Finite Group Databasebeta

← #111151

Subgroups — #111151

Order 512.

605 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21885C211yesmaximal, characteristic
2#20544C211yesmaximal, characteristic
3#17676C211yesmaximal, characteristic
4#17596C211yesmaximal, characteristic
5C2×#1290C211yesmaximal, characteristic
6#7984C211yesmaximal, characteristic
7#7952C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13#2285C2211yescharacteristic
14#2283C2211yescharacteristic
15#1987C412yescomplemented
16#1981C412yescomplemented
17#1892—21no—
18#1892—21no—
19#1803—22no—
20#1771—22no—
21#1768—22no—
22#1752—21no—
23C8×C22⋊C4—21no—
24#1534—22no—
25#1531—22no—
26#1531C414yescomplemented
27#1520—22no—
28#1515C414yescomplemented
29#1506—22no—
30#1290—22no—
31#1289—22no—
32#1285—22no—
33C23×C8D411yescharacteristic
34C23×C8D411yescharacteristic
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533C2×C412yes—
42#533C2×C412yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×#117—21no—
51C2×#117—21no—
52C22×C42C2311yescharacteristic
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432C2×C412yes—
63#432C2×C412yes—
64#432D412yes—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8C2×C412yes—
73C2×C4⋊C8D412yes—
74C2×#103—22no—
75C2×#103—22no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×#97—21no—
80C2×#97—22no—
81C2×#97C2×C412yes—
82C2×#97D412yes—
83C2×C4×C8—22no—
84#384—21no—
85#383C2×C412yes—
86#382—21no—
87#382—21no—
88#381—21no—
89#380—21no—
90#380—21no—
91#380D412yes—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4C2×C412yes—
95C4×C22⋊C4—21no—
96#376D412yes—
97#340—22no—
98#340—22no—
99#336—22no—
100#336—22no—
101#330—28no—
102#328—28no—
103C22×Q8—21no—
104C22×Q8—21no—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4#4111yescharacteristic
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—22no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—22no—
114C22×C8—41no—
115C22×C8—81no—
116C22×C8—81no—
117C22×C8C22⋊C412yes—
118C22×C8C22⋊C412yes—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8C4212yes—
126C4⋊Q8C4212yes—
127#125—22no—
128#125—22no—
129#125—41no—
130#125—41no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×Q8—41no—
146C4×Q8—41no—
147#117—22no—
148#117—22no—
149#117—41no—
150#117—41no—
151#117—82no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—41no—
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C411yescharacteristic
170C2×C42C22×C411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C4⋊C8—41no—
190#103—82no—
191#103—82no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197C22⋊C8—41no—
198#97—22no—
199#97—22no—
200#97—22no—
201#97—22no—
202#97—41no—
203#97—82no—
204#97C22⋊C414yes—
205C4×C8—82no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—41no—
211#95C22⋊C414yes—
212C24#12111yescharacteristic
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—82no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—81no—
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4#11711yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4#12011yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12311yescharacteristic
242C22×C4#12311yescharacteristic
243C22×C4#12411yescharacteristic
244C22×C4#12411yescharacteristic
245C22×C4#12611yescharacteristic
246C22×C4#12711yescharacteristic
247C2×C8—24no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—81no—
259C2×C8—81no—
260C2×C8—82no—
261C2×C8—82no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—82no—
284C42—42no—
285C42—42no—
286C42—42no—
287C42—42no—
288C42—82no—
289C42C2×C4211yescharacteristic
290C42C2×C22⋊C411yescharacteristic
291C42#11711yescharacteristic
292C42#11711yescharacteristic
293C42C4×D411yescharacteristic
294C42C4×D411yescharacteristic
295C42C4×D411yescharacteristic
296C42C4×D411yescharacteristic
297C42C4×D411yescharacteristic
298C42C4×D411yescharacteristic
299C42#12111yescharacteristic
300C42#12111yescharacteristic
301C42#12111yescharacteristic
302C42#12311yescharacteristic
303C42#12411yescharacteristic
304C42#12611yescharacteristic
305C23—41no—
306C23—41no—
307C23#38611yescharacteristic
308C23#38811yescharacteristic
309C23#39011yescharacteristic
310C23#47112yes—
311C23#47312yes—
312Q8—44no—
313Q8—82no—
314Q8—82no—
315Q8—81no—
316Q8—81no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4C4×C22⋊C411yescharacteristic
351C2×C4C4×C22⋊C411yescharacteristic
352C2×C4C4×C22⋊C411yescharacteristic
353C2×C4#38611yescharacteristic
354C2×C4#38611yescharacteristic
355C2×C4#38811yescharacteristic
356C2×C4#38811yescharacteristic
357C2×C4#38811yescharacteristic
358C2×C4#38811yescharacteristic
359C2×C4#38811yescharacteristic
360C2×C4#39011yescharacteristic
361C2×C4#39011yescharacteristic
362C8—44no—
363C8—42no—
364C8—42no—
365C8—82no—
366C8—82no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—42no—
374C22—42no—
375C22#143711yescharacteristic
376C22#190812yes—
377C22#193012yes—
378C22#197711yescharacteristic
379C22#197911yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—84no—
392C4—82no—
393C4—82no—
394C4—162no—
395C4#191111yescharacteristic
396C4#191211yescharacteristic
397C2—22no—
398C2—21no—
399C2—22no—
400C2—21no—
401C2#1094211yescharacteristic
402C2#1094911yescharacteristic
403C2#1095111yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #1987C4semidirect
16. #1981C4semidirect
26. #1531C4semidirect
28. #1515C4semidirect

Maximal subgroups

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

subgroupindex
#218852
#205442
#176762
#175962
C2×#12902
#79842
#79522