Finite Group Databasebeta

← #232155

Subgroups — #232155

Order 512.

397 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19751C211yesmaximal, characteristic
2#9427C211yesmaximal, characteristic
3#9427C211yesmaximal, characteristic
4#9426C211yesmaximal, characteristic
5#9422C211yesmaximal, characteristic
6#9420C211yesmaximal, characteristic
7#9420C211yesmaximal, characteristic
8#9416C211yesmaximal, characteristic
9#9416C211yesmaximal, characteristic
10#9416C211yesmaximal, characteristic
11#9416C211yesmaximal, characteristic
12#9416C211yesmaximal, characteristic
13#9416C211yesmaximal, characteristic
14#8804C211yesmaximal, characteristic
15#8803C211yesmaximal, characteristic
16#3185C2211yescharacteristic
17#3185C2211yescharacteristic
18#3185C2211yescharacteristic
19#3165C2211yescharacteristic
20#3165C2211yescharacteristic
21#3139C2211yescharacteristic
22#2987C2211yescharacteristic
23#2987C2211yescharacteristic
24#2987C2211yescharacteristic
25#2381C2211yescharacteristic
26#1723C411yescharacteristic, complemented
27#1723C411yescharacteristic, complemented
28#1718C411yescharacteristic, complemented
29#1718C411yescharacteristic, complemented
30#1716C411yescharacteristic, complemented
31#1716C411yescharacteristic, complemented
32#1707C411yescharacteristic, complemented
33#1707C411yescharacteristic, complemented
34#1558C2211yescharacteristic
35#1556C2211yescharacteristic
36#1556C2211yescharacteristic
37#1537C2211yescharacteristic
38#1537C2211yescharacteristic
39#1537C2211yescharacteristic
40#1515C2211yescharacteristic
41#1515C2211yescharacteristic
42#1515C2211yescharacteristic
43#1515C2211yescharacteristic
44#1515C2211yescharacteristic
45#1515C2211yescharacteristic
46#1455C2211yescharacteristic
47#1291C2211yescharacteristic
48#1291C2211yescharacteristic
49#1291C2211yescharacteristic
50#1291C2211yescharacteristic
51#1291C2211yescharacteristic
52#1291C2211yescharacteristic
53#1288C2211yescharacteristic
54#1288C2211yescharacteristic
55#1288C2211yescharacteristic
56#1288C2211yescharacteristic
57#1288C2211yescharacteristic
58#1288C2211yescharacteristic
59#533C2311yescharacteristic
60#533C2311yescharacteristic
61#533C2311yescharacteristic
62C2×C4⋊Q8C2311yescharacteristic
63#501—41no—
64#501—41no—
65#500C2×C411yescharacteristic
66#500C2×C411yescharacteristic
67#500C2×C411yescharacteristic
68#500C2×C411yescharacteristic
69#498C2×C411yescharacteristic
70#498C2×C411yescharacteristic
71#498C2×C411yescharacteristic
72#498C2×C411yescharacteristic
73#498C2×C411yescharacteristic
74#498C2×C411yescharacteristic
75#498C2×C411yescharacteristic
76#494C2×C411yescharacteristic
77#494C2×C411yescharacteristic
78#494C2×C411yescharacteristic
79#491—41no—
80#487C2×C411yescharacteristic
81#487C2×C411yescharacteristic
82#487C2×C411yescharacteristic
83#487C2×C411yescharacteristic
84#487C2×C411yescharacteristic
85#487C2×C411yescharacteristic
86#487C2×C411yescharacteristic
87#487C2×C411yescharacteristic
88#487C2×C411yescharacteristic
89#487C2×C411yescharacteristic
90#470—41no—
91#470—41no—
92#470—41no—
93#446—41no—
94#446—41no—
95#446—41no—
96#446—41no—
97#446—41no—
98#446—41no—
99#441—41no—
100#441—41no—
101#441—41no—
102#433—41no—
103#433—41no—
104#433—41no—
105#432C2311yescharacteristic
106#432C2311yescharacteristic
107#432C2311yescharacteristic
108#432C2311yescharacteristic
109#432C2311yescharacteristic
110#432C2311yescharacteristic
111#428—41no—
112#428—41no—
113#419—41no—
114C4×M16C2311yescharacteristic
115C4×M16C2311yescharacteristic
116C4×M16C2311yescharacteristic
117#391—41no—
118#387—41no—
119#387—41no—
120#383C2311yescharacteristic
121#383C2311yescharacteristic
122#333—41no—
123#332—41no—
124#332—41no—
125#330—41no—
126#330—41no—
127#330—41no—
128#328—41no—
129#328—41no—
130#328—41no—
131#328—41no—
132#328—41no—
133#328—41no—
