Finite Group Databasebeta

← #194600

Subgroups — #194600

Order 512.

1250 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#489C211yesmaximal, characteristic
2#34366C211yesmaximal, characteristic
3#21841C211yesmaximal, characteristic, complemented
4#21811C211yesmaximal, characteristic, complemented
5C2×#1533C211yesmaximal, characteristic
6#8999C218yesmaximal, complemented
7#8036C211yesmaximal, characteristic, complemented
8#8036C211yesmaximal, characteristic, complemented
9C22×C4⋊Q8C2211yescharacteristic
10C22×#124C2211yescharacteristic
11C2×#489—22no—
12C2×#489—24no—
13C2×#489C418yescomplemented
14C2×#489C2214yes—
15C2×#432C2211yescharacteristic
16C2×#432C2211yescharacteristic
17C22×#103—21no—
18C22×#102—21no—
19C22×#97C2211yescharacteristic
20#2379C2214yes—
21#2372—22no—
22#2342—22no—
23#2342—22no—
24#2337—22no—
25#2337—22no—
26#2310—21no—
27#2308—21no—
28#2283C2211yescharacteristic
29#2283C2211yescharacteristic
30#1993C2214yescomplemented
31#1988—22no—
32#1981—22no—
33#1960C2214yescomplemented
34#1824—22no—
35#1787—22no—
36#1533—24no—
37C2×#329C2214yes—
38#1295C2214yescomplemented
39#1295C2214yescomplemented
40C23×Q8—21no—
41C23×Q8D411yescharacteristic
42C23×D4D411yescharacteristic
43C23×C8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8C2×C414yescomplemented
51C2×C4⋊Q8C2312yes—
52C2×#124—21no—
53C2×#124—21no—
54C2×#124—22no—
55C2×#124—22no—
56C2×#124C2×C414yes—
57C2×#124D412yes—
58C2×#124D414yes—
59C2×#124C2312yes—
60C2×#117—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—22no—
66C22×C42C2311yescharacteristic
67#489—216no—
68#489—28no—
69#489C2×C4116yescomplemented
70#432—22no—
71#432—22no—
72#432—22no—
73#432—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×#103—28no—
77C2×#103—24no—
78C2×#103—28no—
79C2×#102—28no—
80C2×#102—24no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×#97—21no—
84C2×#97—22no—
85C2×#97C2×C414yescomplemented
86C2×#97C2312yes—
87C2×#97C2312yes—
88C2×#97C2312yes—
89#391—24no—
90#391—24no—
91#388—24no—
92#388—24no—
93#388—24no—
94#388—24no—
95#386—24no—
96#386—24no—
97#385—24no—
98#385—24no—
99#383—21no—
100#383—21no—
101#382C2312yes—
102#382C2312yes—
103#381—22no—
104#381—21no—
105#381—21no—
106#380D412yes—
107#380D412yes—
108#379—22no—
109C4×C4⋊C4D412yes—
110C4×C4⋊C4D412yes—
111C4×C22⋊C4—24no—
112C4×C22⋊C4—24no—
113C4×C22⋊C4—21no—
114C4×C22⋊C4—21no—
115#376—21no—
116#376—21no—
117#340—28no—
118#329—28no—
119C25—21no—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—24no—
123C22×Q8—22no—
124C22×Q8—44no—
125C22×Q8—41no—
126C22×Q8—42no—
127C22×Q8—84no—
128C22×Q8C22⋊C414yes—
129C22×Q8C2×D412yes—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—44no—
133C22×D4C22⋊C414yes—
134C22×D4C2×D412yes—
135C23×C4—21no—
136C23×C4—24no—
137C23×C4—41no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C22×C8—24no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—44no—
146C4⋊Q8—24no—
147C4⋊Q8—24no—
148C4⋊Q8—24no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8C22×C414yes—
156#124—24no—
157#124—24no—
158#124—24no—
159#124—24no—
160#124—24no—
161#124—22no—
162#124—22no—
163#124C22⋊C418yes—
164#124C22×C414yes—
165#124C2×D414yes—
166#117—84no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4C2×D414yes—
177C2×C4⋊C4C2×D414yes—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—84no—
192C2×C42—24no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42C22×C412yes—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D412yes—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42#4112yes—
201C2×C42#4112yes—
