Finite Group Databasebeta

← #220730

Subgroups — #220730

Order 512.

1205 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3283C211yesmaximal, characteristic
2#22670C211yesmaximal, characteristic, complemented
3#22485C212yesmaximal, complemented
4C2×#1580C211yesmaximal, characteristic, complemented
5#9839C211yesmaximal, characteristic, complemented
6#9742C211yesmaximal, characteristic, complemented
7#9217C212yesmaximal, complemented
8#8985C212yesmaximal, complemented
9#8642C211yesmaximal, characteristic, complemented
10#8446C212yesmaximal, complemented
11#8202C211yesmaximal, characteristic, complemented
12C22×#135—21no—
13C2×#547C2212yes—
14#3283—24no—
15#3283C418yescomplemented
16C2×#494C2211yescharacteristic
17C2×#467—21no—
18C2×#461C2212yes—
19C2×#452—22no—
20C2×#442—21no—
21C2×C4×D8C2211yescharacteristic
22C2×C8×D4C2212yes—
23C2×C8×D4C2211yescharacteristic
24D4×C4⋊C4C2212yes—
25#1990C2212yes—
26#1990C2212yes—
27#1988C2212yes—
28#1943—21no—
29#1893—22no—
30#1874—22no—
31#1864—22no—
32#1844C2211yescharacteristic
33#1823—22no—
34#1802C2212yes—
35#1802C2212yes—
36#1770C2211yescharacteristic
37C4×#102C2211yescharacteristic
38C2×#340—22no—
39#1580—24no—
40C2×#334C2211yescharacteristic
41C2×#332C2212yes—
42C2×D4⋊C8C2211yescharacteristic
43C2×D4⋊C8C2212yes—
44C2×#322C2211yescharacteristic
45#1296C2212yes—
46#1279C2211yescharacteristic
47#1277C2211yescharacteristic
48C22×#41—22no—
49C2×#135—28no—
50C2×#135—44no—
51C2×#135—44no—
52C22×Q16—21no—
53C22×Q16—21no—
54C22×SD16—22no—
55C22×D8D411yescharacteristic
56C23×C8—21no—
57C23×C8—22no—
58#547—24no—
59#547—24no—
60#547—42no—
61#547C2×C418yescomplemented
62C2×C4⋊Q8C2312yes—
63C2×#123—42no—
64C2×C22⋊Q8—22no—
65C2×#121—22no—
66C2×C4×D4—22no—
67C2×C4×D4C2312yes—
68C2×C4×D4C2311yescharacteristic
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71#494—22no—
72#494C2×C414yescomplemented
73#467—28no—
74#461—24no—
75#461C2×C418yescomplemented
76#452—216no—
77#442—44no—
78C4×D8C2×C414yes—
79C4×D8D414yes—
80C8×D4—24no—
81C8×D4—22no—
82C8×D4—42no—
83C8×D4C2×C414yescomplemented
84C2×#107D411yescharacteristic
85C2×#106—21no—
86C2×#106—21no—
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2312yes—
89C2×C4⋊C8C2311yescharacteristic
90C2×#103—22no—
91C2×#102—22no—
92C2×#102—22no—
93C2×#102D412yes—
94C2×#102D412yes—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—22no—
97C2×#97—21no—
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#389—22no—
102#385—42no—
103#384C2312yes—
104#380—22no—
105C4×C4⋊C4C2311yescharacteristic
106#340—22no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—22no—
112#340—42no—
113#340D412yes—
114#336—21no—
115#334—22no—
116#332—42no—
117D4⋊C8—24no—
118D4⋊C8—22no—
119#322—22no—
120C2×#41—28no—
121C2×#41—44no—
122C2×#41—44no—
123C2×#41—44no—
124C2×#41—82no—
125C2×#41—82no—
126C22×Q8—22no—
127C22×D4—42no—
128C22×D4C2×D411yescharacteristic
129C22×D4C2×D412yes—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—42no—
133C23×C4—42no—
134#135—432no—
135C2×Q16—24no—
136C2×Q16—24no—
137C2×Q16—42no—
138C2×Q16—42no—
139C2×Q16—81no—
140C2×Q16—81no—
141C2×SD16—28no—
142C2×SD16—44no—
143C2×SD16—44no—
144C2×D8C22⋊C414yes—
145C2×D8D814yes—
146C2×D8SD1614yes—
147C22×C8—21no—
148C22×C8—24no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—44no—
153C22×C8—44no—
154C22×C8—42no—
155C22×C8—42no—
156C22×C8—82no—
157C22×C8C2×D412yes—
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8C22×C414yes—
164#123—48no—
165#123—48no—
166C22⋊Q8—28no—
167C22⋊Q8—44no—
168#121—28no—
169#121—44no—
170C4×D4—21no—
171C4×D4—22no—
172C4×D4—21no—
173C4×D4—22no—
174C4×D4—44no—
175C4×D4—82no—
176C4×D4C22×C414yes—
177C4×D4C22×C412yes—
178C4×D4C2×D414yes—
179C4×D4#4112yes—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C42—21no—
199C2×C42C2411yescharacteristic
200#107C22⋊C414yes—
201#106—42no—
202#106—81no—
203C4⋊C8—21no—
