Finite Group Databasebeta

← #6372180

Subgroups — #6372180

Order 512.

3415 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62813C211yesmaximal, characteristic
2C4×#568C211yesmaximal, characteristic
3C4⋊C4×#41C211yesmaximal, characteristic
4#21307C2112yesmaximal, complemented
5#21306C216yesmaximal, complemented
6C2×#1986C216yesmaximal, complemented
7#20343C214yesmaximal
8C2×#568C2212yes—
9#3504C418yescomplemented
10#3503C414yescomplemented
11#3503C414yescomplemented
12#3502—21no—
13#3502—21no—
14#3501—23no—
15#3501—23no—
16Q8×#41—24no—
17Q8×#41—24no—
18#3458C2216yes—
19C2×#533C2216yes—
20#3428C2212yes—
21C2×C4×#41C2212yes—
22C2×C4×#41C2211yescharacteristic
23#2875—24no—
24C4×#131—24no—
25#2873C2216yes—
26C2×C4×M16C2216yes—
27#2351—21no—
28Q8×C4⋊C4—21no—
29#2349—23no—
30D4×C4⋊C4—23no—
31#2296C2216yes—
32#2285C2213yes—
33C2×C4×C4⋊C4C2213yes—
34#1986C2214yes—
35#1986C22112yes—
36#1986C22124yes—
37#1986C22124yes—
38#1733C22124yes—
39C2×#339C22124yes—
40C22×#41C2311yescharacteristic
41#568—28no—
42#568Q8116yescomplemented
43C2×#131—28no—
44C22×M16C23112yes—
45Q82—24no—
46Q82—24no—
47#557—24no—
48#557—24no—
49#557—28no—
50#557—28no—
51#554—212no—
52#554—212no—
53#554—224no—
54#554—224no—
55D4×Q8—212no—
56D4×Q8—212no—
57#544—26no—
58#544—26no—
59#544C2×C4124yescomplemented
60#544C2×C4124yescomplemented
61#543—212no—
62#533—224no—
63#533—26no—
64#533—26no—
65#533—26no—
66#533—26no—
67#533C2×C4112yescomplemented
68#533C2×C4112yescomplemented
69#533C2314yes—
70#533C23112yes—
71C2×C4⋊Q8C2×C416yescomplemented
72C2×C4⋊Q8C2×C416yescomplemented
73C2×#125C2×C4112yescomplemented
74C2×C22⋊Q8—26no—
75C2×C22⋊Q8—26no—
76#520C2×C418yes—
77#520C2×C418yes—
78#519—22no—
79#518—26no—
80C4×#41—28no—
81C4×#41—216no—
82C4×#41C2×C414yes—
83C4×#41C2×C414yes—
84C4×#41D414yes—
85C4×#41D414yes—
86C2×C4×Q8—22no—
87C2×C4×Q8—26no—
88C2×C4×Q8—26no—
89C2×C4×Q8D411yescharacteristic
90C2×C4×Q8D411yescharacteristic
91C2×C4×D4—26no—
92C2×C4×D4D413yes—
93C2×C4×D4D413yes—
94C2×#117C23112yes—
95C2×#117C2316yes—
96C2×#117C2313yes—
97C22×C4⋊C4—23no—
98C22×C42C2316yes—
99C22×C42C2313yes—
100#405—224no—
101C4×M16—26no—
102C4×M16—26no—
103C4×M16C2314yes—
104C4×M16C23112yes—
105C2×#97—26no—
106C2×C4×C8—26no—
107#389—26no—
108#389—26no—
109#385—212no—
110#384—23no—
111#382—23no—
112#380—23no—
113#380—23no—
114C4×C4⋊C4—23no—
115C4×C4⋊C4—23no—
116C4×C4⋊C4C2311yescharacteristic
117C4×C4⋊C4C2313yes—
118C4×C4⋊C4C2316yes—
119C4×C4⋊C4C2316yes—
120C2×#95—23no—
121#339C23116yes—
122#339C23148yes—
123C2×#41C22×C414yes—
124C2×#41C2×D414yes—
125C2×#41C2×Q818yes—
126C2×#41#4118yes—
127C22×Q8—26no—
128C22×Q8—26no—
129C22×Q8C2×D411yescharacteristic
130C22×D4C2×D413yes—
131C23×C4—26no—
132C23×C4C2×D413yes—
133C23×C4C2×D413yes—
134C23×C4C2413yes—