134#322C2×C411yescharacteristic
135#322C2×C411yescharacteristic
136#322C2×C411yescharacteristic
137C82C2×C411yescharacteristic
138C22×Q8—21no—
139C2×Q16—41no—
140C2×Q16—41no—
141C2×Q16—41no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8C22×C411yescharacteristic
149C4⋊Q8C22×C411yescharacteristic
150C4⋊Q8C22×C411yescharacteristic
151C4⋊Q8C22×C411yescharacteristic
152C4⋊Q8C22×C411yescharacteristic
153C4⋊Q8C22×C411yescharacteristic
154C4⋊Q8C22×C411yescharacteristic
155C4⋊Q8C22×C411yescharacteristic
156#125—21no—
157#125—21no—
158#125—21no—
159C22⋊Q8—21no—
160C22⋊Q8—21no—
161C22⋊Q8—21no—
162C22⋊Q8—21no—
163C22⋊Q8—21no—
164C22⋊Q8—21no—
165C4×Q8—21no—
166C4×Q8—21no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×Q8—21no—
170C4×Q8—21no—
171#117—21no—
172#117—21no—
173#117—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—41no—
177C2×C42—41no—
178C2×C42C2411yescharacteristic
179#107—41no—
180#107—41no—
181#107—41no—
182#107—41no—
183#106—41no—
184#106—41no—
185#106—41no—
186#106—41no—
187#106—41no—
188#106—41no—
189#106—41no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—42no—
197C4⋊C8—42no—
198C4⋊C8—42no—
199#103—41no—
200#103—41no—
201#103—41no—
202#103—41no—
203#103—41no—
204#103—41no—
205#103—41no—
206#103—41no—
207#103—41no—
208#103—41no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8—21no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214C22⋊C8—21no—
215#97—21no—
216#97—21no—
217#97—21no—
218#97—21no—
219#97—21no—
220#97—21no—
221#97—21no—
222#97—21no—
223#97—21no—
224#97—41no—
225#97—41no—
226#97—41no—
227C4×C8—41no—
228C4×C8—41no—
229C4×C8—41no—
230C4×C8C22×C411yescharacteristic
231C4×C8C22×C411yescharacteristic
232C4×C8C22×C411yescharacteristic
233C4×C8C22×C411yescharacteristic
234C4×C8C22×C411yescharacteristic
235C4×C8C22×C411yescharacteristic
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—42no—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—41no—
252C2×Q8—81no—
253C2×Q8—81no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4H(5,𝔽2)11yescharacteristic
258C22×C4H(5,𝔽2)11yescharacteristic
259C22×C4H(5,𝔽2)11yescharacteristic
260Q16—82no—
261Q16—82no—
262Q16—82no—
263M16—42no—
264M16—42no—
265M16—81no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—161no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C42—41no—
307C42—41no—
308C42—41no—
309C42—161no—
310C42C23×C411yescharacteristic
311C42H(5,𝔽2)11yescharacteristic
312C42H(5,𝔽2)11yescharacteristic
313C42H(5,𝔽2)11yescharacteristic
314C23#56011yescharacteristic
315Q8—81no—
316Q8—81no—
317Q8—81no—
318Q8—81no—
319Q8—81no—
320Q8—81no—
321Q8—81no—
322Q8—81no—
323Q8—161no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4#51811yescharacteristic
344C2×C4#51811yescharacteristic
345C2×C4#51811yescharacteristic
346C2×C4#56011yescharacteristic
347C2×C4#56011yescharacteristic
348C2×C4#56011yescharacteristic
349C8—22no—
350C8—22no—
351C8—41no—
352C8—41no—
353C8—41no—
354C8—41no—
355C8—81no—
356C8—81no—
357C8—81no—
358C8—81no—
359C8—81no—
360C8—81no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22#238011yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—81no—
372C4—81no—
373C4—81no—
374C4—81no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4—161no—
380C4—161no—
381C2—41no—
382C2#2336011yescharacteristic
383C2#2336011yescharacteristic
384C2#2336211yescharacteristic
subgroupG/H
center, socleC22#2380
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23#560
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
26. #1723C4semidirect
27. #1723C4semidirect
28. #1718C4semidirect
29. #1718C4semidirect
30. #1716C4semidirect
31. #1716C4semidirect
32. #1707C4semidirect
33. #1707C4semidirect

Maximal subgroups

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

subgroupindex
#197512
#94272
#94262
#94222
#94202
#94162
#88042
#88032