204C4⋊C8—22no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—22no—
208C4⋊C8C22×C412yes—
209#103—28no—
210#102—42no—
211#102—44no—
212#102—82no—
213#102C22⋊C418yes—
214C22⋊C8—24no—
215C22⋊C8—44no—
216#97—42no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8C22×C412yes—
222C4×C8C2×D412yes—
223#95—21no—
224#95—42no—
225C24—22no—
226C24—21no—
227#41—416no—
228#41—88no—
229#41—88no—
230#41—88no—
231C2×Q8—24no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×D4—21no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—48no—
244C2×D4—84no—
245C2×D4—84no—
246C2×D4C2×C22⋊C414yes—
247C2×D4C4×D412yes—
248C2×D4C2×D814yes—
249C2×D4C2×SD1614yes—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—48no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—84no—
269C22×C4—84no—
270C22×C4—84no—
271C22×C4—84no—
272C22×C4—162no—
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4H(5,𝔽2)11yescharacteristic
276Q16—48no—
277Q16—84no—
278SD16—416no—
279D8#102116yescomplemented
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—48no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—41no—
297C2×C8—88no—
298C2×C8C2×C22⋊C412yes—
299C2×C8C2×C22⋊C412yes—
300C2×C8C2×D812yes—
301C2×C8C2×SD1612yes—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—48no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—82no—
316C4⋊C4—84no—
317C4⋊C4C2×C22⋊C414yes—
318C4⋊C4C4×D412yes—
319C22⋊C4—24no—
320C22⋊C4—22no—
321C22⋊C4—24no—
322C22⋊C4—22no—
323C22⋊C4—48no—
324C22⋊C4—84no—
325C42—41no—
326C42—41no—
327C42C23×C411yescharacteristic
328C42C22×D411yescharacteristic
329C42C2×#4111yescharacteristic
330C42H(5,𝔽2)11yescharacteristic
331C23—24no—
332C23—24no—
333C23—24no—
334C23—22no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—41no—
341C23—41no—
342C23—42no—
343C23D4211yescharacteristic
344Q8—44no—
345Q8—44no—
346Q8—82no—
347Q8—82no—
348D4—24no—
349D4—22no—
350D4—22no—
351D4—24no—
352D4—24no—
353D4—42no—
354D4—816no—
355D4C2×#10218yes—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—816no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—168no—
382C2×C4C22×C22⋊C411yescharacteristic
383C2×C4C2×C4×D411yescharacteristic
384C2×C4#51811yescharacteristic
385C2×C4D4211yescharacteristic
386C2×C4#54611yescharacteristic
387C2×C4#54611yescharacteristic
388C2×C4C22×D811yescharacteristic
389C2×C4C22×SD1611yescharacteristic
390C2×C4#57512yes—
391C2×C4#57611yescharacteristic
392C2×C4#57711yescharacteristic
393C2×C4#57712yes—
394C8—22no—
395C8—42no—
396C8—42no—
397C8—81no—
398C8—82no—
399C8—81no—
400C8C2×#10214yes—
401C22—28no—
402C22—44no—
403C22—44no—
404C22—44no—
405C22—44no—
406C22—44no—
407C22—44no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—82no—
412C22—82no—
413C22D4×C22⋊C411yescharacteristic
414C22#326712yes—
415C22D4×D811yescharacteristic
416C22D4×SD1611yescharacteristic
417C22#328912yes—
418C4—21no—
419C4—21no—
420C4—42no—
421C4—44no—
422C4—82no—
423C4—84no—
424C4—162no—
425C4C22×#10211yescharacteristic
426C4#289711yescharacteristic
427C4#294311yescharacteristic
428C4#295211yescharacteristic
429C4#326712yes—
430C4#329411yescharacteristic
431C4#329511yescharacteristic
432C2—48no—
433C2—82no—
434C2#1975812yes—
435C2#1976612yes—
436C2D4×#10211yescharacteristic
437C2#2243811yescharacteristic
438C2#2246511yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3283C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22670C2semidirect
3. #22485C2semidirect
4. C2×#1580C2semidirect
5. #9839C2semidirect
6. #9742C2semidirect
7. #9217C2semidirect
8. #8985C2semidirect
9. #8642C2semidirect
10. #8446C2semidirect
11. #8202C2semidirect
15. #3283C4semidirect
61. #547C2×C4semidirect
72. #494C2×C4semidirect
75. #461C2×C4semidirect
83. C8×D4C2×C4semidirect
279. D8#102semidirect

Maximal subgroups

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

subgroupindex
C2×#32832
#226702
#224852
C2×#15802
#98392
#97422
#92172
#89852
#86422
#84462
#82022