135#131—264no—
136C2×M16—212no—
137C2×M16—224no—
138C2×M16C2×Q8148yescomplemented
139C2×M16C2418yes—
140C22×C8—212no—
141C4⋊Q8—224no—
142C4⋊Q8—224no—
143C4⋊Q8—26no—
144C4⋊Q8—26no—
145C4⋊Q8—26no—
146C4⋊Q8—26no—
147C4⋊Q8C22×C414yes—
148C4⋊Q8C22×C414yes—
149C4⋊Q8C22×C4112yes—
150C4⋊Q8C22×C4112yes—
151#125—224no—
152#125—224no—
153#125—212no—
154#125—212no—
155#125C22×C418yes—
156#125C22×C4124yes—
157C22⋊Q8—212no—
158C22⋊Q8—212no—
159C22⋊Q8—212no—
160C22⋊Q8—212no—
161C22⋊Q8—24no—
162C22⋊Q8—24no—
163C22⋊Q8—212no—
164C22⋊Q8—212no—
165C22⋊Q8—248no—
166C22⋊Q8—248no—
167C4×Q8—28no—
168C4×Q8—28no—
169C4×Q8—212no—
170C4×Q8—212no—
171C4×Q8—212no—
172C4×Q8—212no—
173C4×Q8—24no—
174C4×Q8—24no—
175C4×Q8—212no—
176C4×Q8—212no—
177C4×Q8—48no—
178C4×Q8C22⋊C414yes—
179C4×Q8C22⋊C414yes—
180C4×Q8C22⋊C414yes—
181C4×Q8C22⋊C414yes—
182C4×D4—224no—
183C4×D4—224no—
184C4×D4—424no—
185C4×D4C22⋊C4112yes—
186C4×D4C22⋊C4112yes—
187C4×D4C22⋊C4112yes—
188C4×D4C22⋊C4112yes—
189#117—212no—
190#117—212no—
191#117—224no—
192#117C22×C4124yes—
193#117C22×C4112yes—
194#117C2×D4112yes—
195#117C2418yes—
196C2×C4⋊C4—212no—
197C2×C4⋊C4—212no—
198C2×C4⋊C4—26no—
199C2×C4⋊C4—26no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—23no—
202C2×C4⋊C4—26no—
203C2×C4⋊C4—26no—
204C2×C4⋊C4—26no—
205C2×C4⋊C4—412no—
206C2×C4⋊C4C22×C4112yes—
207C2×C4⋊C4C22×C4112yes—
208C2×C4⋊C4C2×D413yes—
209C2×C4⋊C4C2×D413yes—
210C2×C22⋊C4—212no—
211C2×C22⋊C4—26no—
212C2×C22⋊C4C2×D416yes—
213C2×C42—212no—
214C2×C42—224no—
215C2×C42—26no—
216C2×C42C22×C416yes—
217C2×C42C22×C416yes—
218C2×C42C2×D413yes—
219C2×C42C2×D413yes—
220C2×C42C2×D416yes—
221C2×C42C2×D416yes—
222C2×C42C2416yes—
223C2×C42C2412yes—
224C2×C42C2413yes—
225C2×C42C2411yescharacteristic
226#97—24no—
227#97—212no—
228C4×C8—24no—
229C4×C8—212no—
230#95—21no—
231#95—23no—
232#95—26no—
233#95—26no—
234C24C22×D413yes—
235#41C4×Q8132yes—
236#41C22⋊Q8132yes—
237C2×Q8—24no—
238C2×Q8—24no—
239C2×Q8—212no—
240C2×Q8—212no—
241C2×Q8—812no—
242C2×Q8—812no—
243C2×Q8C2×C22⋊C414yes—
244C2×Q8#12414yes—
245C2×Q8C4⋊Q814yes—
246C2×D4C2×C22⋊C4112yes—
247C2×D4#124112yes—
248C2×D4C4⋊Q8112yes—
249C22×C4—22no—
250C22×C4—212no—
251C22×C4—212no—
252C22×C4—26no—
253C22×C4—46no—
254C22×C4—412no—
255C22×C4—412no—
256C22×C4—83no—
257C22×C4—86no—
258C22×C4C2×C22⋊C416yes—
259C22×C4C2×C22⋊C416yes—
260C22×C4C2×C22⋊C416yes—
261C22×C4C2×C22⋊C416yes—
262C22×C4C23×C416yes—
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D413yes—
266C22×C4C22×D413yes—
267C22×C4C22×D413yes—
268C22×C4C22×D416yes—
269C22×C4C22×Q8112yes—
270C22×C4C2×#41112yes—
271C22×C4C2511yescharacteristic
272M16—248no—
273M16—28no—
274M16C22×Q8116yes—
275C2×C8—248no—
276C2×C8—28no—
277C4⋊C4—24no—
278C4⋊C4—28no—
