Finite Group Databasebeta

← #367549

Subgroups — #367549

Order 512.

868 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1953C211yesmaximal, characteristic
2#21182C212yesmaximal
3#21181C211yesmaximal, characteristic
4C2×#1952C211yesmaximal, characteristic
5#20803C211yesmaximal, characteristic
6#20446C211yesmaximal, characteristic
7#11333C212yesmaximal
8#11253C212yesmaximal
9#10916C212yesmaximal
10#10870C211yesmaximal, characteristic
11#10711C211yesmaximal, characteristic
12C2×#430C2212yes—
13C22×#108C2211yescharacteristic
14C22×#108C2211yescharacteristic
15#2911C2212yes—
16C2×#428C2212yes—
17C2×#428C2212yes—
18C2×#427C2211yescharacteristic
19C22×#107C2211yescharacteristic
20#2873C2212yes—
21C2×#405C2211yescharacteristic
22C2×#405C2211yescharacteristic
23C2×#405C2211yescharacteristic
24#2317C2212yes—
25#1953C418yescomplemented
26#1952—22no—
27#1952—22no—
28#1952—21no—
29#1952—21no—
30#1952—24no—
31#1951—22no—
32#1951—24no—
33#1847—22no—
34#1846—21no—
35#1845—21no—
36#1773—22no—
37#1772—22no—
38C2×#344C2211yescharacteristic
39C2×#344C2212yes—
40C2×#344C2211yescharacteristic
41#1742C2212yes—
42#1742C2211yescharacteristic
43C2×#343C2212yes—
44C2×#343C2212yes—
45#1740C2211yescharacteristic
46#1736C2212yes—
47#1733C2212yes—
48C22×M16C2312yes—
49C22×M16C2311yescharacteristic
50C22×M16C2312yes—
51C22×M16C2311yescharacteristic
52C22×M16C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×#117C2311yescharacteristic
55C2×#117C2312yes—
56C2×#117C2311yescharacteristic
57C2×#117C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59#430—24no—
60#430C2×C418yes—
61C2×#108—24no—
62C2×#108—21no—
63C2×#108—21no—
64C2×#108—22no—
65C2×#108C2×C414yes—
66C2×#108C2×C414yes—
67C2×#108D412yes—
68C2×#108D412yes—
69C2×#108Q814yes—
70#428—24no—
71#428—24no—
72#428—22no—
73#428—22no—
74#428—22no—
75#428—22no—
76#428—42no—
77#428—42no—
78#427—21no—
79#427—21no—
80#427—21no—
81#427—21no—
82#427—42no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—22no—
90C2×#107—22no—
91C2×#107D411yescharacteristic
92C2×#107D411yescharacteristic
93C2×#106—22no—
94C2×#106—22no—
95C2×#106D412yes—
96#405—22no—
97#405—22no—
98#405—24no—
99#405—24no—
100#405—21no—
101#405—21no—
102#405—21no—
103#405—21no—
104#405C2×C414yes—
105#405Q814yes—
106C4×M16—42no—
107C2×#97—21no—
108C2×#97—21no—
109C2×#97—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113#389—42no—
114#385—24no—
115#384—22no—
116#382—22no—
117#344—22no—
118#344C2×C418yes—
119#344Q814yes—
120#343—24no—
121#343—24no—
122#343—22no—
123#343—22no—
124#343D412yes—
125#343D412yes—
126#342—22no—
127#340—42no—
128#339—24no—
129C23×C4—22no—
130C23×C4C2411yescharacteristic
131C2×M16—21no—
132C2×M16—21no—
133C2×M16—22no—
134C2×M16—21no—
135C2×M16—22no—
136C2×M16—21no—
137C2×M16—21no—
138C2×M16—22no—
139C2×M16—21no—
140C2×M16—21no—
141C2×M16—22no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—21no—
149C2×M16—21no—
150C2×M16C22×C414yes—
151C2×M16C22×C414yes—
152C2×M16C22×C412yes—
153C2×M16C2×D414yes—
154C2×M16C2×D412yes—
155C2×M16C2×D412yes—
156C2×M16C2×Q812yes—
157C2×M16C2×Q814yes—
158C2×M16C2×Q812yes—
159C2×M16C2×Q812yes—
160C2×M16#4112yes—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8C22×C412yes—
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×Q812yes—
172C22×C8C2×Q812yes—
173C22×C8#4112yes—
174C22×C8#4111yescharacteristic
175C22×C8#4111yescharacteristic
176#117—21no—
177#117—21no—
178#117—21no—
179#117—21no—
180#117—22no—
181#117—21no—
182#117—21no—
183#117—22no—
184#117—22no—
185#117—21no—
186#117—22no—
187#117—21no—
188#117—21no—
189#117C22×C412yes—
190#117C2×Q812yes—
191#117C2×Q812yes—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4#4112yes—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—22no—
213#108—216no—
214#108—24no—
215#108—24no—
216#108C4⋊C418yescomplemented
217#108C4⋊C418yescomplemented
218#107—21no—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—42no—
223#107—42no—
224#107—44no—
225#107—44no—
226#107—41no—
227#107—41no—
228#107—41no—
229#107—41no—
230#106—22no—
231#106—22no—
232#106—42no—
233#106—42no—
234#106—44no—
235#106—44no—
236#97—24no—
237#97—24no—
238#97—44no—
239#97—41no—
240#97—41no—
241C4×C8—24no—
242C4×C8—24no—
243C4×C8—44no—
244C4×C8—41no—
245C4×C8—41no—
246#95—44no—
247C24C2×#4111yescharacteristic
