Finite Group Databasebeta

← #203086

Subgroups — #203086

Order 512.

628 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17820C211yesmaximal, characteristic
2#17715C211yesmaximal, characteristic
3#17501C211yesmaximal, characteristic
4#17486C211yesmaximal, characteristic
5#17326C211yesmaximal, characteristic
6#9587C214yesmaximal, complemented
7#9585C214yesmaximal, complemented
8C2×#1299C211yesmaximal, characteristic
9#7888C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#1574C2214yes—
18#1571—21no—
19#1570—21no—
20#1569—22no—
21#1552C414yes—
22#1549C412yes—
23#1546C412yes—
24#1544C2212yes—
25#1544C2212yes—
26#1493—21no—
27#1493—21no—
28#1493C2214yes—
29#1491—22no—
30#1487—22no—
31#1485C2214yes—
32#1481—21no—
33#1481—21no—
34#1454—22no—
35#1452C2212yes—
36#1451—21no—
37#1450C2212yes—
38C8×M16—21no—
39#1299C2212yes—
40#1299C2212yes—
41#1278C2214yes—
42C22×M16—21no—
43C22×M16—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533C2×C412yes—
51#533C2312yes—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×#117—21no—
60C22×C42C2311yescharacteristic
61#432—21no—
62#432—21no—
63#432C2×C412yes—
64#432D412yes—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431C2×C412yes—
72#431D411yescharacteristic
73#431D411yescharacteristic
74#431D412yes—
75#431C2311yescharacteristic
76#431C2311yescharacteristic
77#431C2312yes—
78#431C2312yes—
79#423—21no—
80#423—21no—
81#423—21no—
82#423C2×C412yes—
83#423D411yescharacteristic
84#423D411yescharacteristic
85#423C2312yes—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2×C412yes—
90#407—21no—
91#407—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C4×M16—21no—
95C4×M16—21no—
96C4×M16—21no—
97C4×M16—21no—
98C4×M16—21no—
99C4×M16C2312yes—
100C2×#97—21no—
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—22no—
128C2×M16—44no—
129C2×M16—41no—
130C2×M16—41no—
131C2×M16—82no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—41no—
135C22×C8—41no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8C22×C412yes—
143C4⋊Q8C22×C412yes—
144#125—22no—
145#125—22no—
146#125—41no—
147#125—41no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C22⋊Q8—41no—
153C22⋊Q8—41no—
154C22⋊Q8—41no—
155C22⋊Q8—41no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—41no—
161C4×Q8—41no—
162C4×Q8—41no—
163C4×Q8—41no—
164#117—22no—
165#117—22no—
166#117—41no—
167#117—41no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C42—41no—
176C2×C42C22×C411yescharacteristic
177C2×C42C22×C411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42C2411yescharacteristic
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—22no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—41no—
198C4⋊C8—41no—
199C4⋊C8C22×C412yes—
200C4⋊C8C22×C412yes—
201C4⋊C8C22×C414yes—
202C4⋊C8C2×D412yes—
203C4⋊C8C2×D412yes—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—21no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8—21no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214#97—22no—
215#97—22no—
216#97—22no—
217#97—41no—
218C4×C8—22no—
219C4×C8—21no—
220C4×C8—22no—
221C4×C8—21no—
222C4×C8—22no—
223C4×C8—22no—
224C4×C8—22no—
225C4×C8—41no—
226C4×C8C2×D412yes—
227C4×C8C2×D412yes—
228C24#12611yescharacteristic
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—82no—
234C2×Q8—82no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4#11711yescharacteristic
242C22×C4#11711yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12311yescharacteristic
250C22×C4#12311yescharacteristic
251C22×C4#12411yescharacteristic
252C22×C4#12411yescharacteristic
253C22×C4#12611yescharacteristic
254C22×C4#12611yescharacteristic
255C22×C4#12611yescharacteristic
256C22×C4C2×#4111yescharacteristic
257C22×C4C2×#4111yescharacteristic
258C22×C4C2×#4111yescharacteristic
259M16—24no—
260M16—42no—
261M16—44no—
262M16—42no—
263M16—42no—
264M16—82no—
265M16—82no—
266M16—82no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—24no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—44no—
275C2×C8—44no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—82no—
279C2×C8—82no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C42—42no—
295C42—42no—
296C42#11711yescharacteristic
297C42#11711yescharacteristic
298C42C4×D411yescharacteristic
299C42C4×D411yescharacteristic
300C42C4×D411yescharacteristic
301C42C4×D411yescharacteristic
302C42#12111yescharacteristic
303C42#12111yescharacteristic
304C42#12311yescharacteristic
305C42#12311yescharacteristic
306C42#12411yescharacteristic
307C42#12411yescharacteristic
308C42C23×C411yescharacteristic
309C42C22×D411yescharacteristic
310C42C22×D411yescharacteristic
311C42C2×#4111yescharacteristic
312C23—22no—
313C23—41no—
314C23—41no—
315C23#38811yescharacteristic
316C23#38811yescharacteristic
317C23C2×#12611yescharacteristic
318Q8—44no—
319Q8—44no—
320Q8—82no—
321Q8—82no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4#38811yescharacteristic
346C2×C4#38811yescharacteristic
347C2×C4#38811yescharacteristic
348C2×C4#38811yescharacteristic
349C2×C4#38811yescharacteristic
350C2×C4#38811yescharacteristic
351C2×C4#47112yes—
352C2×C4#47312yes—
353C2×C4C2×#11711yescharacteristic
354C2×C4C2×C4×D411yescharacteristic
355C2×C4C2×C4×D411yescharacteristic
356C2×C4C2×#12111yescharacteristic
357C2×C4C2×#12311yescharacteristic
358C2×C4C2×#12411yescharacteristic
359C8—42no—
360C8—42no—
361C8—44no—
362C8—42no—
363C8—42no—
364C8—44no—
365C8—42no—
366C8—42no—
367C8—82no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—82no—
377C22#193012yes—
378C22C2×#38811yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—82no—
388C4—82no—
389C4#193012yes—
390C4#295612yes—
391C4#295712yes—
392C4C2×#47111yescharacteristic
393C4C2×#47311yescharacteristic
394C2—22no—
395C2—41no—
396C2—41no—
397C2#2107111yescharacteristic
398C2C2×#193011yescharacteristic
399C2#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
#178202
#177152
#175012
#174862
#173262
#95872
#95852
C2×#12992
#78882