Finite Group Databasebeta

← #370615

Subgroups — #370615

Order 512.

1048 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3163C211yesmaximal, characteristic
2#21326C212yesmaximal
3#21156C211yesmaximal, characteristic
4C2×#1773C211yesmaximal, characteristic
5#11392C212yesmaximal
6#11217C212yesmaximal
7#10896C212yesmaximal
8#10870C211yesmaximal, characteristic
9#10869C211yesmaximal, characteristic
10#10863C211yesmaximal, characteristic
11#10707C211yesmaximal, characteristic
12C2×#544C2212yes—
13#3163—24no—
14#3163C418yescomplemented
15C2×#501—22no—
16C2×#499—21no—
17C2×#498—21no—
18C22×#108C2211yescharacteristic
19C2×#428C2212yes—
20C2×#427C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C2×#405C2211yescharacteristic
24C2×#405C2212yes—
25C2×#405C2211yescharacteristic
26C22×#97C2211yescharacteristic
27#2317C2212yes—
28#1993—22no—
29#1992—22no—
30#1991—22no—
31#1990—22no—
32#1944—21no—
33#1944—21no—
34#1942—22no—
35#1773—24no—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#341C2211yescharacteristic
39#1736C2212yes—
40#1736C2212yes—
41#1736C2212yes—
42#1735C2211yescharacteristic
43C2×#340C2212yes—
44C2×#339C2211yescharacteristic
45C2×#339C2212yes—
46C2×#339C2211yescharacteristic
47C4⋊#95C2212yes—
48C22×M16C2312yes—
49C22×M16C2311yescharacteristic
50C22×M16C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53#544—24no—
54#544—24no—
55#544—42no—
56#544C2×C418yes—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×#125—22no—
60C2×#125—22no—
61C2×C22⋊Q8—42no—
62C2×#117C2311yescharacteristic
63C2×#117C2312yes—
64C2×#117C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C42C2311yescharacteristic
68#501—216no—
69#499—28no—
70#498—28no—
71C2×#108—22no—
72C2×#108—22no—
73C2×#108Q814yescomplemented
74#428—24no—
75#428C2×C418yes—
76#427C2×C414yes—
77#427D414yes—
78C2×#107—22no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—22no—
82C2×#107—22no—
83C2×#107D412yes—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—22no—
87C2×#106—22no—
88C2×#106C2×C414yes—
89C2×#106D411yescharacteristic
90C2×#106D411yescharacteristic
91#405—22no—
92#405—22no—
93#405—22no—
94#405—24no—
95#405—42no—
96#405C2×C414yes—
97C2×#97—22no—
98C2×#97—21no—
99C2×#97—22no—
100C2×#97—21no—
101C2×#97—22no—
102C2×#97Q814yescomplemented
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—22no—
106#389—22no—
107#389—22no—
108#385—42no—
109#384—22no—
110#382—22no—
111C2×#95—42no—
112#343—24no—
113#343—42no—
114#341Q814yescomplemented
115#340—22no—
116#340—22no—
117#340—22no—
118#340—22no—
119#340—22no—
120#340—22no—
121#340—22no—
122#340—22no—
123#340D412yes—
124#340D412yes—
125#339—22no—
126#339—24no—
127#339—22no—
128C22×Q8—42no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4C2411yescharacteristic
133C2×M16—21no—
134C2×M16—21no—
135C2×M16—21no—
136C2×M16—22no—
137C2×M16—21no—
138C2×M16—22no—
139C2×M16—21no—
140C2×M16—22no—
141C2×M16—21no—
142C2×M16—21no—
143C2×M16—22no—
144C2×M16—21no—
145C2×M16—22no—
146C2×M16C22×C412yes—
147C2×M16C2×Q812yes—
148C2×M16C2×Q814yes—
149C2×M16C2×Q812yes—
150C2×M16#4112yes—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8C22×C412yes—
159C22×C8C2×D412yes—
160C22×C8C2×D412yes—
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2×Q812yes—
164C22×C8C2×Q812yes—
165C22×C8C2×Q812yes—
166C22×C8C2×Q812yes—
167C22×C8#4112yes—
168C22×C8#4112yes—
169C4⋊Q8—28no—
170C4⋊Q8—28no—
171C4⋊Q8—44no—
172C4⋊Q8—44no—
173#125—28no—
174#125—28no—
175#125—44no—
176#125—44no—
177C22⋊Q8—48no—
178C22⋊Q8—48no—
179#117—21no—
180#117—22no—
181#117—21no—
182#117—22no—
183#117—21no—
184#117—21no—
185#117—21no—
186#117—21no—
187#117C22×C414yes—
188#117C22×C412yes—
189#117C2×D414yes—
190#117#4112yes—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
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—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4C22×C414yes—
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—21no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42C2×Q812yes—
225#108—28no—
226#107—28no—
227#107—42no—
228#107—42no—
229#107C22⋊C418yes—
230#106—24no—
231#106—24no—
232#106—28no—
233#106C22⋊C414yes—
234#106C22⋊C414yes—
235#97—24no—
236#97—28no—
237#97—44no—
