Finite Group Databasebeta

← #203080

Subgroups — #203080

Order 512.

634 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17820C211yesmaximal, characteristic
2#17739C211yesmaximal, characteristic
3#17491C211yesmaximal, characteristic
4C2×#1493C211yesmaximal, characteristic
5#17323C211yesmaximal, characteristic
6#9587C214yesmaximal, complemented
7#9585C214yesmaximal, complemented
8#8061C211yesmaximal, characteristic
9#7885C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13#2920C2211yescharacteristic
14#2920C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#1574C2214yes—
18#1571—21no—
19#1570—21no—
20#1569—22no—
21#1561C414yes—
22#1556C412yes—
23#1556C412yes—
24#1544C2214yes—
25#1497—22no—
26#1493—21no—
27#1493—21no—
28#1493C2212yes—
29#1493C2212yes—
30#1487—21no—
31#1487—21no—
32#1485C2214yes—
33C2×Q8⋊C8—21no—
34C2×Q8⋊C8—21no—
35#1452C2212yes—
36#1451—21no—
37#1451—21no—
38#1450C2212yes—
39C8×M16—21no—
40C8×M16—21no—
41#1299C2214yes—
42#1278C2212yes—
43#1278C2212yes—
44C22×M16—21no—
45C23×C8—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533C2×C412yes—
53#533C2312yes—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×#117—21no—
62C22×C42C2311yescharacteristic
63#432—21no—
64#432—21no—
65#432—21no—
66#432—21no—
67#432—21no—
68#432C2×C412yes—
69#432C2×C412yes—
70#432D412yes—
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431—21no—
76#431C2312yes—
77#431C2312yes—
78#431C2312yes—
79#423—21no—
80#423—21no—
81#423—21no—
82#423—21no—
83#423C2×C412yes—
84#423C2×C412yes—
85#423D411yescharacteristic
86#423D411yescharacteristic
87#423C2312yes—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90#407—21no—
91#407—21no—
92#407—21no—
93#407—21no—
94C4×M16—21no—
95C4×M16—21no—
96C4×M16—21no—
97C4×M16C2312yes—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8D411yescharacteristic
102C2×C4×C8D411yescharacteristic
103C2×C4×C8D411yescharacteristic
104C2×C4×C8D411yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#335—24no—
108#334—24no—
109#332C2×C418yes—
110#332C2×C418yes—
111Q8⋊C8—24no—
112Q8⋊C8—24no—
113Q8⋊C8—22no—
114Q8⋊C8—22no—
115Q8⋊C8—22no—
116Q8⋊C8—22no—
117#322—22no—
118#322—22no—
119#322—22no—
120C82—22no—
121C22×Q8—21no—
122C22×Q8—21no—
123C23×C4#4111yescharacteristic
124C23×C4#4111yescharacteristic
125C23×C4#4111yescharacteristic
126C2×M16—22no—
127C2×M16—21no—
128C2×M16—21no—
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—41no—
133C2×M16—82no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—44no—
141C22×C8—41no—
142C4⋊Q8—21no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8C22×C412yes—
149C4⋊Q8C22×C412yes—
150#125—22no—
151#125—22no—
152#125—41no—
153#125—41no—
154C22⋊Q8—22no—
155C22⋊Q8—22no—
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158C22⋊Q8—41no—
159C22⋊Q8—41no—
160C22⋊Q8—41no—
161C22⋊Q8—41no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—41no—
167C4×Q8—41no—
168C4×Q8—41no—
169C4×Q8—41no—
170#117—22no—
171#117—22no—
172#117—41no—
173#117—41no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C42—41no—
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42C2411yescharacteristic
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—41no—
201C4⋊C8—41no—
202C4⋊C8—41no—
203C4⋊C8—41no—
204C4⋊C8C22×C414yes—
205C4⋊C8C22×C414yes—
206C4⋊C8C2×D412yes—
207C4⋊C8C2×D412yes—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214C22⋊C8—41no—
215C22⋊C8—41no—
216#97—22no—
217#97—22no—
218#97—22no—
219#97—22no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—22no—
225C4×C8—22no—
226C4×C8—22no—
227C4×C8—22no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8C2×D411yescharacteristic
231C4×C8C2×D411yescharacteristic
232C4×C8C2×D411yescharacteristic
233C4×C8C2×D411yescharacteristic
234C24#12611yescharacteristic
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—44no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—82no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4#11711yescharacteristic
248C22×C4#11711yescharacteristic
249C22×C4C4×D411yescharacteristic
250C22×C4C4×D411yescharacteristic
251C22×C4C4×D411yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12311yescharacteristic
256C22×C4#12311yescharacteristic
257C22×C4#12411yescharacteristic
258C22×C4#12411yescharacteristic
259C22×C4#12611yescharacteristic
260C22×C4#12611yescharacteristic
261C22×C4#12611yescharacteristic
262C22×C4C2×#4111yescharacteristic
263C22×C4C2×#4111yescharacteristic
264C22×C4C2×#4111yescharacteristic
265M16—44no—
266M16—42no—
267M16—42no—
268M16—44no—
269M16—82no—
270M16—82no—
271M16—82no—
272C2×C8—24no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—82no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C42—42no—
304C42—42no—
305C42#11711yescharacteristic
306C42#11711yescharacteristic
307C42C4×D411yescharacteristic
308C42C4×D411yescharacteristic
309C42C4×D411yescharacteristic
310C42C4×D411yescharacteristic
311C42#12111yescharacteristic
312C42#12111yescharacteristic
313C42#12311yescharacteristic
314C42#12311yescharacteristic
315C42#12411yescharacteristic
316C42#12411yescharacteristic
317C42C23×C411yescharacteristic
318C42C22×D411yescharacteristic
319C42C22×D411yescharacteristic
320C42C2×#4111yescharacteristic
321C23—22no—
322C23—41no—
323C23—41no—
324C23#38811yescharacteristic
325C23#38811yescharacteristic
326C23C2×#12611yescharacteristic
327Q8—44no—
328Q8—44no—
329Q8—82no—
330Q8—82no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4#38811yescharacteristic
355C2×C4#38811yescharacteristic
356C2×C4#38811yescharacteristic
357C2×C4#38811yescharacteristic
358C2×C4#38811yescharacteristic
359C2×C4#38811yescharacteristic
360C2×C4#47112yes—
361C2×C4#47312yes—
362C2×C4C2×#11711yescharacteristic
363C2×C4C2×C4×D411yescharacteristic
364C2×C4C2×C4×D411yescharacteristic
365C2×C4C2×#12111yescharacteristic
366C2×C4C2×#12311yescharacteristic
367C2×C4C2×#12411yescharacteristic
368C8—22no—
369C8—22no—
370C8—41no—
371C8—41no—
372C8—44no—
373C8—44no—
374C8—42no—
375C8—42no—
376C8—82no—
377C8—82no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—44no—
387C22#193012yes—
388C22C2×#38811yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—82no—
398C4—82no—
399C4#193012yes—
400C4#295512yes—
401C4#295612yes—
402C4C2×#47111yescharacteristic
403C4C2×#47311yescharacteristic
404C2—22no—
405C2—41no—
406C2—41no—
407C2#2107011yescharacteristic
408C2C2×#193011yescharacteristic
409C2#2109911yescharacteristic
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
#177392
#174912
C2×#14932
#173232
#95872
#95852
#80612
#78852