#8768C214yesmaximal, complemented
6#8094C212yesmaximal
7#7961C212yesmaximal
8C22×C4×Q8C2211yescharacteristic
9#2920C2212yes—
10#2920C2212yes—
11#2356C414yes—
12#2356C2212yes—
13#2356C2212yes—
14#2355—21no—
15#2355—21no—
16#2355—22no—
17#2313—22no—
18#2310—21no—
19#2285C2211yescharacteristic
20#2285C2211yescharacteristic
21#2283C412yes—
22#2283C412yes—
23#2274—21no—
24#1821—21no—
25#1801C2212yes—
26#1801C2212yes—
27#1784—21no—
28C4×#101—21no—
29C4×#101—21no—
30#1467—22no—
31#1463—22no—
32#1461C2212yes—
33#1461C2212yes—
34#1303C2218yes—
35#1286C2218yes—
36C23×Q8D411yescharacteristic
37C2×C4×Q8—22no—
38C2×C4×Q8—22no—
39C2×C4×Q8—22no—
40C2×C4×Q8—22no—
41C2×C4×Q8C2×C412yes—
42C2×C4×Q8D411yescharacteristic
43C2×C4×Q8D411yescharacteristic
44C2×C4×Q8D412yes—
45C2×C4×Q8C2311yescharacteristic
46C2×C4×Q8C2311yescharacteristic
47C2×#117—21no—
48C2×#117—21no—
49C22×C4⋊C4D412yes—
50C22×C4⋊C4D411yescharacteristic
51C22×C42C2×C412yes—
52C22×C42C2311yescharacteristic
53#432—22no—
54#432—22no—
55#432—22no—
56#432—22no—
57#431—22no—
58#431—22no—
59#431—22no—
60#431C2314yes—
61#423—22no—
62#423C2314yes—
63C2×#101—22no—
64C2×#101—22no—
65#407—22no—
66#407—22no—
67C4×M16—22no—
68C4×M16—22no—
69#391—24no—
70#391—22no—
71#391—22no—
72#387—24no—
73#387—24no—
74#387—24no—
75#387—24no—
76#384—21no—
77#384—21no—
78#383C2×C414yes—
79#383C2×C414yes—
80#382—22no—
81#382—22no—
82#382C2×C412yes—
83#382C2312yes—
84#381—22no—
85#381—22no—
86#380—21no—
87#380—21no—
88#380C2×C412yes—
89#380C2×C411yescharacteristic
90#380C2×C411yescharacteristic
91C4×C4⋊C4—24no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4C2×C412yes—
99C4×C4⋊C4C2312yes—
100C4×C22⋊C4—22no—
101C4×C22⋊C4—22no—
102#376C2×C414yes—
103#376C2×C414yes—
104#343—22no—
105#343—22no—
106#324—24no—
107#324—24no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—42no—
111C22×Q8—42no—
112C22×Q8C22⋊C412yes—
113C22×Q8C2×D411yescharacteristic
114C22×Q8C2×D411yescharacteristic
115C23×C4C22⋊C412yes—
116C23×C4C22⋊C412yes—
117C23×C4C22×C411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C2×M16—24no—
121C2×M16—42no—
122C4×Q8—24no—
123C4×Q8—24no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—24no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—24no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134#117—44no—
135#117—44no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4C22⋊C412yes—
148C2×C4⋊C4C22⋊C412yes—
149C2×C4⋊C4C22⋊C412yes—
150C2×C4⋊C4C2×D412yes—
151C2×C4⋊C4C2×D412yes—
152C2×C4⋊C4C2×D411yescharacteristic
153C2×C4⋊C4C2×D411yescharacteristic
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—22no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—24no—
164C2×C42—24no—
165C2×C42C22⋊C412yes—
166C2×C42C22⋊C412yes—
167C2×C42C22×C412yes—
168C2×C42C22×C412yes—
169C2×C42C22×C411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42#4112yes—
173C2×C42#4112yes—
174C2×C42#4112yes—
175C2×C42#4112yes—
176C2×C42C2411yescharacteristic
177C4⋊C8—24no—
178C4⋊C8—24no—
179C4⋊C8—24no—
180C4⋊C8—42no—
181#101—44no—
182#101—44no—
183C22⋊C8—24no—
184C22⋊C8—42no—
185C22⋊C8—42no—
186C22⋊C8—42no—
187#97—42no—
188#97—42no—
189C4×C8—24no—
190C4×C8—42no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—24no—
195#95—22no—
196#95—24no—
197#95—24no—
198#95—22no—
199#95C22×C414yes—
200#95C22×C412yes—
201#95C22×C412yes—
202C24C2×C22⋊C411yescharacteristic
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—24no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—48no—
213C2×Q8—48no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—81no—
227C22×C4—82no—
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4#11711yescharacteristic
236C22×C4#11711yescharacteristic
237C22×C4#11711yescharacteristic
238C22×C4#11711yescharacteristic
239C22×C4#12312yes—
240C22×C4#12312yes—
241C22×C4#12312yes—
242C22×C4#12312yes—
243C22×C4C23×C411yescharacteristic
244C22×C4C22×D411yescharacteristic
245C22×C4C22×D411yescharacteristic
246M16—44no—
247M16—44no—
248M16—84no—
249C2×C8—44no—
250C2×C8—44no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—44no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C22⋊C4—44no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C42—22no—
269C42—22no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—44no—
277C42—44no—
278C42—84no—
279C42—84no—
280C42#11712yes—
281C42#11712yes—
282C42#12312yes—
283C42#12312yes—
284C42#12312yes—
285C42#12312yes—
286C42C2×#4112yes—
287C42C2×#4112yes—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23#38111yescharacteristic
293C23#38111yescharacteristic
294C23C22×C22⋊C411yescharacteristic
295Q8—48no—
296Q8—44no—
297Q8—44no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4#38111yescharacteristic
336C2×C4#38112yes—
337C2×C4#38112yes—
338C2×C4#38111yescharacteristic
339C2×C4C2×#11711yescharacteristic
340C2×C4C2×#11711yescharacteristic
341C2×C4C2×#12312yes—
342C2×C4C2×#12312yes—
343C2×C4#57512yes—
344C2×C4#57512yes—
345C8—84no—
346C8—84no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—42no—
356C22—42no—
357C22C2×#38111yescharacteristic
358C22#289611yescharacteristic
359C22#289611yescharacteristic
360C4—22no—
361C4—22no—
362C4—22no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—42no—
367C4—42no—
368C4—42no—
369C4—42no—
370C4—84no—
371C4—82no—
372C4—82no—
373C4#183012yes—
374C4#309314yes—
375C2—21no—
376C2—21no—
377C2—41no—
378C2—41no—
379C2#2070111yescharacteristic
380C2#2070811yescharacteristic
381C2C2×#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. #8769C2semidirect
5. #8768C2semidirect

Maximal subgroups

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

subgroupindex
#341872
#207272
#173652
#87692
#87682
#80942
#79612