Finite Group Databasebeta

← #190082

Subgroups — #190082

Order 512.

911 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34121C211yesmaximal, characteristic
2#17366C211yesmaximal, characteristic
3#17366C211yesmaximal, characteristic
4#8744C218yesmaximal, complemented
5#7953C212yesmaximal, complemented
6#7950C212yesmaximal, complemented
7C2×C4×#41C2211yescharacteristic
8#2920C2212yes—
9#2920C2212yes—
10#2357—21no—
11#2356—21no—
12#2355—21no—
13#2354—21no—
14#2353—22no—
15#2352—22no—
16#2298C2214yes—
17#2297C412yes—
18#2285C2211yescharacteristic
19#2283C412yes—
20C4×#117C412yes—
21C4×#117C412yes—
22C2×C4×C4⋊C4C2211yescharacteristic
23#1465—22no—
24#1465—22no—
25#1464C2214yes—
26#1464C2214yes—
27#1461—21no—
28#1461—21no—
29#1460—21no—
30#1460—21no—
31#1285C2218yes—
32#1285C2218yes—
33C22×#41C2×C411yescharacteristic
34C4×#41—22no—
35C4×#41—22no—
36C4×#41—22no—
37C4×#41—22no—
38C2×C4×Q8D411yescharacteristic
39C2×C4×Q8D411yescharacteristic
40C2×C4×D4D412yes—
41C2×C4×D4D412yes—
42C2×C4×D4D411yescharacteristic
43C2×C4×D4D411yescharacteristic
44C2×#117—22no—
45C2×#117C2×C412yes—
46C2×#117C2×C411yescharacteristic
47C2×#117C2312yes—
48C22×C4⋊C4—21no—
49C22×C42C2×C412yes—
50C22×C42C2×C412yes—
51C22×C42C2311yescharacteristic
52#432—22no—
53#432—22no—
54#432C2314yes—
55#432C2314yes—
56#431—22no—
57#431—22no—
58#431—22no—
59#431—22no—
60#423—22no—
61#423—22no—
62#407—22no—
63#407—22no—
64C4×M16—22no—
65C4×M16—22no—
66#391—22no—
67#390—22no—
68#388—24no—
69#388—24no—
70#387—24no—
71#387—24no—
72#386—24no—
73#384—21no—
74#383—22no—
75#382—22no—
76#382—21no—
77#381—22no—
78#380—22no—
79#380—21no—
80#380—21no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4C2×C411yescharacteristic
87C4×C4⋊C4C2×C411yescharacteristic
88C4×C4⋊C4C2×C412yes—
89C4×C4⋊C4C2312yes—
90C4×C4⋊C4C2312yes—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—24no—
93C4×C22⋊C4C2×C414yes—
94C4×C22⋊C4C2×C414yes—
95#376—22no—
96#376C2×C414yes—
97C2×#95—21no—
98C43C2×C414yes—
99#324—24no—
100#324—24no—
101#323—24no—
102#323—24no—
103C2×#41—22no—
104C2×#41—22no—
105C2×#41—42no—
106C2×#41—42no—
107C22×Q8C22⋊C411yescharacteristic
108C22×D4C22⋊C411yescharacteristic
109C22×D4C22⋊C412yes—
110C23×C4—22no—
111C23×C4C22⋊C412yes—
112C23×C4C22⋊C412yes—
113C23×C4C22×C411yescharacteristic
114C23×C4C22×C412yes—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C2×M16—42no—
118C2×M16—42no—
119C4×Q8—22no—
120C4×Q8—22no—
121C4×Q8—22no—
122C4×Q8—22no—
123C4×D4—24no—
124C4×D4—24no—
125C4×D4—24no—
126C4×D4—24no—
127C4×D4—22no—
128C4×D4—22no—
129C4×D4—22no—
130C4×D4—22no—
131#117—22no—
132#117—22no—
133#117—24no—
134#117—24no—
135#117—48no—
136#117—44no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4C22⋊C411yescharacteristic
145C2×C4⋊C4C22⋊C411yescharacteristic
146C2×C4⋊C4C2×D412yes—
147C2×C4⋊C4C2×D412yes—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4C22⋊C412yes—
152C2×C22⋊C4C2×D412yes—
153C2×C22⋊C4C2×D412yes—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—24no—
159C2×C42—22no—
160C2×C42—24no—
161C2×C42—44no—
162C2×C42C22⋊C412yes—
163C2×C42C22⋊C412yes—
164C2×C42C22×C412yes—
165C2×C42C22×C414yes—
166C2×C42C22×C412yes—
167C2×C42C22×C412yes—
168C2×C42C22×C411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42#4112yes—
