Finite Group Databasebeta

← #128202

Subgroups — #128202

Order 512.

552 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20721C211yesmaximal, characteristic
2#17363C211yesmaximal, characteristic
3#17363C211yesmaximal, characteristic
4#7327C214yesmaximal, complemented
5C2×C4×#41C2211yescharacteristic
6C2×#432—21no—
7C2×#432—21no—
8C2×#432C411yescharacteristic
9C2×#431C411yescharacteristic
10#1818C2212yes—
11#1797—21no—
12#1797—21no—
13#1780C412yes—
14#1750—21no—
15#1750—21no—
16#1467C412yes—
17#1467C412yes—
18#1466—21no—
19#1466—21no—
20#1464—21no—
21#1464—21no—
22#1464C2212yes—
23#1464C2212yes—
24#1459—21no—
25#1459—21no—
26#1459C412yes—
27#1459C412yes—
28C22×#41C2×C411yescharacteristic
29C4×#41—41no—
30C4×#41—41no—
31C4×#41—41no—
32C4×#41—41no—
33C2×C4×Q8C2×C411yescharacteristic
34C2×C4×Q8C2×C411yescharacteristic
35C2×C4×D4C2×C411yescharacteristic
36C2×C4×D4C2×C411yescharacteristic
37C2×C4×D4D411yescharacteristic
38C2×C4×D4D411yescharacteristic
39C2×C4×D4Q811yescharacteristic
40C2×C4×D4Q811yescharacteristic
41C2×#117D411yescharacteristic
42C2×#117D411yescharacteristic
43C2×#117C2311yescharacteristic
44C22×C42C2×C411yescharacteristic
45C22×C42D411yescharacteristic
46C22×C42D411yescharacteristic
47#432—22no—
48#432—22no—
49#432—22no—
50#432—22no—
51#432—41no—
52#432—41no—
53#432—41no—
54#432—41no—
55#432—41no—
56#432—41no—
57#431—41no—
58#431—41no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—41no—
62C2×C4⋊C8—41no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8C2×C412yes—
68C2×C22⋊C8C2×C412yes—
69C2×#97—21no—
70C2×#97—41no—
71C2×#97—41no—
72C2×C4×C8—21no—
73#336—22no—
74#336—22no—
75#324—22no—
76#324—22no—
77#324C2×C414yes—
78#324C2×C414yes—
79#323—22no—
80#323—22no—
81#323—22no—
82#323—22no—
83C2×#41—41no—
84C2×#41—41no—
85C2×#41—42no—
86C2×#41—42no—
87C22×Q8C4211yescharacteristic
88C22×D4C4211yescharacteristic
89C22×D4C4⋊C411yescharacteristic
90C22×D4C4⋊C411yescharacteristic
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C4⋊C411yescharacteristic
96C23×C4C4⋊C411yescharacteristic
97C23×C4C4⋊C411yescharacteristic
98C23×C4C4⋊C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—42no—
103C22×C8—42no—
104C4×Q8—41no—
105C4×Q8—41no—
106C4×Q8—41no—
107C4×Q8—41no—
108C4×D4—41no—
109C4×D4—41no—
110C4×D4—41no—
111C4×D4—41no—
112C4×D4—41no—
113C4×D4—41no—
114C4×D4—41no—
115C4×D4—41no—
116C4×D4—41no—
117C4×D4—41no—
118C4×D4—41no—
119C4×D4—41no—
120#117—41no—
121#117—41no—
122#117—41no—
123#117—41no—
124#117—41no—
125#117—41no—
126C2×C4⋊C4C22⋊C411yescharacteristic
127C2×C4⋊C4C22⋊C411yescharacteristic
128C2×C4⋊C4C4⋊C411yescharacteristic
129C2×C4⋊C4C4⋊C411yescharacteristic
130C2×C4⋊C4C22×C411yescharacteristic
131C2×C4⋊C4C22×C411yescharacteristic
132C2×C22⋊C4C22⋊C411yescharacteristic
133C2×C22⋊C4C22⋊C411yescharacteristic
134C2×C22⋊C4C4⋊C411yescharacteristic
135C2×C22⋊C4C4⋊C411yescharacteristic
136C2×C22⋊C4C2×D411yescharacteristic
137C2×C22⋊C4C2×Q811yescharacteristic
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42C4211yescharacteristic
151C2×C42C4211yescharacteristic
152C2×C42C22⋊C411yescharacteristic
153C2×C42C22⋊C411yescharacteristic
154C2×C42C22⋊C411yescharacteristic
155C2×C42C22⋊C411yescharacteristic
156C2×C42C4⋊C411yescharacteristic
157C2×C42C4⋊C411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C4⋊C8—42no—
161C4⋊C8—42no—
162C4⋊C8—42no—
163C4⋊C8—42no—
164C4⋊C8—41no—
165C4⋊C8—41no—
166C4⋊C8—41no—
167C4⋊C8—41no—
168C22⋊C8—24no—
169C22⋊C8—24no—
170C22⋊C8—44no—
171C22⋊C8—44no—
172C22⋊C8—42no—
173C22⋊C8—42no—
174C22⋊C8—42no—
175C22⋊C8—42no—
176#97—42no—
177#97—42no—
178#97—42no—
179#97—42no—
180#97—42no—
181C4×C8—42no—
182C24#9511yescharacteristic
183C24#9511yescharacteristic
184C24C2×C4⋊C411yescharacteristic
185#41—48no—
186#41—48no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—41no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—81no—
227C22×C4—81no—
228C22×C4—81no—
229C22×C4#9511yescharacteristic
230C22×C4#9511yescharacteristic
231C22×C4#9511yescharacteristic
232C22×C4#9511yescharacteristic
233C22×C4#9511yescharacteristic
234C22×C4#9511yescharacteristic
235C22×C4C2×C4211yescharacteristic
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C4⋊C411yescharacteristic
240C22×C4C2×C4⋊C411yescharacteristic
241C2×C8—44no—
242C2×C8—44no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—84no—
250C2×C8—84no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—21no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42#9911yescharacteristic
282C42#9911yescharacteristic
283C42#9911yescharacteristic
284C42#9911yescharacteristic
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23#34412yes—
299C23#34412yes—
300C23C2×#9511yescharacteristic
301Q8—44no—
302D4—44no—
303D4—44no—
304D4—44no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4#34211yescharacteristic
349C2×C4#34211yescharacteristic
350C2×C4#34211yescharacteristic
351C2×C4#34211yescharacteristic
352C2×C4C2×#9911yescharacteristic
353C2×C4C2×#9911yescharacteristic
354C8—82no—
355C8—82no—
356C8—84no—
357C8—84no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—22no—
362C22—42no—
363C22—42no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22C2×#34211yescharacteristic
376C22C2×#34411yescharacteristic
377C22C2×#34411yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4#139512yes—
393C2—21no—
394C2—21no—
395C2—42no—
396C2—42no—
397C2—81no—
398C2#811511yescharacteristic
399C2C2×#139511yescharacteristic
400C2#1258911yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×#117C23
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7327C2semidirect

Maximal subgroups

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

subgroupindex
#207212
#173632
#73272