Finite Group Databasebeta

← #110915

Subgroups — #110915

Order 512.

617 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20469C211yesmaximal, characteristic
2#17755C211yesmaximal, characteristic
3#17575C211yesmaximal, characteristic
4C2×#1506C211yesmaximal, characteristic
5#8089C211yesmaximal, characteristic
6#8022C211yesmaximal, characteristic
7#7952C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#376C2211yescharacteristic
15#1768—22no—
16#1749—21no—
17C4×C22⋊C8—21no—
18#1746—21no—
19#1726—21no—
20#1556C412yescomplemented
21#1556C412yescomplemented
22#1552—22no—
23#1552—22no—
24#1520—21no—
25#1520—21no—
26#1518—22no—
27#1506—21no—
28#1506—21no—
29#1506C414yescomplemented
30#1503C414yescomplemented
31C2×#328—22no—
32#1302—22no—
33#1294—22no—
34#1285—22no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533C2×C412yes—
42#533C2×C412yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×#117—21no—
51C2×#117—21no—
52C22×C42—21no—
53C22×C42C2311yescharacteristic
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432C2×C412yes—
63#432C2×C412yes—
64#432C2×C412yes—
65#432D412yes—
66#431—21no—
67#431—21no—
68#431C2×C412yes—
69#431D412yes—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8D412yes—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8D411yescharacteristic
78C2×C22⋊C8D411yescharacteristic
79C2×#97—21no—
80C2×#97—21no—
81C2×#97—21no—
82C2×#97—22no—
83C2×C4×C8—21no—
84C2×C4×C8—22no—
85#384—21no—
86#382—21no—
87#380—21no—
88#380—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4C2×C412yes—
91#376—21no—
92#376—21no—
93#376C2×C412yes—
94#376D411yescharacteristic
95#376D411yescharacteristic
96C2×#95—21no—
97C2×#95D412yes—
98#336—22no—
99#336—22no—
100#332—24no—
101#332—24no—
102#328—24no—
103#328—24no—
104#328—24no—
105#328—24no—
106C22×Q8—21no—
107C22×Q8—21no—
108C23×C4—22no—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C22×C8—21no—
113C22×C8—22no—
114C22×C8—21no—
115C22×C8—41no—
116C22×C8—41no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8C4212yes—
124C4⋊Q8C4212yes—
125#125—22no—
126#125—22no—
127#125—41no—
128#125—41no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—41no—
142C4×Q8—41no—
143C4×Q8—41no—
144C4×Q8—41no—
145#117—22no—
146#117—22no—
147#117—41no—
148#117—41no—
149#117—82no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C42—21no—
162C2×C42—22no—
163C2×C42—21no—
164C2×C42—22no—
165C2×C42—44no—
166C2×C42—41no—
167C2×C42C22×C411yescharacteristic
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—41no—
185C4⋊C8—41no—
186C4⋊C8C22⋊C414yes—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—21no—
190C22⋊C8—21no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195C22⋊C8C22⋊C412yes—
196C22⋊C8C22⋊C412yes—
197#97—22no—
198#97—22no—
199#97—22no—
200#97—22no—
201#97—44no—
202#97—41no—
203#97—41no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—44no—
207#95—22no—
208#95—22no—
209#95—21no—
210#95—21no—
211#95—22no—
212#95—41no—
213#95C22⋊C412yes—
214#95C22⋊C412yes—
215C24#12111yescharacteristic
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—82no—
221C2×Q8—82no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—81no—
234C22×C4—82no—
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4#11711yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4C4×D411yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4#12011yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12311yescharacteristic
248C22×C4#12311yescharacteristic
249C22×C4#12411yescharacteristic
250C22×C4#12411yescharacteristic
251C22×C4#12611yescharacteristic
252C22×C4#12711yescharacteristic
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—24no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—82no—
263C2×C8—82no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—82no—
282C42—24no—
283C42—42no—
284C42—44no—
285C42—42no—
286C42—42no—
287C42—82no—
288C42C2×C4211yescharacteristic
289C42C2×C22⋊C411yescharacteristic
290C42#11711yescharacteristic
291C42#11711yescharacteristic
292C42C4×D411yescharacteristic
293C42C4×D411yescharacteristic
294C42C4×D411yescharacteristic
295C42C4×D411yescharacteristic
296C42C4×D411yescharacteristic
297C42C4×D411yescharacteristic
298C42#12111yescharacteristic
299C42#12111yescharacteristic
300C42#12111yescharacteristic
301C42#12311yescharacteristic
302C42#12411yescharacteristic
303C42#12611yescharacteristic
304C23—41no—
305C23—41no—
306C23#38611yescharacteristic
307C23#38811yescharacteristic
308C23#39011yescharacteristic
309C23#47112yes—
310C23#47312yes—
311Q8—82no—
312Q8—82no—
313Q8—82no—
314Q8—82no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—84no—
344C2×C4—84no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4C4×C22⋊C411yescharacteristic
351C2×C4C4×C22⋊C411yescharacteristic
352C2×C4C4×C22⋊C411yescharacteristic
353C2×C4#38611yescharacteristic
354C2×C4#38611yescharacteristic
355C2×C4#38811yescharacteristic
356C2×C4#38811yescharacteristic
357C2×C4#38811yescharacteristic
358C2×C4#38811yescharacteristic
359C2×C4#38811yescharacteristic
360C2×C4#39011yescharacteristic
361C2×C4#39011yescharacteristic
362C8—42no—
363C8—42no—
364C8—44no—
365C8—82no—
366C8—82no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—42no—
374C22—42no—
375C22#143711yescharacteristic
376C22#190812yes—
377C22#193012yes—
378C22#197711yescharacteristic
379C22#197911yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—44no—
392C4—82no—
393C4—82no—
394C4—82no—
395C4—162no—
396C4#190612yes—
397C2—22no—
398C2—21no—
399C2—22no—
400C2—21no—
401C2#1094211yescharacteristic
402C2#1094511yescharacteristic
403C2#1095111yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
20. #1556C4semidirect
21. #1556C4semidirect
29. #1506C4semidirect
30. #1503C4semidirect

Maximal subgroups

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

subgroupindex
#204692
#177552
#175752
C2×#15062
#80892
#80222
#79522