Finite Group Databasebeta

← #110679

Subgroups — #110679

Order 512.

604 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#21845C211yesmaximal, characteristic
2#21389C211yesmaximal, characteristic
3#17562C211yesmaximal, characteristic
4#17324C211yesmaximal, characteristic
5#8052C211yesmaximal, characteristic
6#7980C211yesmaximal, characteristic
7#7940C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2283C2211yescharacteristic
13#2283C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1995C414yescomplemented
16#1988C412yescomplemented
17#1986—22no—
18#1981C412yescomplemented
19#1873—22no—
20#1826—22no—
21C4×#103—21no—
22C4×#103—21no—
23#1515C414yescomplemented
24#1511—22no—
25#1503—22no—
26#1454—22no—
27#1453—22no—
28#1451—21no—
29C8×M16—21no—
30#1297—22no—
31#1289—22no—
32#1284—22no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533C2×C412yes—
40#533C2×C412yes—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#125—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×#117—21no—
49C2×#117—21no—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432C2×C412yes—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431D412yes—
63#423—21no—
64#423C2×C412yes—
65C2×C4⋊C8C2×C412yes—
66C2×#103—22no—
67C2×#103—22no—
68#407D412yes—
69#407D412yes—
70C2×C22⋊C8D412yes—
71C4×M16—21no—
72C4×M16D412yes—
73C2×C4×C8D411yescharacteristic
74C2×C4×C8D411yescharacteristic
75#383—21no—
76#383C2×C412yes—
77#382—21no—
78#382—21no—
79#381—21no—
80#381—21no—
81#380—21no—
82#380—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4C2×C412yes—
87C4×C22⋊C4—21no—
88C4×C22⋊C4—21no—
89C4×C22⋊C4—21no—
90C4×C22⋊C4—21no—
91#376—21no—
92#376—21no—
93#376C2×C412yes—
94C43—21no—
95#340—22no—
96#340—22no—
97#339—28no—
98#328—28no—
99#322—22no—
100#322—24no—
101C82—22no—
102C22×Q8—21no—
103C22×Q8—21no—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C2×M16—22no—
108C2×M16C22⋊C414yes—
109C22×C8C22⋊C412yes—
110C22×C8C22⋊C412yes—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8C4212yes—
118C4⋊Q8C4212yes—
119#125—22no—
120#125—22no—
121#125—41no—
122#125—41no—
123C22⋊Q8—22no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C22⋊Q8—41no—
128C22⋊Q8—41no—
129C22⋊Q8—41no—
130C22⋊Q8—41no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—41no—
138C4×Q8—41no—
139#117—22no—
140#117—22no—
141#117—44no—
142#117—41no—
143#117—41no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—41no—
160C2×C42—41no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42C22×C411yescharacteristic
164C2×C42C22×C411yescharacteristic
165C2×C42C22×C411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181#103—44no—
182#103—44no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8C22⋊C414yes—
189#97—22no—
190#97—22no—
191C4×C8—21no—
192C4×C8—22no—
193C4×C8—22no—
194C4×C8—21no—
195C4×C8—21no—
196C4×C8—21no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—41no—
204#95—41no—
205#95—41no—
206#95—41no—
207C24#12711yescharacteristic
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—82no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4C2×C22⋊C411yescharacteristic
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4C4×D411yescharacteristic
228C22×C4C4×D411yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12411yescharacteristic
239C22×C4#12411yescharacteristic
240C22×C4#12411yescharacteristic
241M16—44no—
242M16—44no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—24no—
246C2×C8—24no—
247C2×C8—42no—
248C2×C8—42no—
249C4⋊C4—24no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C22⋊C4—44no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—82no—
269C22⋊C4—82no—
270C42—44no—
271C42—44no—
272C42—42no—
273C42—42no—
274C42—82no—
275C42—82no—
276C42—82no—
277C42C2×C4211yescharacteristic
278C42#11711yescharacteristic
279C42#11711yescharacteristic
280C42#11711yescharacteristic
281C42C4×D411yescharacteristic
282C42C4×D411yescharacteristic
283C42C4×D411yescharacteristic
284C42C4×D411yescharacteristic
285C42C4×D411yescharacteristic
286C42C4×D411yescharacteristic
287C42#12011yescharacteristic
288C42#12311yescharacteristic
289C42#12311yescharacteristic
290C42#12311yescharacteristic
291C42#12611yescharacteristic
292C42#12611yescharacteristic
293C23—22no—
294C23—41no—
295C23—41no—
296C23#39011yescharacteristic
297C23#39011yescharacteristic
298C23#39011yescharacteristic
299Q8—42no—
300Q8—42no—
301Q8—82no—
302Q8—82no—
303Q8—82no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4C4×C22⋊C411yescharacteristic
336C2×C4C4×C22⋊C411yescharacteristic
337C2×C4C4×C22⋊C411yescharacteristic
338C2×C4#38611yescharacteristic
339C2×C4#38611yescharacteristic
340C2×C4#38611yescharacteristic
341C2×C4#38811yescharacteristic
342C2×C4#38811yescharacteristic
343C2×C4#38811yescharacteristic
344C2×C4#38811yescharacteristic
345C2×C4#38811yescharacteristic
346C2×C4#38811yescharacteristic
347C8—44no—
348C8—42no—
349C8—42no—
350C8—44no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—82no—
360C22#143711yescharacteristic
361C22#197612yes—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—84no—
373C4—82no—
374C4—82no—
375C4—162no—
376C4—162no—
377C4#188412yes—
378C4#190912yes—
379C2—22no—
380C2—41no—
381C2—41no—
382C2#1094411yescharacteristic
383C2#1094711yescharacteristic
384C2#1095911yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radicalC24#127
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #1995C4semidirect
16. #1988C4semidirect
18. #1981C4semidirect
23. #1515C4semidirect

Maximal subgroups

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

subgroupindex
#218452
#213892
#175622
#173242
#80522
#79802
#79402