172C2×C42#4112yes—
173C2×C42#4112yes—
174C2×C42#4112yes—
175C2×C42C2411yescharacteristic
176C4⋊C8—24no—
177C4⋊C8—24no—
178C4⋊C8—42no—
179C4⋊C8—42no—
180C22⋊C8—24no—
181C22⋊C8—24no—
182C22⋊C8—42no—
183C22⋊C8—42no—
184#97—24no—
185#97—24no—
186C4×C8—42no—
187C4×C8—42no—
188#95—21no—
189#95—21no—
190#95—22no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—24no—
195#95—22no—
196#95—22no—
197C24C2×C22⋊C412yes—
198C24C2×C22⋊C411yescharacteristic
199#41—48no—
200#41—48no—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—22no—
204C2×D4—24no—
205C2×D4—22no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—42no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—81no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4#11712yes—
237C22×C4#11712yes—
238C22×C4#12312yes—
239C22×C4#12312yes—
240C22×C4#12312yes—
241C22×C4#12312yes—
242C22×C4C23×C411yescharacteristic
243C22×C4C22×D411yescharacteristic
244C22×C4C22×D411yescharacteristic
245M16—84no—
246M16—84no—
247C2×C8—44no—
248C2×C8—44no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—42no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—82no—
259C22⋊C4—24no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C22⋊C4—48no—
263C22⋊C4—44no—
264C22⋊C4—84no—
265C42—22no—
266C42—22no—
267C42—22no—
268C42—22no—
269C42—22no—
270C42—22no—
271C42—42no—
272C42—42no—
273C42—44no—
274C42—48no—
275C42—44no—
276C42—84no—
277C42#11711yescharacteristic
278C42#11711yescharacteristic
279C42#11711yescharacteristic
280C42#11711yescharacteristic
281C42#12312yes—
282C42#12312yes—
283C42#12312yes—
284C42#12312yes—
285C42C2×#4112yes—
286C42C2×#4112yes—
287C23—21no—
288C23—21no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—41no—
293C23—41no—
294C23—42no—
295C23—42no—
296C23#38112yes—
297C23C22×C22⋊C411yescharacteristic
298Q8—44no—
299D4—48no—
300D4—44no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—24no—
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—42no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—88no—
336C2×C4#38111yescharacteristic
337C2×C4#38111yescharacteristic
338C2×C4#38111yescharacteristic
339C2×C4#38112yes—
340C2×C4#38111yescharacteristic
341C2×C4#41312yes—
342C2×C4#41312yes—
343C2×C4C2×#11711yescharacteristic
344C2×C4C2×#11711yescharacteristic
345C2×C4C2×#12312yes—
346C2×C4C2×#12312yes—
347C8—84no—
348C8—84no—
349C22—21no—
350C22—21no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—42no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—82no—
360C22—82no—
361C22C2×#38111yescharacteristic
362C22C2×#41311yescharacteristic
363C22C2×#41311yescharacteristic
364C4—21no—
365C4—21no—
366C4—22no—
367C4—21no—
368C4—22no—
369C4—21no—
370C4—21no—
371C4—44no—
372C4—42no—
373C4—42no—
374C4—42no—
375C4—84no—
376C4—82no—
377C4#182214yes—
378C4#182712yes—
379C2—21no—
380C2—21no—
381C2—42no—
382C2—82no—
383C2C2×#182211yescharacteristic
384C2C2×#182711yescharacteristic
385C2#2070811yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×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. #8744C2semidirect
5. #7953C2semidirect
6. #7950C2semidirect

Maximal subgroups

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

subgroupindex
#341212
#173662
#87442
#79532
#79502