Finite Group Databasebeta

← #110413

Subgroups — #110413

Order 512.

655 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#21303C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic
3#17365C211yesmaximal, characteristic
4#17334C211yesmaximal, characteristic
5#8128C211yesmaximal, characteristic
6#8119C211yesmaximal, characteristic
7#7920C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#1985—22no—
16#1984—21no—
17#1984—21no—
18#1821C412yes—
19#1801—22no—
20#1784C412yes—
21C4×#101—22no—
22#1728C414yes—
23#1467—22no—
24#1463C414yes—
25#1461—22no—
26#1455—22no—
27#1453—22no—
28#1452—21no—
29#1450—21no—
30#1309—22no—
31#1307—22no—
32#1281—22no—
33C23×Q8C2×C411yescharacteristic
34C22×M16C2×C412yes—
35C22×M16C2×C412yes—
36C2×C4×Q8—21no—
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C412yes—
50C22×C42C2×C412yes—
51C22×C42C2311yescharacteristic
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432D412yes—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431D412yes—
68#423—21no—
69#423—21no—
70#423—21no—
71#423—21no—
72#423—21no—
73#423—21no—
74#423—21no—
75#423D412yes—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×#101—24no—
79#407C2×C412yes—
80#407C2×C412yes—
81#407C2×C412yes—
82#407C2×C412yes—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16—21no—
86C4×M16—21no—
87C4×M16—21no—
88C4×M16—21no—
89C4×M16—21no—
90C4×M16D412yes—
91C2×#97D412yes—
92C2×C4×C8D411yescharacteristic
93C2×C4×C8D411yescharacteristic
94#343—24no—
95#336—24no—
96#336—24no—
97#324—28no—
98#322—22no—
99#322—24no—
100C82—22no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—44no—
108C23×C4C4212yes—
109C23×C4C4212yes—
110C23×C4C22×C411yescharacteristic
111C23×C4C22×C411yescharacteristic
112C23×C4C22×C411yescharacteristic
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—24no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—42no—
122C2×M16—44no—
123C2×M16C22⋊C414yes—
124C22×C8—22no—
125C22×C8C22⋊C412yes—
126C22×C8C22⋊C412yes—
127C4×Q8—41no—
128C4×Q8—41no—
129C4×Q8—41no—
130C4×Q8—41no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—41no—
134C4×Q8—41no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—41no—
138C4×Q8—41no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8—41no—
183C4⋊C8—41no—
184C4⋊C8—41no—
185#101—48no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8C22⋊C414yes—
194#97—22no—
195#97—22no—
196#97—22no—
197#97—22no—
198#97—41no—
199#97—41no—
200C4×C8—21no—
201C4×C8—22no—
202C4×C8—22no—
203C4×C8—21no—
204C4×C8—21no—
205C4×C8—21no—
206C4×C8—41no—
207C4×C8—41no—
208C24C2×C4211yescharacteristic
209C2×Q8—24no—
210C2×Q8—24no—
211C2×Q8—24no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—44no—
216C2×Q8—44no—
217C2×Q8—44no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—41no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—44no—
231C22×C4—42no—
232C22×C4—81no—
233C22×C4—82no—
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4#11711yescharacteristic
238C22×C4#11711yescharacteristic
239C22×C4#11711yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4C4×D411yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4C4×D411yescharacteristic
250C22×C4C4×D411yescharacteristic
251C22×C4C4×D411yescharacteristic
252M16—44no—
253M16—44no—
254M16—44no—
255M16—44no—
256M16—82no—
257M16—82no—
258M16—82no—
259M16—82no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—44no—
271C2×C8#10114yescomplemented
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C42—42no—
281C42—42no—
282C42—42no—
283C42—42no—
284C42#12011yescharacteristic
285C42#12111yescharacteristic
286C42#12111yescharacteristic
287C42#12111yescharacteristic
288C42#12111yescharacteristic
289C42#12111yescharacteristic
290C42#12111yescharacteristic
291C42#12311yescharacteristic
292C42#12311yescharacteristic
293C42#12311yescharacteristic
294C42#12411yescharacteristic
295C42#12411yescharacteristic
296C42#12411yescharacteristic
297C42#12611yescharacteristic
298C42#12611yescharacteristic
299C42#12711yescharacteristic
300C23—22no—
301C23—41no—
302C23—41no—
303C23C4×C22⋊C411yescharacteristic
304C23C4×C22⋊C411yescharacteristic
305C23C4×C22⋊C411yescharacteristic
306Q8—44no—
307Q8—44no—
308Q8—44no—
309Q8—44no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—41no—
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—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4#38611yescharacteristic
341C2×C4#38611yescharacteristic
342C2×C4#38611yescharacteristic
343C2×C4#38811yescharacteristic
344C2×C4#38811yescharacteristic
345C2×C4#38811yescharacteristic
346C2×C4#38811yescharacteristic
347C2×C4#38811yescharacteristic
348C2×C4#38811yescharacteristic
349C2×C4#39011yescharacteristic
350C2×C4#39011yescharacteristic
351C2×C4#39011yescharacteristic
352C2×C4C2×#10112yes—
353C2×C4#41312yes—
354C8—24no—
355C8—44no—
356C8—44no—
357C8—42no—
358C8—82no—
359C8—82no—
360C8—82no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—44no—
370C22#143711yescharacteristic
371C22C4×#10112yes—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—44no—
382C4—82no—
383C4#189112yes—
384C4#190112yes—
385C4#196611yescharacteristic
386C4#196711yescharacteristic
387C2—22no—
388C2—41no—
389C2—41no—
390C2#1093711yescharacteristic
391C2#1094111yescharacteristic
392C2#1096311yescharacteristic
subgroupG/H
centerC2×C4#386
commutator, prime radicalC24C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
LCSG ▷ C24 ▷ 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
271. C2×C8#101semidirect

Maximal subgroups

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

subgroupindex
#213032
#207272
#173652
#173342
#81282
#81192
#79202