Finite Group Databasebeta

← #357043

Subgroups — #357043

Order 512.

1014 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1952C211yesmaximal, characteristic
2C2×#1951C211yesmaximal, characteristic
3#20455C211yesmaximal, characteristic
4#20454C211yesmaximal, characteristic
5#20453C212yesmaximal
6C2×#1773C211yesmaximal, characteristic
7#10899C212yesmaximal
8#10899C212yesmaximal
9#10863C211yesmaximal, characteristic
10#10827C212yesmaximal
11#10707C211yesmaximal, characteristic
12C22×#108C2211yescharacteristic
13C2×#428C2212yes—
14C2×#428C2212yes—
15C2×#427C2211yescharacteristic
16C22×#107C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C2×#405C2212yes—
19C2×#405C2211yescharacteristic
20C2×#405C2211yescharacteristic
21C22×#97C2211yescharacteristic
22#2274C2212yes—
23#2274C2212yes—
24#1952C418yescomplemented
25#1951C418yescomplemented
26#1776—22no—
27#1776—21no—
28#1776—21no—
29#1776—22no—
30#1776—22no—
31C4×#107—22no—
32C4×#107—22no—
33C4×#106—21no—
34C4×#106—21no—
35C4×#106—22no—
36#1773C418yescomplemented
37C2×#343C2212yes—
38C2×#343C2212yes—
39C2×#341C2211yescharacteristic
40#1736C2212yes—
41#1736C2212yes—
42#1735C2211yescharacteristic
43C2×#340C2212yes—
44C2×#339C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2212yes—
47C4⋊#95C2212yes—
48C22×M16C2311yescharacteristic
49C22×M16C2311yescharacteristic
50C22×M16C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×#117C2312yes—
54C2×#117C2311yescharacteristic
55C2×#117C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C42C2311yescharacteristic
59C2×#108C2×C414yes—
60C2×#108C2×C414yes—
61C2×#108C2×C414yes—
62#428—24no—
63#428—24no—
64#428C2×C418yes—
65#428C2×C418yes—
66#427C2×C414yes—
67#427Q814yes—
68C2×#107—21no—
69C2×#107—21no—
70C2×#107—22no—
71C2×#107—22no—
72C2×#107—22no—
73C2×#107C2×C414yes—
74C2×#107D412yes—
75C2×#106—21no—
76C2×#106—21no—
77C2×#106—22no—
78C2×#106—22no—
79C2×#106—22no—
80C2×#106C2×C414yes—
81C2×#106D411yescharacteristic
82C2×#106D411yescharacteristic
83#405—24no—
84#405—22no—
85#405—22no—
86#405C2×C414yes—
87#405C2×C414yes—
88#405C2×C418yes—
89C2×#97—22no—
90C2×#97—21no—
91C2×#97—21no—
92C2×#97—22no—
93C2×#97—22no—
94C2×#97C2×C414yes—
95C2×C4×C8—21no—
96C2×C4×C8—22no—
97C2×C4×C8—21no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C22⋊C4—42no—
107C4×C22⋊C4—42no—
108C2×#95—42no—
109#343C2×C418yes—
110#343C2×C418yes—
111#341—22no—
112#340—22no—
113#340—22no—
114#340—22no—
115#340—22no—
116#340—22no—
117#340—22no—
118#340D412yes—
119#340D412yes—
120#339—22no—
121#339—22no—
122#339C2×C418yes—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4C2411yescharacteristic
127C2×M16—22no—
128C2×M16—22no—
129C2×M16—22no—
130C2×M16—21no—
131C2×M16—21no—
132C2×M16—21no—
133C2×M16—21no—
134C2×M16—21no—
135C2×M16—21no—
136C2×M16—21no—
137C2×M16—21no—
138C2×M16—21no—
139C2×M16—21no—
140C2×M16C22×C412yes—
141C2×M16C22×C412yes—
142C2×M16C22×C414yes—
143C2×M16C22×C414yes—
144C2×M16C22×C414yes—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8C22×C412yes—
153C22×C8C22×C412yes—
154C22×C8C22×C412yes—
155C22×C8C22×C412yes—
156C22×C8C2×D412yes—
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159C22×C8C2×Q812yes—
160C22×C8#4112yes—
161C22×C8#4112yes—
162C22×C8#4112yes—
163#117—21no—
164#117—21no—
165#117—22no—
166#117—21no—
167#117—21no—
168#117—22no—
169#117C22×C414yes—
170#117C22×C412yes—
171#117C22×C412yes—
172#117C2×Q814yes—
173#117#4112yes—
174#117#4112yes—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
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⋊C4C22×C414yes—
186C2×C4⋊C4C22×C414yes—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42C22×C412yes—
208#108C42116yescomplemented
209#107—24no—
210#107—24no—
211#107—28no—
212#107—28no—
213#107C4⋊C418yes—
214#106—24no—
215#106—24no—
216#106—28no—
217#106—28no—
218#106C4⋊C414yes—
219#106C4⋊C414yes—
220#97—28no—
221#97—28no—
222#97—24no—
223#97—24no—
224C4×C8—28no—
225C4×C8—24no—
226C4×C8—24no—
227#95—42no—
228#95—42no—
229#95—42no—
230#95—42no—
231#95—84no—