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—162no—
258C22×C4C23×C411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C22×Q811yescharacteristic
263C22×C4C22×Q811yescharacteristic
264C22×C4C22×Q811yescharacteristic
265C22×C4C2×#4111yescharacteristic
266C22×C4C2×#4111yescharacteristic
267C22×C4C2×#4111yescharacteristic
268C22×C4H(5,𝔽2)11yescharacteristic
269C22×C4H(5,𝔽2)11yescharacteristic
270C22×C42−1+411yescharacteristic
271C22×C42−1+411yescharacteristic
272M16—24no—
273M16—22no—
274M16—22no—
275M16—24no—
276M16—22no—
277M16—22no—
278M16—22no—
279M16—24no—
280M16—22no—
281M16—24no—
282M16—24no—
283M16—22no—
284M16—22no—
285M16—42no—
286M16—41no—
287M16—42no—
288M16—41no—
289M16C2×C4⋊C418yes—
290M16C4×D414yes—
291M16C4⋊Q814yes—
292C2×C8—22no—
293C2×C8—24no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—24no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—24no—
300C2×C8—22no—
301C2×C8—24no—
302C2×C8—22no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8C2×C4⋊C412yes—
308C2×C8C2×C4⋊C412yes—
309C2×C8C4×D412yes—
310C2×C8C4×D412yes—
311C2×C8C4×Q812yes—
312C2×C8C4×Q812yes—
313C2×C8#12111yescharacteristic
314C2×C8#12111yescharacteristic
315C2×C8#12111yescharacteristic
316C2×C8#12111yescharacteristic
317C2×C8C22⋊Q812yes—
318C2×C8C22⋊Q812yes—
319C2×C8#12512yes—
320C2×C8#12512yes—
321C2×C8C4⋊Q812yes—
322C2×C8C4⋊Q812yes—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—22no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—41no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—164no—
340C22⋊C4—22no—
341C22⋊C4—22no—
342C22⋊C4—42no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—42no—
346C22⋊C4—41no—
347C22⋊C4C4×Q814yes—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—41no—
352C42—41no—
353C42—42no—
354C42—42no—
355C42—41no—
356C23—41no—
357C23—41no—
358C23C2×C4×Q811yescharacteristic
359C23#53211yescharacteristic
360C23#53311yescharacteristic
361C23#54611yescharacteristic
362C23#55011yescharacteristic
363C23#55211yescharacteristic
364C23#55711yescharacteristic
365C2×C4—24no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4—81no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4—84no—
380C2×C4—82no—
381C2×C4—162no—
382C2×C4—162no—
383C2×C4C22×C4⋊C411yescharacteristic
384C2×C4C2×C4×D411yescharacteristic
385C2×C4C2×C4×D411yescharacteristic
386C2×C4C2×C4×Q811yescharacteristic
387C2×C4#52011yescharacteristic
388C2×C4#52011yescharacteristic
389C2×C4C2×#12111yescharacteristic
390C2×C4C2×C22⋊Q811yescharacteristic
391C2×C4C2×#12511yescharacteristic
392C2×C4C2×C4⋊Q811yescharacteristic
393C2×C4C2×C4⋊Q811yescharacteristic
394C2×C4#53311yescharacteristic
395C2×C4#54011yescharacteristic
396C2×C4#54111yescharacteristic
397C2×C4#54411yescharacteristic
398C2×C4#54411yescharacteristic
399C2×C4D4211yescharacteristic
400C2×C4D4211yescharacteristic
401C2×C4#54711yescharacteristic
402C2×C4#54711yescharacteristic
403C2×C4#54811yescharacteristic
404C2×C4D4×Q811yescharacteristic
405C2×C4D4×Q811yescharacteristic
406C2×C4D4×Q811yescharacteristic
407C2×C4#55311yescharacteristic
408C2×C4#55411yescharacteristic
409C2×C4#55411yescharacteristic
410C2×C4Q8211yescharacteristic
411C8—28no—
412C8—24no—
413C8—24no—
414C8—44no—
415C8—42no—
416C8#38912yes—
417C8#38912yes—
418C8#38912yes—
419C8#38912yes—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22C4×C4⋊Q811yescharacteristic
431C22Q8×C4⋊C411yescharacteristic
432C22#236011yescharacteristic
433C22#242811yescharacteristic
434C22#244111yescharacteristic
435C22#250611yescharacteristic
436C22#255011yescharacteristic
437C4—44no—
438C4—82no—
439C4—82no—
440C4—81no—
441C4—81no—
442C4—82no—
443C4—162no—
444C4C2×#38911yescharacteristic
445C4#233011yescharacteristic
446C4D4×C4⋊C411yescharacteristic
447C4#236111yescharacteristic
448C4#242911yescharacteristic
449C4#244011yescharacteristic
450C4#250711yescharacteristic
451C4#255111yescharacteristic
452C2—21no—
453C2—21no—
454C2—21no—
455C2—21no—
456C2—21no—
457C2—21no—
458C2#1400511yescharacteristic
459C2#2544711yescharacteristic
460C2#2544811yescharacteristic
subgroupG/H
center, socleC22#2428
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #1953C4semidirect
216. #108C4⋊C4semidirect
217. #108C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19532
#211822
#211812
C2×#19522
#208032
#204462
#113332
#112532
#109162
#108702
#107112