Finite Group Databasebeta

← #232156

Subgroups — #232156

Order 512.

407 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19752C211yesmaximal, characteristic
2#9430C211yesmaximal, characteristic
3#9427C211yesmaximal, characteristic, complemented
4#9426C211yesmaximal, characteristic, complemented
5#9422C211yesmaximal, characteristic, complemented
6#9420C211yesmaximal, characteristic, complemented
7#9419C211yesmaximal, characteristic
8#9418C211yesmaximal, characteristic
9#9418C211yesmaximal, characteristic
10#9416C211yesmaximal, characteristic, complemented
11#9416C211yesmaximal, characteristic, complemented
12#9415C211yesmaximal, characteristic
13#9415C211yesmaximal, characteristic
14#8803C211yesmaximal, characteristic, complemented
15#8803C211yesmaximal, characteristic, complemented
16#3185C2211yescharacteristic
17#3182C2211yescharacteristic
18#3182C2211yescharacteristic
19#3165C2211yescharacteristic
20#3141C2211yescharacteristic
21#3139C2211yescharacteristic
22#2987C2211yescharacteristic
23#2987C2211yescharacteristic
24#2980C2211yescharacteristic
25#2381C2211yescharacteristic
26#1723C411yescharacteristic, complemented
27#1720C411yescharacteristic, complemented
28#1719C411yescharacteristic, complemented
29#1718C411yescharacteristic, complemented
30#1718C411yescharacteristic, complemented
31#1714C411yescharacteristic, complemented
32#1710C411yescharacteristic, complemented
33#1708C411yescharacteristic, complemented
34#1558C2211yescharacteristic
35#1556C2211yescharacteristic
36#1556C2211yescharacteristic
37#1537C2211yescharacteristic
38#1537C2211yescharacteristic
39#1536C2211yescharacteristic
40#1515C2211yescharacteristic
41#1515C2211yescharacteristic
42#1515C2211yescharacteristic
43#1515C2211yescharacteristic
44#1514C2211yescharacteristic
45#1514C2211yescharacteristic
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#532C2311yescharacteristic
62C2×C4⋊Q8C2311yescharacteristic
63#501—41no—
64#500C2×C411yescharacteristic
65#500C2×C411yescharacteristic
66#500C2×C411yescharacteristic
67#500C2×C411yescharacteristic
68#498C2×C411yescharacteristic
69#498C2×C411yescharacteristic
70#498C2×C411yescharacteristic
71#494C2×C411yescharacteristic
72#492C2×C411yescharacteristic
73#492C2×C411yescharacteristic
74#491—41no—
75#489—41no—
76#487C2×C411yescharacteristic
77#487C2×C411yescharacteristic
78#487C2×C411yescharacteristic
79#487C2×C411yescharacteristic
80#487C2×C411yescharacteristic
81#487C2×C411yescharacteristic
82#487C2×C411yescharacteristic
83#487C2×C411yescharacteristic
84#487C2×C411yescharacteristic
85#487C2×C411yescharacteristic
86#486C2×C411yescharacteristic
87#486C2×C411yescharacteristic
88#486C2×C411yescharacteristic
89#486C2×C411yescharacteristic
90#470—41no—
91#468—41no—
92#468—41no—
93#446—41no—
94#446—41no—
95#446—41no—
96#446—41no—
97#441—41no—
98#440—41no—
99#440—41no—
100#435—41no—
101#435—41no—
102#433—41no—
103#433—41no—
104#432C2311yescharacteristic
105#432C2311yescharacteristic
106#432C2311yescharacteristic
107#432C2311yescharacteristic
108#432C2311yescharacteristic
109#432C2311yescharacteristic
110#428—41no—
111#419—41no—
112#417—41no—
113#408—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#329—41no—
128#328—41no—
129#328—41no—
130#328—41no—
131#328—41no—
132#327—41no—
133#327—41no—
134#322C2×C411yescharacteristic
135#322C2×C411yescharacteristic
136#322C2×C411yescharacteristic
137C82C2×C411yescharacteristic
138C2×#41—21no—
139C22×Q8—21no—
140C2×Q16—41no—
141C2×SD16—41no—
142C2×SD16—41no—
143C2×M16—21no—
144C2×M16—21no—
145C2×M16—21no—
146C2×M16—41no—
147C22×C8—41no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8C22×C411yescharacteristic
