Finite Group Databasebeta

← #367569

Subgroups — #367569

Order 512.

924 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1952C211yesmaximal, characteristic
2C2×#1951C211yesmaximal, characteristic
3#21160C212yesmaximal
4#21158C211yesmaximal, characteristic
5#20802C211yesmaximal, characteristic
6C2×#1773C211yesmaximal, characteristic
7#11342C212yesmaximal
8#11217C212yesmaximal
9#10896C212yesmaximal
10#10863C211yesmaximal, characteristic
11#10707C211yesmaximal, characteristic
12C22×#108C2211yescharacteristic
13C2×#428C2212yes—
14C2×#428C2212yes—
15C2×#428C2212yes—
16C2×#427C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#106C2211yescharacteristic
19C2×#405C2211yescharacteristic
20C2×#405C2212yes—
21C2×#405C2211yescharacteristic
22C22×#97C2211yescharacteristic
23#2317C2212yes—
24#2316C2212yes—
25#1952C418yescomplemented
26#1951—24no—
27#1944—22no—
28#1943—21no—
29#1943—21no—
30#1943—22no—
31#1942—22no—
32#1942—22no—
33#1941—22no—
34#1847—22no—
35#1846—21no—
36#1844—21no—
37#1773—24no—
38C2×#343C2212yes—
39C2×#343C2212yes—
40C2×#341C2211yescharacteristic
41#1736C2212yes—
42#1735C2211yescharacteristic
43C2×#340C2212yes—
44C2×#339C2211yescharacteristic
45C2×#339C2212yes—
46C2×#339C2211yescharacteristic
47C4⋊#95C2212yes—
48C22×M16C2311yescharacteristic
49C22×M16C2312yes—
50C22×M16C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×#117C2311yescharacteristic
54C2×#117C2312yes—
55C2×#117C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C42C2311yescharacteristic
59C2×#108C2×C414yes—
60C2×#108D414yes—
61C2×#108Q814yes—
62#428—24no—
63#428—24no—
64#428—24no—
65#428—42no—
66#428—42no—
67#428C2×C418yes—
68#427—22no—
69#427—22no—
70C2×#107—21no—
71C2×#107—21no—
72C2×#107—22no—
73C2×#107—22no—
74C2×#107—22no—
75C2×#107—22no—
76C2×#107C2×C414yes—
77C2×#107D412yes—
78C2×#106—22no—
79C2×#106—21no—
80C2×#106—21no—
81C2×#106—22no—
82C2×#106—22no—
83C2×#106—22no—
84C2×#106—22no—
85C2×#106D411yescharacteristic
86C2×#106D411yescharacteristic
87#405—22no—
88#405—24no—
89#405—22no—
90#405—22no—
91#405—42no—
92#405C2×C414yes—
93C2×#97—22no—
94C2×#97—21no—
95C2×#97—22no—
96C2×#97—21no—
97C2×#97—22no—
98C2×#97Q814yes—
99C2×C4×C8—21no—
100C2×C4×C8—22no—
101C2×C4×C8—21no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#385—42no—
109#384—22no—
110#382—22no—
111#380—42no—
112C2×#95—42no—
113#343—24no—
114#343C2×C418yes—
115#341Q814yes—
116#340—22no—
117#340—22no—
118#340—22no—
119#340—22no—
120#340D412yes—
121#340D412yes—
122#339—24no—
123#339—22no—
124#339—22no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4C2411yescharacteristic
129C2×M16—21no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—21no—
133C2×M16—22no—
134C2×M16—21no—
135C2×M16—21no—
136C2×M16—21no—
137C2×M16—21no—
138C2×M16—21no—
139C2×M16—21no—
140C2×M16C22×C414yes—
141C2×M16C22×C412yes—
142C2×M16C2×D414yes—
143C2×M16C2×Q812yes—
144C2×M16C2×Q814yes—
145C2×M16C2×Q812yes—
146C2×M16#4112yes—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8C22×C412yes—
155C22×C8C2×D412yes—
156C22×C8C2×D412yes—
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159C22×C8C2×Q812yes—
160C22×C8C2×Q812yes—
161C22×C8C2×Q812yes—
162C22×C8C2×Q812yes—
163C22×C8#4112yes—
164C22×C8#4112yes—
165#117—21no—
166#117—21no—
167#117—22no—
168#117—21no—
169#117—21no—
170#117—21no—
171#117—22no—
172#117—22no—
173#117—22no—
174#117—21no—
175#117C22×C412yes—
176#117#4112yes—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4C22×C414yes—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—21no—
209C2×C42C2×Q812yes—
210#108C4⋊C4116yescomplemented
211#107—24no—
212#107—24no—
213#107—24no—
214#107—28no—
215#107—44no—
216#107—44no—
217#106—22no—
218#106—22no—
219#106—28no—
220#106—44no—
221#106—44no—
222#106—42no—
223#106—42no—
224#97—24no—
225#97—28no—
226#97—44no—
227#97—42no—
228C4×C8—24no—
229C4×C8—44no—
230C4×C8—42no—
231#95—42no—
232#95—42no—
233#95—42no—
234#95—42no—
235#95—84no—