238#97—42no—
239C4×C8—24no—
240C4×C8—44no—
241C4×C8—42no—
242#95—42no—
243#95—42no—
244#95—84no—
245C24H(5,𝔽2)11yescharacteristic
246C2×Q8—48no—
247C2×Q8—162no—
248C2×Q8—162no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4C23×C411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×Q811yescharacteristic
274C22×C4C22×Q811yescharacteristic
275C22×C4C22×Q811yescharacteristic
276C22×C4C2×#4111yescharacteristic
277C22×C4C2×#4111yescharacteristic
278C22×C4C2×#4111yescharacteristic
279C22×C4C2×#4111yescharacteristic
280C22×C4H(5,𝔽2)11yescharacteristic
281C22×C42−1+411yescharacteristic
282C22×C42−1+411yescharacteristic
283M16—22no—
284M16—22no—
285M16—22no—
286M16—24no—
287M16—22no—
288M16—24no—
289M16—22no—
290M16—42no—
291M16—41no—
292M16—42no—
293M16—41no—
294M16C4×Q814yes—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—24no—
305C2×C8—22no—
306C2×C8—41no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—41no—
310C2×C8C2×C22⋊C412yes—
311C2×C8C2×C22⋊C412yes—
312C2×C8C4×Q814yes—
313C2×C8C4×Q814yes—
314C2×C8C22⋊Q814yes—
315C2×C8C22⋊Q814yes—
316C2×C8#12412yes—
317C2×C8#12412yes—
318C2×C8C4⋊Q814yes—
319C2×C8C4⋊Q812yes—
320C2×C8C4⋊Q812yes—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—48no—
331C4⋊C4—48no—
332C4⋊C4—48no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—162no—
340C4⋊C4—162no—
341C4⋊C4C2×C22⋊C414yes—
342C4⋊C4C2×C22⋊C414yes—
343C4⋊C4C4×D414yes—
344C22⋊C4—24no—
345C22⋊C4—22no—
346C22⋊C4—24no—
347C22⋊C4—22no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C42—22no—
351C42—24no—
352C42—24no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—41no—
360C23—22no—
361C23—22no—
362C23#51811yescharacteristic
363C23#54211yescharacteristic
364C23#54411yescharacteristic
365C23D4211yescharacteristic
366C23#54611yescharacteristic
367C23#55511yescharacteristic
368C23Q8211yescharacteristic
369Q8—168no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—84no—
392C2×C4—84no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—162no—
397C2×C4—162no—
398C2×C4C22×C22⋊C411yescharacteristic
399C2×C4C2×C4×D411yescharacteristic
400C2×C4C2×C4×Q811yescharacteristic
401C2×C4C2×C4×Q811yescharacteristic
402C2×C4C2×C4×Q811yescharacteristic
403C2×C4#51911yescharacteristic
404C2×C4C2×C22⋊Q811yescharacteristic
405C2×C4C2×C22⋊Q811yescharacteristic
406C2×C4C2×#12411yescharacteristic
407C2×C4C2×C4⋊Q811yescharacteristic
408C2×C4C2×C4⋊Q811yescharacteristic
409C2×C4#53211yescharacteristic
410C2×C4#53311yescharacteristic
411C2×C4#53311yescharacteristic
412C2×C4#54211yescharacteristic
413C2×C4#54411yescharacteristic
414C2×C4D4211yescharacteristic
415C2×C4#54611yescharacteristic
416C2×C4#54711yescharacteristic
417C2×C4#54811yescharacteristic
418C2×C4#54811yescharacteristic
419C2×C4D4×Q811yescharacteristic
420C2×C4D4×Q811yescharacteristic
421C2×C4D4×Q811yescharacteristic
422C2×C4#55411yescharacteristic
423C2×C4#55411yescharacteristic
424C2×C4#55611yescharacteristic
425C2×C4#55711yescharacteristic
426C8—24no—
427C8—24no—
428C8—44no—
429C8—42no—
430C8#39114yes—
431C8#39114yes—
432C22—22no—
433C22—22no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—22no—
439C22—22no—
440C22#233111yescharacteristic
441C22D4×C22⋊C411yescharacteristic
442C22#236911yescharacteristic
443C22#243911yescharacteristic
444C22#245111yescharacteristic
445C22#249111yescharacteristic
446C22#255111yescharacteristic
447C22#328812yes—
448C22#328912yes—
449C4—44no—
450C4—44no—
451C4—44no—
452C4—82no—
453C4—81no—
454C4—81no—
455C4—84no—
456C4—162no—
457C4C2×#39111yescharacteristic
458C4C4×C4⋊Q811yescharacteristic
459C4Q8×C22⋊C411yescharacteristic
460C4#236511yescharacteristic
461C4#244011yescharacteristic
462C4#245211yescharacteristic
463C4#249211yescharacteristic
464C4#255011yescharacteristic
465C2—21no—
466C2—21no—
467C2—21no—
468C2—21no—
469C2#1405311yescharacteristic
470C2#2246412yes—
471C2#2781011yescharacteristic
472C2#2781111yescharacteristic
473C2#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
14. #3163C4semidirect
73. C2×#108Q8semidirect
102. C2×#97Q8semidirect
114. #341Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31632
#213262
#211562
C2×#17732
#113922
#112172
#108962
#108702
#108692
#108632
#107072