Finite Group Databasebeta

← #203069

Subgroups — #203069

Order 512.

630 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17806C211yesmaximal, characteristic
2#17715C211yesmaximal, characteristic
3#17479C211yesmaximal, characteristic
4C2×#1485C211yesmaximal, characteristic
5#17326C211yesmaximal, characteristic
6#9587C214yesmaximal, complemented
7#9585C214yesmaximal, complemented
8#8064C211yesmaximal, characteristic
9#7882C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#1574C2212yes—
18#1574C2212yes—
19#1573—22no—
20#1564—22no—
21#1552C414yes—
22#1549C411yescharacteristic
23#1549C411yescharacteristic
24#1546C411yescharacteristic
25#1546C411yescharacteristic
26#1544C2214yes—
27#1497—22no—
28#1493C2214yes—
29#1485—22no—
30#1485C2212yes—
31#1485C2212yes—
32#1479—21no—
33#1479—21no—
34C2×Q8⋊C8—22no—
35#1454—22no—
36#1452C2212yes—
37#1451—21no—
38#1450C2212yes—
39C8×M16—21no—
40#1299C2214yes—
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#432C2×C412yes—
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×C411yescharacteristic
74#431C2×C411yescharacteristic
75#431D412yes—
76#431D411yescharacteristic
77#431D411yescharacteristic
78#431D412yes—
79#431C2312yes—
80#431C2312yes—
81#431C2311yescharacteristic
82#431C2311yescharacteristic
83#423C2×C411yescharacteristic
84#423C2×C411yescharacteristic
85#423C2312yes—
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2×C412yes—
88C2×C4⋊C8D412yes—
89#407—21no—
90#407—21no—
91C2×C22⋊C8—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×C8—21no—
104C2×C4×C8C2312yes—
105#335—22no—
106#335—22no—
107#334—22no—
108#334—22no—
109#332C2×C414yes—
110#332C2×C414yes—
111#332C2×C414yes—
112#332C2×C414yes—
113Q8⋊C8—24no—
114Q8⋊C8—24no—
115Q8⋊C8—24no—
116Q8⋊C8—24no—
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—21no—
127C2×M16—21no—
128C2×M16—24no—
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—42no—
132C2×M16—41no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—41no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8—21no—
143C4⋊Q8C22×C412yes—
144C4⋊Q8C22×C412yes—
145#125—22no—
146#125—22no—
147#125—41no—
148#125—41no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C22⋊Q8—41no—
154C22⋊Q8—41no—
155C22⋊Q8—41no—
156C22⋊Q8—41no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—41no—
162C4×Q8—41no—
163C4×Q8—41no—
164C4×Q8—41no—
165#117—22no—
166#117—22no—
167#117—41no—
168#117—41no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C42—41no—
177C2×C42C22×C411yescharacteristic
178C2×C42C22×C411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42C2411yescharacteristic
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—22no—
198C4⋊C8C22×C412yes—
199C4⋊C8C22×C412yes—
200C4⋊C8C22×C412yes—
201C4⋊C8C22×C412yes—
202C4⋊C8C2×D412yes—
203C4⋊C8C2×D412yes—
204C22⋊C8—21no—
205C22⋊C8—21no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—21no—
209C22⋊C8—21no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214C22⋊C8—22no—
215C22⋊C8—41no—
216C22⋊C8—41no—
217#97—22no—
218#97—22no—
219#97—41no—
220#97—41no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—22no—
224C4×C8—21no—
225C4×C8—22no—
226C4×C8—21no—
227C4×C8—22no—
228C4×C8—22no—
229C4×C8—22no—
230C4×C8—41no—
231C4×C8—41no—
232C4×C8C2×D412yes—
233C4×C8C2×D412yes—
234C24#12611yescharacteristic
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—44no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—42no—
241C2×Q8—42no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4#11711yescharacteristic
249C22×C4#11711yescharacteristic
250C22×C4C4×D411yescharacteristic
251C22×C4C4×D411yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4C4×D411yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12311yescharacteristic
257C22×C4#12311yescharacteristic
258C22×C4#12411yescharacteristic
259C22×C4#12411yescharacteristic
260C22×C4#12611yescharacteristic
261C22×C4#12611yescharacteristic
262C22×C4#12611yescharacteristic
263C22×C4C2×#4111yescharacteristic
264C22×C4C2×#4111yescharacteristic
265C22×C4C2×#4111yescharacteristic
266M16—24no—
267M16—44no—
268M16—42no—
269M16—42no—
270M16—42no—
271M16—82no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—44no—
281C2×C8—44no—
282C2×C8—44no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—82no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C42—42no—
301C42—42no—
302C42#11711yescharacteristic
303C42#11711yescharacteristic
304C42C4×D411yescharacteristic
305C42C4×D411yescharacteristic
306C42C4×D411yescharacteristic
307C42C4×D411yescharacteristic
308C42#12111yescharacteristic
309C42#12111yescharacteristic
310C42#12311yescharacteristic
311C42#12311yescharacteristic
312C42#12411yescharacteristic
313C42#12411yescharacteristic
314C42C23×C411yescharacteristic
315C42C22×D411yescharacteristic
316C42C22×D411yescharacteristic
317C42C2×#4111yescharacteristic
318C23—22no—
319C23—41no—
320C23—41no—
321C23#38811yescharacteristic
322C23#38811yescharacteristic
323C23C2×#12611yescharacteristic
324Q8—42no—
325Q8—42no—
326Q8—44no—
327Q8—82no—
328Q8—82no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4#38811yescharacteristic
353C2×C4#38811yescharacteristic
354C2×C4#38811yescharacteristic
355C2×C4#38811yescharacteristic
356C2×C4#38811yescharacteristic
357C2×C4#38811yescharacteristic
358C2×C4#44312yes—
359C2×C4#44412yes—
360C2×C4C2×#11711yescharacteristic
361C2×C4C2×C4×D411yescharacteristic
362C2×C4C2×C4×D411yescharacteristic
363C2×C4C2×#12111yescharacteristic
364C2×C4C2×#12311yescharacteristic
365C2×C4C2×#12411yescharacteristic
366C8—42no—
367C8—42no—
368C8—44no—
369C8—42no—
370C8—42no—
371C8—42no—
372C8—42no—
373C8—42no—
374C8—42no—
375C8—82no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—82no—
385C22#192912yes—
386C22C2×#38811yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—82no—
396C4—82no—
397C4#192912yes—
398C4C2×#44311yescharacteristic
399C4C2×#44411yescharacteristic
400C4#295512yes—
401C4#306912yes—
402C2—22no—
403C2—41no—
404C2—41no—
405C2C2×#192911yescharacteristic
406C2#2107711yescharacteristic
407C2#2110011yescharacteristic
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
#178062
#177152
#174792
C2×#14852
#173262
#95872
#95852
#80642
#78822