Finite Group Databasebeta

← #512435

Subgroups — #512435

Order 512.

562 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2255C211yesmaximal, characteristic, complemented
2C2×#2108C211yesmaximal, characteristic
3#33216C211yesmaximal, characteristic
4#33215C211yesmaximal, characteristic
5#13282C211yesmaximal, characteristic, complemented
6#13281C211yesmaximal, characteristic, complemented
7#13168C211yesmaximal, characteristic, complemented
8#13167C211yesmaximal, characteristic, complemented
9#13166C211yesmaximal, characteristic, complemented
10#13165C211yesmaximal, characteristic, complemented
11#12908C211yesmaximal, characteristic, complemented
12#12836C211yesmaximal, characteristic
13#12836C211yesmaximal, characteristic
14#12835C211yesmaximal, characteristic
15#12834C211yesmaximal, characteristic
16C22×M32C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×#499C2211yescharacteristic
20C2×#405C2211yescharacteristic
21#2629—21no—
22#2628—21no—
23#2255—24no—
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×#366C2211yescharacteristic
32#2108—24no—
33C2×#346C2211yescharacteristic
34C2×#346C2211yescharacteristic
35C2×C4×C16C2211yescharacteristic
36C2×C4×C16C2211yescharacteristic
37#1993—21no—
38#1992—21no—
39#1944—21no—
40#1943—21no—
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1846C2211yescharacteristic
46#1845C2211yescharacteristic
47#1809—21no—
48#1809—21no—
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51#1808C2211yescharacteristic
52#1808C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×M16D411yescharacteristic
62C23×C8C2311yescharacteristic
63C2×C4⋊Q8—21no—
64C2×#125—21no—
65C2×#117D411yescharacteristic
66C2×M32—22no—
67C2×M32—22no—
68C2×M32Q814yescomplemented
69C22×C16—22no—
70C22×C16—22no—
71C22×C16Q814yescomplemented
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76#500—22no—
77#500—22no—
78#499—22no—
79#499—22no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#106—21no—
91C2×#106—21no—
92#405—22no—
93#405—22no—
94C2×#97D411yescharacteristic
95C2×#97D411yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#400—41no—
99#399—41no—
100#398—41no—
101#398—41no—
102#396—41no—
103#396—41no—
104#395—41no—
105#393—41no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#384—21no—
111#382—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116#367—22no—
117#367—22no—
118#367—22no—
119#367—22no—
120#366—22no—
121#366—22no—
122#366—22no—
123#366—22no—
124#346—22no—
125#346D414yes—
126C4×C16—22no—
127C4×C16—22no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136C22×Q8—41no—
137C23×C4C2×D411yescharacteristic
138C2×M16—22no—
139C2×M16—22no—
140C2×M16Q1614yes—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8C2×Q812yes—
149C22×C8C2×Q812yes—
150C22×C8C2411yescharacteristic
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153#125—42no—
154#125—42no—
155#117—22no—
156#117—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4D811yescharacteristic
175C2×C4⋊C4D811yescharacteristic
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4D811yescharacteristic
181C2×C22⋊C4D811yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184M32—28no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—28no—
190C2×C16C2×D412yes—
191C2×C16C2×D412yes—
192C2×C16C2×Q812yes—
193C2×C16C2×Q812yes—
194C2×C16C2×Q812yes—
195C2×C16C2×Q812yes—
196C2×C16#4112yes—
197C2×C16#4112yes—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—42no—
215#107—42no—
216#106—42no—
217#106—42no—
218#97—22no—
219#97—22no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8C2×D412yes—
223C4×C8#4112yes—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234C24C2×D811yescharacteristic
235C2×Q8—82no—
236C2×Q8—161no—
237C2×Q8—161no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4C4⋊Q812yes—
251C22×C4C2×D811yescharacteristic
252C22×C4C2×D811yescharacteristic
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C2×Q1612yes—
257C22×C4C22×D411yescharacteristic
258M16—28no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8#12112yes—
268C2×C8C4⋊Q812yes—
269C2×C8C2×Q1612yes—
270C2×C8C2×Q1612yes—
271C2×C8#13512yes—
272C2×C8C22×D411yescharacteristic
273C2×C8C22×Q811yescharacteristic
274C2×C8C22×Q811yescharacteristic
275C2×C8C2×#4111yescharacteristic
276C2×C8C2×#4111yescharacteristic
277C2×C8C2×#4111yescharacteristic
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—161no—
300C4⋊C4—161no—
301C4⋊C4—161no—
302C4⋊C4—161no—
303C4⋊C4—161no—
304C4⋊C4—161no—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—161no—
308C22⋊C4—161no—
309C22⋊C4—161no—
310C22⋊C4—161no—
311C42—21no—
312C42—21no—
313C42#12112yes—
314C42#13512yes—
315C16—22no—
316C16—22no—
317C16C22⋊Q814yes—
318C16C22⋊Q814yes—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23#50012yes—
326C23C22×D811yescharacteristic
327Q8—164no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4#46612yes—
369C2×C4#46612yes—
370C2×C4#50012yes—
371C2×C4C2×#12111yescharacteristic
372C2×C4C2×C4⋊Q811yescharacteristic
373C2×C4#53211yescharacteristic
374C2×C4C22×Q1611yescharacteristic
375C2×C4C2×#13511yescharacteristic
376C2×C4C2×#13511yescharacteristic
377C8—22no—
378C8—22no—
379C8#46714yes—
380C8C2×C22⋊Q811yescharacteristic
381C8C2×C22⋊Q811yescharacteristic
382C8#53311yescharacteristic
383C8#53311yescharacteristic
384C22—22no—
385C22—22no—
386C22—22no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22C2×#46611yescharacteristic
392C22C2×#50011yescharacteristic
393C22#316911yescharacteristic
394C22#341511yescharacteristic
395C22#341611yescharacteristic
396C22#341611yescharacteristic
397C22#341711yescharacteristic
398C4—22no—
399C4—22no—
400C4—22no—
401C4—41no—
402C4—41no—
403C4—161no—
404C4—161no—
405C4—161no—
406C4—161no—
407C4—161no—
408C4—161no—
409C4—161no—
410C4—161no—
411C4#285711yescharacteristic
412C4C2×#46711yescharacteristic
413C4#316011yescharacteristic
414C4#317011yescharacteristic
415C2—42no—
416C2#3320911yescharacteristic
417C2#3361211yescharacteristic
418C2#3361311yescharacteristic
419C2#3372911yescharacteristic
420C2#3373011yescharacteristic
421C2#3374411yescharacteristic
422C2#3374511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2255C2semidirect
5. #13282C2semidirect
6. #13281C2semidirect
7. #13168C2semidirect
8. #13167C2semidirect
9. #13166C2semidirect
10. #13165C2semidirect
11. #12908C2semidirect
68. C2×M32Q8semidirect
71. C22×C16Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#22552
C2×#21082
#332162
#332152
#132822
#132812
#131682
#131672
#131662
#131652
#129082
#128362
#128352
#128342