232C24H(5,𝔽2)11yescharacteristic
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4C23×C411yescharacteristic
254C22×C4C23×C411yescharacteristic
255C22×C4C23×C411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C22×D411yescharacteristic
258C22×C4C22×Q811yescharacteristic
259C22×C4C2×#4111yescharacteristic
260C22×C4C2×#4111yescharacteristic
261C22×C4C2×#4111yescharacteristic
262C22×C4C2×#4111yescharacteristic
263C22×C4C2×#4111yescharacteristic
264C22×C42−1+411yescharacteristic
265C22×C42−1+411yescharacteristic
266C22×C42−1+411yescharacteristic
267M16—24no—
268M16—24no—
269M16—24no—
270M16—22no—
271M16—22no—
272M16—22no—
273M16—22no—
274M16—41no—
275M16—41no—
276M16—41no—
277M16—41no—
278M16C2×C4218yes—
279C2×C8—22no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8C2×C4214yes—
295C2×C8C2×C4⋊C412yes—
296C2×C8C2×C4⋊C412yes—
297C2×C8#11714yes—
298C2×C8C4×D414yes—
299C2×C8C4×D412yes—
300C2×C8C4×D412yes—
301C2×C8C4×D412yes—
302C2×C8C4×D412yes—
303C2×C8C4×Q814yes—
304C2×C8C4×Q814yes—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4C2×C4⋊C414yes—
319C4⋊C4C2×C4⋊C414yes—
320C4⋊C4C4×D412yes—
321C4⋊C4C4×D414yes—
322C4⋊C4C4×D412yes—
323C22⋊C4—24no—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—24no—
329C42—24no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—24no—
335C42—22no—
336C42—22no—
337C42—41no—
338C42—41no—
339C42—162no—
340C42—162no—
341C42—162no—
342C42—162no—
343C23—22no—
344C23—22no—
345C23#52011yescharacteristic
346C23#52011yescharacteristic
347C23#52011yescharacteristic
348C23D4211yescharacteristic
349C23#55311yescharacteristic
350C23#55311yescharacteristic
351C23#55411yescharacteristic
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4—162no—
379C2×C4—162no—
380C2×C4C22×C4211yescharacteristic
381C2×C4C22×C4⋊C411yescharacteristic
382C2×C4C2×#11711yescharacteristic
383C2×C4C2×C4×D411yescharacteristic
384C2×C4C2×C4×D411yescharacteristic
385C2×C4C2×C4×D411yescharacteristic
386C2×C4C2×C4×D411yescharacteristic
387C2×C4C2×C4×D411yescharacteristic
388C2×C4C2×C4×Q811yescharacteristic
389C2×C4C2×C4×Q811yescharacteristic
390C2×C4C4×#4111yescharacteristic
391C2×C4C4×#4111yescharacteristic
392C2×C4C4×#4111yescharacteristic
393C2×C4#51911yescharacteristic
394C2×C4#51911yescharacteristic
395C2×C4#51911yescharacteristic
396C2×C4#54711yescharacteristic
397C2×C4#54711yescharacteristic
398C2×C4#54711yescharacteristic
399C2×C4#54811yescharacteristic
400C2×C4#54811yescharacteristic
401C2×C4D4×Q811yescharacteristic
402C2×C4#55211yescharacteristic
403C2×C4#55211yescharacteristic
404C2×C4#55611yescharacteristic
405C2×C4#55611yescharacteristic
406C2×C4#55711yescharacteristic
407C2×C4#55711yescharacteristic
408C8—24no—
409C8—28no—
410C8—42no—
411C8—42no—
412C8C4×C4⋊C414yes—
413C8C4×C4⋊C414yes—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22#227611yescharacteristic
423C22#234711yescharacteristic
424C22D4×C4⋊C411yescharacteristic
425C22#236111yescharacteristic
426C22#236111yescharacteristic
427C22#237011yescharacteristic
428C22#237011yescharacteristic
429C22#328812yes—
430C22#328912yes—
431C4—42no—
432C4—42no—
433C4—44no—
434C4—44no—
435C4—44no—
436C4—82no—
437C4—84no—
438C4—84no—
439C4C2×C4×C4⋊C411yescharacteristic
440C4C42×D411yescharacteristic
441C4#234411yescharacteristic
442C4#235111yescharacteristic
443C4#236211yescharacteristic
444C4#236211yescharacteristic
445C4#236511yescharacteristic
446C4#236511yescharacteristic
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2#1382111yescharacteristic
452C2#2269312yes—
453C2#2304711yescharacteristic
454C2#2304811yescharacteristic
455C2#2304912yes—
subgroupG/H
center, socleC23D42
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC24H(5,𝔽2)
LCSG ▷ C2×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
24. #1952C4semidirect
25. #1951C4semidirect
36. #1773C4semidirect
208. #108C42semidirect

Maximal subgroups

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

subgroupindex
C2×#19522
C2×#19512
#204552
#204542
#204532
C2×#17732
#108992
#108632
#108272
#107072