202C2×C42C2411yescharacteristic
203C4⋊C8—42no—
204C4⋊C8—42no—
205#103—216no—
206#103—88no—
207#102—216no—
208C22⋊C8—42no—
209C22⋊C8—42no—
210#97—24no—
211#97—22no—
212#97—22no—
213#97—22no—
214#97—22no—
215#97—22no—
216#97C22×C414yes—
217#95—24no—
218#95—24no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227C24—22no—
228C24—24no—
229C24—44no—
230C24—82no—
231C24—82no—
232C24#12011yescharacteristic
233C2×Q8—22no—
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—48no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—48no—
240C2×Q8—42no—
241C2×Q8—44no—
242C2×Q8—84no—
243C2×Q8—82no—
244C2×Q8—84no—
245C2×Q8—88no—
246C2×Q8—88no—
247C2×Q8C2×C22⋊C414yes—
248C2×D4—22no—
249C2×D4—24no—
250C2×D4—48no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—84no—
254C2×D4—82no—
255C2×D4C2×C22⋊C414yes—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—84no—
267C22×C4—84no—
268C22×C4—84no—
269C22×C4—84no—
270C22×C4C2×C22⋊C412yes—
271C22×C4C4×D412yes—
272C22×C4C4×D412yes—
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12011yescharacteristic
276C22×C4#12112yes—
277C22×C4#12112yes—
278C22×C4#12312yes—
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C2×C8—24no—
283C2×C8—48no—
284C2×C8—44no—
285C2×C8—44no—
286C2×C8—44no—
287C2×C8—42no—
288C2×C8—42no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—84no—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—24no—
301C22⋊C4—24no—
302C22⋊C4—28no—
303C22⋊C4—28no—
304C22⋊C4—44no—
305C22⋊C4—44no—
306C22⋊C4—84no—
307C22⋊C4—84no—
308C22⋊C4—84no—
309C42—82no—
310C42—82no—
311C42—84no—
312C42C2×C22⋊C412yes—
313C42C4×D412yes—
314C42C4×D412yes—
315C42#12112yes—
316C42#12112yes—
317C42#12312yes—
318C42C23×C411yescharacteristic
319C42C22×D411yescharacteristic
320C42C2×#4111yescharacteristic
321C42C2×#4111yescharacteristic
322C23—22no—
323C23—22no—
324C23—24no—
325C23—42no—
326C23—42no—
327C23—44no—
328C23—44no—
329C23—48no—
330C23—84no—
331C23—84no—
332C23—168no—
333C23#38612yes—
334C23#45514yes—
335C23C2×#12011yescharacteristic
336Q8—44no—
337Q8—48no—
338Q8—82no—
339Q8—84no—
340Q8—84no—
341Q8—88no—
342D4—44no—
343D4—82no—
344D4—84no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—48no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—88no—
369C2×C4—84no—
370C2×C4—168no—
371C2×C4#38612yes—
372C2×C4#38612yes—
373C2×C4#38612yes—
374C2×C4C22×C22⋊C411yescharacteristic
375C2×C4C2×C4×D411yescharacteristic
376C2×C4C2×C4×D411yescharacteristic
377C2×C4C2×#12111yescharacteristic
378C2×C4C2×#12111yescharacteristic
379C2×C4C2×#12311yescharacteristic
380C8—44no—
381C8—82no—
382C8—82no—
383C22—28no—
384C22—22no—
385C22—22no—
386C22—44no—
387C22—44no—
388C22—44no—
389C22—84no—
390C22—84no—
391C22—168no—
392C22#188514yes—
393C22C2×#38611yescharacteristic
394C22C2×#45512yes—
395C22#302112yes—
396C22#302212yes—
397C4—21no—
398C4—21no—
399C4—22no—
400C4—21no—
401C4—21no—
402C4—22no—
403C4—21no—
404C4—21no—
405C4—22no—
406C4—44no—
407C4—84no—
408C4—84no—
409C4—84no—
410C2—24no—
411C2—84no—
412C2#1905514yes—
413C2#2094411yescharacteristic
414C2#2094511yescharacteristic
415C2C2×#188511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21841C2semidirect
4. #21811C2semidirect
6. #8999C2semidirect
7. #8036C2semidirect
8. #8036C2semidirect
13. C2×#489C4semidirect
30. #1993C22semidirect
33. #1960C22semidirect
38. #1295C22semidirect
39. #1295C22semidirect
50. C2×C4⋊Q8C2×C4semidirect
69. #489C2×C4semidirect
85. C2×#97C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4892
#343662
#218412
#218112
C2×#15332
#89992
#80362