236C24H(5,𝔽2)11yescharacteristic
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4C23×C411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C22×C4C22×Q811yescharacteristic
262C22×C4C22×Q811yescharacteristic
263C22×C4C22×Q811yescharacteristic
264C22×C4C2×#4111yescharacteristic
265C22×C4C2×#4111yescharacteristic
266C22×C4C2×#4111yescharacteristic
267C22×C4C2×#4111yescharacteristic
268C22×C4H(5,𝔽2)11yescharacteristic
269C22×C42−1+411yescharacteristic
270C22×C42−1+411yescharacteristic
271M16—24no—
272M16—22no—
273M16—24no—
274M16—22no—
275M16—24no—
276M16—22no—
277M16—22no—
278M16—22no—
279M16—41no—
280M16—41no—
281M16C2×C4⋊C418yes—
282M16C4×Q814yes—
283C2×C8—22no—
284C2×C8—24no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—24no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—24no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8C2×C4⋊C414yes—
299C2×C8C4×D412yes—
300C2×C8C4×D412yes—
301C2×C8C4×Q814yes—
302C2×C8#12112yes—
303C2×C8#12112yes—
304C2×C8C22⋊Q814yes—
305C2×C8#12514yes—
306C2×C8C4⋊Q814yes—
307C2×C8C4⋊Q812yes—
308C2×C8C4⋊Q812yes—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—162no—
326C4⋊C4—162no—
327C4⋊C4—162no—
328C4⋊C4—162no—
329C4⋊C4C4×D412yes—
330C4⋊C4C4×D412yes—
331C22⋊C4—22no—
332C22⋊C4—22no—
333C22⋊C4—41no—
334C22⋊C4—42no—
335C22⋊C4—41no—
336C22⋊C4—42no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—41no—
342C42—42no—
343C42—42no—
344C42—41no—
345C42—41no—
346C42—41no—
347C23—22no—
348C23—22no—
349C23#52011yescharacteristic
350C23#54011yescharacteristic
351C23#54411yescharacteristic
352C23D4211yescharacteristic
353C23#55011yescharacteristic
354C23#55311yescharacteristic
355C23Q8211yescharacteristic
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—162no—
383C2×C4—162no—
384C2×C4C22×C4⋊C411yescharacteristic
385C2×C4C2×C4×D411yescharacteristic
386C2×C4C2×C4×D411yescharacteristic
387C2×C4C2×C4×Q811yescharacteristic
388C2×C4C2×C4×Q811yescharacteristic
389C2×C4#52011yescharacteristic
390C2×C4C2×#12111yescharacteristic
391C2×C4C2×C22⋊Q811yescharacteristic
392C2×C4C2×#12511yescharacteristic
393C2×C4C2×C4⋊Q811yescharacteristic
394C2×C4C2×C4⋊Q811yescharacteristic
395C2×C4#53211yescharacteristic
396C2×C4#53311yescharacteristic
397C2×C4#53311yescharacteristic
398C2×C4#54111yescharacteristic
399C2×C4#54411yescharacteristic
400C2×C4D4211yescharacteristic
401C2×C4#54611yescharacteristic
402C2×C4#54711yescharacteristic
403C2×C4#54711yescharacteristic
404C2×C4#54811yescharacteristic
405C2×C4D4×Q811yescharacteristic
406C2×C4D4×Q811yescharacteristic
407C2×C4D4×Q811yescharacteristic
408C2×C4#55211yescharacteristic
409C2×C4#55411yescharacteristic
410C2×C4#55411yescharacteristic
411C2×C4#55711yescharacteristic
412C8—28no—
413C8—24no—
414C8—24no—
415C8—42no—
416C8#38914yes—
417C8#38914yes—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22#233011yescharacteristic
427C22Q8×C4⋊C411yescharacteristic
428C22#236111yescharacteristic
429C22#242811yescharacteristic
430C22#244111yescharacteristic
431C22#250711yescharacteristic
432C22#255111yescharacteristic
433C22#328812yes—
434C22#328912yes—
435C4—42no—
436C4—42no—
437C4—82no—
438C4—84no—
439C4—82no—
440C4—82no—
441C4—82no—
442C4—162no—
443C4C2×#38911yescharacteristic
444C4C4×C4⋊Q811yescharacteristic
445C4D4×C4⋊C411yescharacteristic
446C4#236011yescharacteristic
447C4#242911yescharacteristic
448C4#244011yescharacteristic
449C4#250611yescharacteristic
450C4#255011yescharacteristic
451C2—21no—
452C2—21no—
453C2—21no—
454C2—21no—
455C2#1400511yescharacteristic
456C2#2304711yescharacteristic
457C2#2304811yescharacteristic
458C2#2544312yes—
459C2#2882612yes—
subgroupG/H
center, socleC23D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24H(5,𝔽2)
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #1952C4semidirect
210. #108C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19522
C2×#19512
#211602
#211582
#208022
C2×#17732
#113422
#112172
#108962
#108632
#107072