152C4⋊Q8C22×C411yescharacteristic
153C4⋊Q8C22×C411yescharacteristic
154C4⋊Q8C22×C411yescharacteristic
155C4⋊Q8C22×C411yescharacteristic
156C4⋊Q8C22×C411yescharacteristic
157C4⋊Q8C22×C411yescharacteristic
158#127C22×C411yescharacteristic
159#125—21no—
160#125—21no—
161#124—21no—
162C22⋊Q8—21no—
163C22⋊Q8—21no—
164C22⋊Q8—21no—
165C22⋊Q8—21no—
166#121—21no—
167#121—21no—
168C4×Q8—21no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×Q8—21no—
172C4×D4—21no—
173C4×D4—21no—
174#117—21no—
175#117—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—41no—
179C2×C42—41no—
180C2×C42C2411yescharacteristic
181#107—41no—
182#107—41no—
183#107—41no—
184#107—41no—
185#106—41no—
186#106—41no—
187#106—41no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—42no—
195C4⋊C8—42no—
196#103—41no—
197#103—41no—
198#103—41no—
199#103—41no—
200#103—41no—
201#103—41no—
202#103—41no—
203#103—41no—
204#103—41no—
205#103—41no—
206#102—41no—
207#102—41no—
208#102—41no—
209#102—41no—
210C22⋊C8—21no—
211C22⋊C8—21no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214C22⋊C8—21no—
215C22⋊C8—21no—
216C22⋊C8—42no—
217#97—21no—
218#97—21no—
219#97—21no—
220#97—21no—
221#97—21no—
222#97—21no—
223#97—21no—
224#97—21no—
225#97—21no—
226#97—41no—
227#97—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—
243#41—81no—
244#41—81no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—41no—
253C2×Q8—81no—
254C2×Q8—81no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4H(5,𝔽2)11yescharacteristic
264C22×C4H(5,𝔽2)11yescharacteristic
265C22×C4H(5,𝔽2)11yescharacteristic
266Q16—82no—
267SD16—82no—
268SD16—82no—
269M16—42no—
270M16—81no—
271M16—81no—
272M16—161no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—161no—
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—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—161no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C42—41no—
312C42—41no—
313C42—161no—
314C42C23×C411yescharacteristic
315C42H(5,𝔽2)11yescharacteristic
316C42H(5,𝔽2)11yescharacteristic
317C42H(5,𝔽2)11yescharacteristic
318C23—41no—
319C23#56011yescharacteristic
320Q8—81no—
321Q8—81no—
322Q8—81no—
323Q8—81no—
324Q8—81no—
325Q8—81no—
326Q8—81no—
327Q8—161no—
328D4—81no—
329D4—81no—
330D4—81no—
331D4—161no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—81no—
343C2×C4—81no—
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#51811yescharacteristic
353C2×C4#51811yescharacteristic
354C2×C4#51811yescharacteristic
355C2×C4#56011yescharacteristic
356C2×C4#56011yescharacteristic
357C2×C4#56011yescharacteristic
358C8—22no—
359C8—41no—
360C8—41no—
361C8—41no—
362C8—41no—
363C8—41no—
364C8—81no—
365C8—81no—
366C8—81no—
367C8—81no—
368C8—81no—
369C8—81no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22#238011yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—161no—
391C4—161no—
392C2—41no—
393C2—161no—
394C2#2336011yescharacteristic
395C2#2336211yescharacteristic
396C2#2336211yescharacteristic
subgroupG/H
center, socleC22#2380
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#532C23
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
3. #9427C2semidirect
4. #9426C2semidirect
5. #9422C2semidirect
6. #9420C2semidirect
10. #9416C2semidirect
11. #9416C2semidirect
14. #8803C2semidirect
15. #8803C2semidirect
26. #1723C4semidirect
27. #1720C4semidirect
28. #1719C4semidirect
29. #1718C4semidirect
30. #1718C4semidirect
31. #1714C4semidirect
32. #1710C4semidirect
33. #1708C4semidirect

Maximal subgroups

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

subgroupindex
#197522
#94302
#94272
#94262
#94222
#94202
#94192
#94182
#94162
#94152
#88032