Finite Group Databasebeta

← #110535

Subgroups — #110535

Order 512.

635 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#21332C211yesmaximal, characteristic
2#20479C211yesmaximal, characteristic
3#17755C211yesmaximal, characteristic
4C2×#1531C211yesmaximal, characteristic
5#8034C211yesmaximal, characteristic
6#7952C211yesmaximal, characteristic
7#7930C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C22×#97C2211yescharacteristic
13#2285C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1996—22no—
16#1996—21no—
17#1996—21no—
18#1771—22no—
19#1752—21no—
20C8×C22⋊C4—21no—
21#1745—21no—
22#1740C414yes—
23C4×#97—21no—
24#1556C412yescomplemented
25#1556C412yescomplemented
26#1552—22no—
27#1552—22no—
28#1531—22no—
29#1531—21no—
30#1531—21no—
31C2×#330C414yescomplemented
32#1295—22no—
33#1285—22no—
34#1282—22no—
35C23×C8—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125C2×C412yes—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×#117C2×C412yes—
52C2×#117C2×C412yes—
53C2×#117C2×C412yes—
54C22×C42C2311yescharacteristic
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432C2×C412yes—
65#432C2×C412yes—
66#432D412yes—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8D412yes—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×#97—21no—
74C2×#97—21no—
75C2×#97—21no—
76C2×#97—21no—
77C2×#97—21no—
78C2×#97—22no—
79C2×#97C2×C412yes—
80C2×#97C2×C412yes—
81C2×#97D412yes—
82C2×#97D411yescharacteristic
83C2×#97D411yescharacteristic
84C2×C4×C8—22no—
85#384—21no—
86#382—21no—
87#380—21no—
88#380—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C22⋊C4D412yes—
94C4×C22⋊C4D411yescharacteristic
95C4×C22⋊C4D411yescharacteristic
96#376—21no—
97C43—21no—
98#342—24no—
99#342—24no—
100#336—22no—
101#336—22no—
102#332—24no—
103#332—24no—
104#330—24no—
105#330—24no—
106C22×Q8—21no—
107C22×Q8—21no—
108C23×C4C22×C411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C22×C8—21no—
112C22×C8—22no—
113C22×C8—21no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—44no—
118C22×C8—41no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127#125—22no—
128#125—22no—
129#125—22no—
130#125—22no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×Q8—41no—
146C4×Q8—41no—
147#117—24no—
148#117—22no—
149#117—22no—
150#117—22no—
151#117—22no—
152#117—42no—
153#117—44no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4C4212yes—
163C2×C4⋊C4C4212yes—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4C22⋊C412yes—
167C2×C22⋊C4C22⋊C412yes—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42C22×C411yescharacteristic
175C2×C42C22×C411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—41no—
189C4⋊C8—41no—
190C4⋊C8—41no—
191C4⋊C8C22⋊C414yes—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197C22⋊C8—41no—
198#97—22no—
199#97—22no—
200#97—22no—
201#97—22no—
202#97—22no—
203#97—22no—
204#97—22no—
205#97—21no—
206#97—21no—
207#97—44no—
208#97—41no—
209#97C22⋊C412yes—
210#97C22⋊C412yes—
211C4×C8—44no—
212#95—22no—
213#95—21no—
214#95—21no—
215#95—41no—
216#95—41no—
217C24C2×C22⋊C411yescharacteristic
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—82no—
223C2×Q8—82no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—81no—
233C22×C4C2×C4211yescharacteristic
234C22×C4#11711yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4C4×D411yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4#12011yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12311yescharacteristic
249C22×C4#12411yescharacteristic
250C22×C4#12711yescharacteristic
251C2×C8—24no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—24no—
255C2×C8—44no—
256C2×C8—44no—
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—44no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—22no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—44no—
286C22⋊C4#10114yes—
287C42—44no—
288C42—42no—
289C42—42no—
290C42—42no—
291C42—42no—
292C42—82no—
293C42—82no—
294C42C2×C22⋊C411yescharacteristic
295C42C2×C22⋊C411yescharacteristic
296C42#11711yescharacteristic
297C42#11711yescharacteristic
298C42C4×D411yescharacteristic
299C42C4×D411yescharacteristic
300C42C4×D411yescharacteristic
301C42C4×D411yescharacteristic
302C42#12111yescharacteristic
303C42#12111yescharacteristic
304C42#12311yescharacteristic
305C42#12311yescharacteristic
306C42#12411yescharacteristic
307C42#12411yescharacteristic
308C42#12611yescharacteristic
309C42#12611yescharacteristic
310C23—41no—
311C23—41no—
312C23C4×C22⋊C411yescharacteristic
313C23#38611yescharacteristic
314C23#39011yescharacteristic
315C23C2×#10112yes—
316C23#41312yes—
317Q8—82no—
318Q8—82no—
319Q8—82no—
320Q8—82no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4C4×C22⋊C411yescharacteristic
355C2×C4C4×C22⋊C411yescharacteristic
356C2×C4#38611yescharacteristic
357C2×C4#38611yescharacteristic
358C2×C4#38811yescharacteristic
359C2×C4#38811yescharacteristic
360C2×C4#38811yescharacteristic
361C2×C4#38811yescharacteristic
362C2×C4#38811yescharacteristic
363C2×C4#38811yescharacteristic
364C2×C4#39011yescharacteristic
365C2×C4#39011yescharacteristic
366C8—24no—
367C8—42no—
368C8—44no—
369C8—82no—
370C8—82no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—42no—
378C22—42no—
379C22#143711yescharacteristic
380C22C4×#10112yes—
381C22#190112yes—
382C22#196611yescharacteristic
383C22#196711yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—44no—
396C4—82no—
397C4—82no—
398C4—82no—
399C4—162no—
400C4#190612yes—
401C2—22no—
402C2—21no—
403C2—21no—
404C2—22no—
405C2#1094511yescharacteristic
406C2#1095111yescharacteristic
407C2#1096311yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ 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
24. #1556C4semidirect
25. #1556C4semidirect
31. C2×#330C4semidirect

Maximal subgroups

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

subgroupindex
#213322
#204792
#177552
C2×#15312
#80342
#79522
#79302