279C4⋊C4—28no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—212no—
283C4⋊C4—212no—
284C4⋊C4—224no—
285C4⋊C4—224no—
286C4⋊C4—212no—
287C4⋊C4—212no—
288C4⋊C4—46no—
289C4⋊C4—412no—
290C4⋊C4—412no—
291C4⋊C4—86no—
292C4⋊C4C2×C22⋊C4112yes—
293C4⋊C4C2×C22⋊C4112yes—
294C4⋊C4C23×C418yes—
295C4⋊C4C23×C418yes—
296C22⋊C4—224no—
297C22⋊C4—28no—
298C22⋊C4—412no—
299C22⋊C4C2×C22⋊C4124yes—
300C42—224no—
301C42—28no—
302C42—24no—
303C42—212no—
304C42—412no—
305C42C2×C22⋊C416yes—
306C42C2×C22⋊C416yes—
307C42C2×C22⋊C416yes—
308C42C2×C22⋊C416yes—
309C42C23×C411yescharacteristic
310C42C23×C411yescharacteristic
311C42C23×C416yes—
312C42C22×D411yescharacteristic
313C42C22×D416yes—
314C42C22×D411yescharacteristic
315C23—43no—
316C23—43no—
317C23C22×C22⋊C416yes—
318C23C2×#12416yes—
319C23C2×C4⋊Q816yes—
320C23C23×D411yescharacteristic
321Q8—48no—
322Q8—48no—
323Q8—812no—
324Q8—812no—
325Q8#391116yes—
326D4#391148yes—
327C2×C4—22no—
328C2×C4—212no—
329C2×C4—24no—
330C2×C4—28no—
331C2×C4—28no—
332C2×C4—43no—
333C2×C4—43no—
334C2×C4—46no—
335C2×C4—41no—
336C2×C4—46no—
337C2×C4—46no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—46no—
341C2×C4—46no—
342C2×C4—46no—
343C2×C4—812no—
344C2×C4—812no—
345C2×C4—824no—
346C2×C4C22×C22⋊C411yescharacteristic
347C2×C4C22×C22⋊C411yescharacteristic
348C2×C4C22×C22⋊C411yescharacteristic
349C2×C4C22×C22⋊C416yes—
350C2×C4C22×C22⋊C416yes—
351C2×C4C22×C22⋊C411yescharacteristic
352C2×C4C22×C22⋊C416yes—
353C2×C4C2×C4×Q8124yes—
354C2×C4C2×C22⋊Q8124yes—
355C2×C4C2×#12416yes—
356C2×C4C2×C4⋊Q816yes—
357C2×C4C24×C411yescharacteristic
358C2×C4C23×D411yescharacteristic
359C2×C4C23×Q812yes—
360C2×C4C22×#4112yes—
361C8—216no—
362C22—21no—
363C22—26no—
364C22—21no—
365C22—26no—
366C22—46no—
367C22—46no—
368C22C2×#391112yes—
369C22C23×C22⋊C411yescharacteristic
370C22C22×#12411yescharacteristic
371C22C22×C4⋊Q811yescharacteristic
372C4—21no—
373C4—26no—
374C4—21no—
375C4—21no—
376C4—26no—
377C4—44no—
378C4—48no—
379C4—48no—
380C4—86no—
381C4C2×#391112yes—
382C4#290112yes—
383C4C22×C4×Q812yes—
384C4C22×C22⋊Q812yes—
385C2—21no—
386C2—21no—
387C2—212no—
388C2C22×#39111yescharacteristic
389C2C2×#290111yescharacteristic
390C2#6009311yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC22C23×C22⋊C4
prime radicalC22×#41C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #21307C2semidirect
5. #21306C2semidirect
6. C2×#1986C2semidirect
9. #3504C4semidirect
10. #3503C4semidirect
11. #3503C4semidirect
42. #568Q8semidirect
59. #544C2×C4semidirect
60. #544C2×C4semidirect
67. #533C2×C4semidirect
68. #533C2×C4semidirect
71. C2×C4⋊Q8C2×C4semidirect
72. C2×C4⋊Q8C2×C4semidirect
73. C2×#125C2×C4semidirect
138. C2×M16C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#628132
C4×#5682
C4⋊C4×#412
#213072
#213062
C2×#19862
#203432