Finite Group Databasebeta

← #110509

Subgroups — #110509

Order 512.

594 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21306C211yesmaximal, characteristic
2#17715C211yesmaximal, characteristic
3#17385C211yesmaximal, characteristic
4#17327C211yesmaximal, characteristic
5#8105C211yesmaximal, characteristic
6#8038C211yesmaximal, characteristic
7#7888C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2285C2211yescharacteristic
15#1986—21no—
16#1986—21no—
17#1986—22no—
18#1733C414yes—
19#1552—22no—
20#1549C412yescomplemented
21#1546C412yescomplemented
22#1544—21no—
23#1544—21no—
24#1472—22no—
25#1470C414yescomplemented
26#1461—22no—
27#1453—22no—
28#1451—21no—
29#1451—21no—
30C8×M16—21no—
31C8×M16—21no—
32#1305—22no—
33#1296—22no—
34#1278—22no—
35C22×M16C2×C412yes—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125C2×C412yes—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×#117C2×C412yes—
52C2×#117C2×C412yes—
53C22×C42C2311yescharacteristic
54#432—21no—
55#432—21no—
56#432—21no—
57#432C2×C412yes—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431C2×C412yes—
67#431C2×C412yes—
68#431D411yescharacteristic
69#431D411yescharacteristic
70#431D412yes—
71#423—21no—
72#423C2×C412yes—
73C2×C4⋊C8—21no—
74C2×C4⋊C8D412yes—
75#407—21no—
76#407D412yes—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C4×M16—21no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16—21no—
83C4×M16D411yescharacteristic
84C4×M16D411yescharacteristic
85C4×M16D412yes—
86C2×#97—21no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90#384—21no—
91#382—21no—
92#380—21no—
93#380—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96#339—24no—
97#339—24no—
98#332—22no—
99#332—22no—
100#332—22no—
101#332—22no—
102#324—28no—
103#322—22no—
104#322—22no—
105#322—22no—
106C82—22no—
107C22×Q8—21no—
108C22×Q8—21no—
109C23×C4C22×C411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—24no—
115C2×M16—22no—
116C2×M16—42no—
117C2×M16C22⋊C412yes—
118C2×M16C22⋊C412yes—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—41no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130#125—22no—
131#125—22no—
132#125—22no—
133#125—22no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C22⋊Q8—41no—
141C22⋊Q8—41no—
142C4×Q8—41no—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148C4×Q8—41no—
149C4×Q8—41no—
150#117—22no—
151#117—22no—
152#117—22no—
153#117—22no—
154#117—44no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4C4212yes—
163C2×C4⋊C4C4212yes—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—41no—
187C4⋊C8C22⋊C412yes—
188C4⋊C8C22⋊C412yes—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—21no—
192C22⋊C8—21no—
193C22⋊C8—22no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197C22⋊C8C22⋊C414yes—
198#97—22no—
199#97—22no—
200#97—21no—
201#97—21no—
202#97—41no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—21no—
206C4×C8—21no—
207C4×C8—22no—
208C4×C8—21no—
209C4×C8—21no—
210C4×C8—22no—
211C4×C8—41no—
212C4×C8—41no—
213#95—41no—
214#95—41no—
215C24C4×D411yescharacteristic
216C2×Q8—44no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—82no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4C2×C4211yescharacteristic
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4#11711yescharacteristic
233C22×C4#11711yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4#12311yescharacteristic
243C22×C4#12411yescharacteristic
244C22×C4#12411yescharacteristic
245C22×C4#12611yescharacteristic
246C22×C4#12711yescharacteristic
247M16—22no—
248M16—22no—
249M16—22no—
250M16—44no—
251M16—44no—
252M16—82no—
253M16#10114yes—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—24no—
258C2×C8—22no—
259C2×C8—44no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—82no—
263C2×C8—82no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—41no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—82no—
282C42—42no—
283C42—42no—
284C42—44no—
285C42C2×C22⋊C411yescharacteristic
286C42#11711yescharacteristic
287C42C4×D411yescharacteristic
288C42C4×D411yescharacteristic
289C42C4×D411yescharacteristic
290C42C4×D411yescharacteristic
291C42C4×D411yescharacteristic
292C42C4×D411yescharacteristic
293C42#12011yescharacteristic
294C42#12111yescharacteristic
295C42#12111yescharacteristic
296C42#12111yescharacteristic
297C42#12311yescharacteristic
298C42#12311yescharacteristic
299C42#12411yescharacteristic
300C42#12611yescharacteristic
301C23—21no—
302C23—21no—
303C23—41no—
304C23—41no—
305C23C4×C22⋊C411yescharacteristic
306C23#38811yescharacteristic
307C23#39011yescharacteristic
308Q8—82no—
309Q8—82no—
310Q8—82no—
311Q8—82no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4C4×C22⋊C411yescharacteristic
338C2×C4C4×C22⋊C411yescharacteristic
339C2×C4#38611yescharacteristic
340C2×C4#38611yescharacteristic
341C2×C4#38611yescharacteristic
342C2×C4#38811yescharacteristic
343C2×C4#38811yescharacteristic
344C2×C4#38811yescharacteristic
345C2×C4#38811yescharacteristic
346C2×C4#38811yescharacteristic
347C2×C4#39011yescharacteristic
348C2×C4#39011yescharacteristic
349C2×C4C2×#10112yes—
350C2×C4#41312yes—
351C8—24no—
352C8—44no—
353C8—42no—
354C8—42no—
355C8—42no—
356C8—82no—
357C8—82no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—81no—
367C22—81no—
368C22#143711yescharacteristic
369C22#196611yescharacteristic
370C22#196711yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—82no—
380C4—82no—
381C4—162no—
382C4C4×#10112yes—
383C4#188112yes—
384C4#190112yes—
385C4#197612yes—
386C2—21no—
387C2—21no—
388C2—41no—
389C2—41no—
390C2#1094411yescharacteristic
391C2#1095611yescharacteristic
392C2#1096311yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ 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
20. #1549C4semidirect
21. #1546C4semidirect
25. #1470C4semidirect

Maximal subgroups

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

subgroupindex
#213062
#177152
#173852
#173272
#81052
#80382
#78882