Finite Group Databasebeta

← #238418

Subgroups — #238418

Order 512.

722 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37563C211yesmaximal, characteristic
2#21826C211yesmaximal, characteristic
3#17584C211yesmaximal, characteristic
4#17562C211yesmaximal, characteristic
5#10383C218yesmaximal
6#8110C211yesmaximal, characteristic
7#8050C211yesmaximal, characteristic
8#7945C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2844—22no—
13#2665—22no—
14#2665—22no—
15#2665C2214yes—
16#2605—22no—
17#2605—22no—
18#2605—22no—
19#2283C2211yescharacteristic
20#2283C2211yescharacteristic
21#2283C2211yescharacteristic
22#2283C2211yescharacteristic
23#1996—22no—
24#1996C2214yes—
25#1820—22no—
26#1518—22no—
27#1515—22no—
28#1515—22no—
29#1511C2214yes—
30#1508C2214yes—
31#1503—22no—
32#1305C2214yes—
33#1297C2214yes—
34#1284C2214yes—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43C2×C4⋊Q8Q811yescharacteristic, complemented
44C2×C4⋊Q8Q811yescharacteristic, complemented
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8C2312yes—
48C2×C4×Q8—21no—
49C2×C4×Q8Q812yescomplemented
50C2×#117—21no—
51C22×C42C2311yescharacteristic
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#423—21no—
61#423—21no—
62#407C2312yes—
63#407C2312yes—
64C4×M16—21no—
65C4×M16—21no—
66#400—24no—
67#400—24no—
68#400—24no—
69#396—24no—
70#395—24no—
71#395—24no—
72#395—24no—
73#393—24no—
74#391—24no—
75#391—24no—
76#391—24no—
77#387—24no—
78#387—24no—
79#387—24no—
80#383—21no—
81#383—21no—
82#383D412yes—
83#383D412yes—
84#382—21no—
85#382—21no—
86#382—21no—
87#382D412yes—
88#381—21no—
89#381—21no—
90#381D412yes—
91#381C2312yes—
92#380—21no—
93#380—21no—
94#380—21no—
95#380C2312yes—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4D412yes—
100C4×C22⋊C4—21no—
101C4×C22⋊C4D412yes—
102C4×C22⋊C4C2312yes—
103C4×C22⋊C4C2312yes—
104#376—21no—
105#376—21no—
106#376—21no—
107#376—21no—
108#342—28no—
109#328—28no—
110#328—28no—
111C22×Q8—24no—
112C22×Q8—21no—
113C22×Q8C2×Q812yes—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2411yescharacteristic
117C2×M16—22no—
118C2×M16—22no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127#125—41no—
128#125—41no—
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—41no—
148#117—41no—
149#117—41no—
150#117—41no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4C2×Q812yes—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4C2×D414yes—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×Q811yescharacteristic
179C2×C42C2×Q811yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C4⋊C8—41no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C4⋊C8—41no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194#97—41no—
195#97—41no—
196C4×C8—41no—
197C4×C8—41no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—41no—
214#95—41no—
215#95—41no—
216#95C2×D414yes—
217#95C2×D414yes—
218C24C22×D411yescharacteristic
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—84no—
224C2×Q8—84no—
225C2×Q8—84no—
226C2×Q8—82no—
227C2×Q8—82no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4#12011yescharacteristic
238C22×C4#12011yescharacteristic
239C22×C4#12011yescharacteristic
240C22×C4#12011yescharacteristic
241C22×C4C22⋊Q811yescharacteristic
242C22×C4C22⋊Q811yescharacteristic
243C22×C4C22⋊Q811yescharacteristic
244C22×C4C22⋊Q811yescharacteristic
245C22×C4#12411yescharacteristic
246C22×C4#12411yescharacteristic
247C22×C4#12411yescharacteristic
248C22×C4#12411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C22×C4C22×Q811yescharacteristic
252C22×C4C2×#4111yescharacteristic
253C22×C4C2×#4111yescharacteristic
254C22×C4C2×#4111yescharacteristic
255M16—82no—
256M16—82no—
257C2×C8—42no—
258C2×C8—42no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—82no—
272C4⋊C4—82no—
273C4⋊C4—82no—
274C4⋊C4—84no—
275C4⋊C4—84no—
276C4⋊C4—84no—
277C4⋊C4—82no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—44no—
281C22⋊C4—44no—
282C22⋊C4—82no—
283C22⋊C4—82no—
284C42—42no—
285C42—42no—
286C42—82no—
287C42—82no—
288C42—82no—
289C42—82no—
290C42C22⋊Q811yescharacteristic
291C42C22⋊Q811yescharacteristic
292C42C22⋊Q811yescharacteristic
293C42C22⋊Q811yescharacteristic
294C42C22⋊Q811yescharacteristic
295C42C22⋊Q811yescharacteristic
296C42C22⋊Q811yescharacteristic
297C42C22⋊Q811yescharacteristic
298C42#12411yescharacteristic
299C42#12411yescharacteristic
300C42#12411yescharacteristic
301C42#12411yescharacteristic
302C42#12411yescharacteristic
303C42#12411yescharacteristic
304C42#12411yescharacteristic
305C42#12411yescharacteristic
306C23—22no—
307C23—41no—
308C23—41no—
309C23C2×#12011yescharacteristic
310C23C2×C22⋊Q811yescharacteristic
311C23C2×#12411yescharacteristic
312Q8—82no—
313Q8—82no—
314Q8—82no—
315Q8—82no—
316Q8—168no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4#39311yescharacteristic
352C2×C4#39311yescharacteristic
353C2×C4#39311yescharacteristic
354C2×C4#39311yescharacteristic
355C2×C4#39311yescharacteristic
356C2×C4#39311yescharacteristic
357C2×C4#39311yescharacteristic
358C2×C4#39311yescharacteristic
359C2×C4C2×C22⋊Q811yescharacteristic
360C2×C4C2×C22⋊Q811yescharacteristic
361C2×C4C2×#12411yescharacteristic
362C2×C4C2×#12411yescharacteristic
363C8—82no—
364C8—82no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—82no—
374C22C2×#39311yescharacteristic
375C22#302212yes—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—84no—
389C4—84no—
390C4—84no—
391C4—82no—
392C4—82no—
393C4—162no—
394C2—22no—
395C2—41no—
396C2—41no—
397C2#2346411yescharacteristic
398C2#2347211yescharacteristic
399C2#2347411yescharacteristic
subgroupG/H
center, socleC22C2×#393
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
43. C2×C4⋊Q8Q8semidirect
44. C2×C4⋊Q8Q8semidirect
49. C2×C4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#375632
#218262
#175842
#175622
#103832
#81102
#80502
#79452