Finite Group Databasebeta

← #203101

Subgroups — #203101

Order 512.

612 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17817C211yesmaximal, characteristic
2C2×#1544C211yesmaximal, characteristic
3#17491C211yesmaximal, characteristic
4#17486C211yesmaximal, characteristic
5#17326C211yesmaximal, characteristic
6#9587C214yesmaximal, complemented
7#9585C214yesmaximal, complemented
8#8065C211yesmaximal, characteristic
9#7884C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#423C2211yescharacteristic
17#1574C2214yes—
18#1573—22no—
19#1567—21no—
20#1566—21no—
21#1544C414yes—
22#1544C2212yes—
23#1544C2212yes—
24C2×#332C412yes—
25C2×#332C412yes—
26#1497—22no—
27#1493C2214yes—
28#1487—21no—
29#1487—21no—
30#1487—21no—
31#1487—21no—
32#1485C2214yes—
33#1481—22no—
34#1454—22no—
35#1452C2212yes—
36#1451—21no—
37#1450C2212yes—
38C8×M16—21no—
39#1299C2212yes—
40#1299C2212yes—
41#1278C2214yes—
42C22×M16—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533C2×C412yes—
50#533C2312yes—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#125—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×#117—21no—
59C22×C42C2311yescharacteristic
60#432—21no—
61#432—21no—
62#432—21no—
63#432D412yes—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431C2×C412yes—
74#431C2×C412yes—
75#431D411yescharacteristic
76#431D411yescharacteristic
77#431D412yes—
78#431C2311yescharacteristic
79#431C2311yescharacteristic
80#431C2312yes—
81#431C2312yes—
82#423—21no—
83#423—21no—
84#423—21no—
85#423—21no—
86#423D411yescharacteristic
87#423D411yescharacteristic
88#423C2312yes—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8C2×C412yes—
93C2×C4⋊C8C2×C412yes—
94#407—21no—
95#407—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C4×M16—21no—
100C4×M16C2312yes—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8C2312yes—
104#335—24no—
105#334—24no—
106#332C2×C414yes—
107#332C2×C414yes—
108#332C2×C414yes—
109#332C2×C414yes—
110Q8⋊C8—24no—
111Q8⋊C8—24no—
112Q8⋊C8—24no—
113Q8⋊C8—24no—
114#322—22no—
115#322—22no—
116#322—22no—
117C82—22no—
118C22×Q8—21no—
119C22×Q8—21no—
120C23×C4#4111yescharacteristic
121C23×C4#4111yescharacteristic
122C23×C4#4111yescharacteristic
123C2×M16—21no—
124C2×M16—21no—
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—41no—
128C2×M16—82no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—41no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8C22×C412yes—
140C4⋊Q8C22×C412yes—
141#125—22no—
142#125—22no—
143#125—41no—
144#125—41no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8—41no—
150C22⋊Q8—41no—
151C22⋊Q8—41no—
152C22⋊Q8—41no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—41no—
158C4×Q8—41no—
159C4×Q8—41no—
160C4×Q8—41no—
161#117—22no—
162#117—22no—
163#117—41no—
164#117—41no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C42—41no—
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42C2411yescharacteristic
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—41no—
195C4⋊C8—41no—
196C4⋊C8—41no—
197C4⋊C8—41no—
198C4⋊C8C22×C412yes—
199C4⋊C8C22×C412yes—
200C4⋊C8C22×C414yes—
201C4⋊C8C2×D412yes—
202C4⋊C8C2×D412yes—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—21no—
207C22⋊C8—21no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—21no—
211C22⋊C8—21no—
212C22⋊C8—22no—
213C22⋊C8—41no—
214C22⋊C8—41no—
215#97—22no—
216#97—22no—
217C4×C8—22no—
218C4×C8—22no—
219C4×C8—21no—
220C4×C8—22no—
221C4×C8—21no—
222C4×C8—22no—
223C4×C8—22no—
224C4×C8—22no—
225C4×C8C2×D412yes—
226C4×C8C2×D412yes—
227C24#12611yescharacteristic
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—82no—
233C2×Q8—82no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4#11711yescharacteristic
241C22×C4#11711yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12311yescharacteristic
249C22×C4#12311yescharacteristic
250C22×C4#12411yescharacteristic
251C22×C4#12411yescharacteristic
252C22×C4#12611yescharacteristic
253C22×C4#12611yescharacteristic
254C22×C4#12611yescharacteristic
255C22×C4C2×#4111yescharacteristic
256C22×C4C2×#4111yescharacteristic
257C22×C4C2×#4111yescharacteristic
258M16—42no—
259M16—42no—
260M16—44no—
261M16—82no—
262M16—82no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—24no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—44no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—82no—
274C2×C8—82no—
275C2×C8—82no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C42—42no—
291C42—42no—
292C42#11711yescharacteristic
293C42#11711yescharacteristic
294C42C4×D411yescharacteristic
295C42C4×D411yescharacteristic
296C42C4×D411yescharacteristic
297C42C4×D411yescharacteristic
298C42#12111yescharacteristic
299C42#12111yescharacteristic
300C42#12311yescharacteristic
301C42#12311yescharacteristic
302C42#12411yescharacteristic
303C42#12411yescharacteristic
304C42C23×C411yescharacteristic
305C42C22×D411yescharacteristic
306C42C22×D411yescharacteristic
307C42C2×#4111yescharacteristic
308C23—22no—
309C23—41no—
310C23—41no—
311C23#38811yescharacteristic
312C23#38811yescharacteristic
313C23C2×#12611yescharacteristic
314Q8—44no—
315Q8—44no—
316Q8—82no—
317Q8—82no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4#38811yescharacteristic
342C2×C4#38811yescharacteristic
343C2×C4#38811yescharacteristic
344C2×C4#38811yescharacteristic
345C2×C4#38811yescharacteristic
346C2×C4#38811yescharacteristic
347C2×C4#47112yes—
348C2×C4#47312yes—
349C2×C4C2×#11711yescharacteristic
350C2×C4C2×C4×D411yescharacteristic
351C2×C4C2×C4×D411yescharacteristic
352C2×C4C2×#12111yescharacteristic
353C2×C4C2×#12311yescharacteristic
354C2×C4C2×#12411yescharacteristic
355C8—42no—
356C8—42no—
357C8—42no—
358C8—42no—
359C8—44no—
360C8—42no—
361C8—42no—
362C8—82no—
363C8—82no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—82no—
373C22#193012yes—
374C22C2×#38811yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—82no—
384C4—82no—
385C4#193012yes—
386C4#295712yes—
387C4#295712yes—
388C4C2×#47111yescharacteristic
389C4C2×#47311yescharacteristic
390C2—22no—
391C2—41no—
392C2—41no—
393C2#2107211yescharacteristic
394C2C2×#193011yescharacteristic
395C2#2109811yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#388
prime radicalC24#126
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9587C2semidirect
7. #9585C2semidirect

Maximal subgroups

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

subgroupindex
#178172
C2×#15442
#174912
#174862
#173262
#95872
#95852
#80652
#78842