Finite Group Databasebeta

← #120428

Subgroups — #120428

Order 512.

490 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21993C211yesmaximal, characteristic
2#17809C211yesmaximal, characteristic
3#10847C211yesmaximal, characteristic
4#8098C211yesmaximal, characteristic
5C2×#1303C211yesmaximal, characteristic
6#8004C211yesmaximal, characteristic
7#8004C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1950—21no—
16#1950—21no—
17#1855—22no—
18#1848—21no—
19#1837—21no—
20C2×#343—22no—
21#1574C414yescomplemented
22#1571—21no—
23#1570—21no—
24#1565—22no—
25#1303—22no—
26#1303—21no—
27#1303—21no—
28#1292—21no—
29#1292—21no—
30#1292—21no—
31#1292—21no—
32C22×M16—21no—
33C22×M16—21no—
34C22×M16D412yes—
35C22×C4⋊C4—21no—
36C22×C42—21no—
37C22×C42C2311yescharacteristic
38#432—21no—
39#432—21no—
40#432—21no—
41#432Q812yes—
42#431—21no—
43#431—21no—
44#431—21no—
45#431—21no—
46#431—21no—
47#431C2×C412yes—
48#431C2×C412yes—
49#431D412yes—
50C2×#108—22no—
51C2×#108—22no—
52#423—21no—
53#423—21no—
54#423—21no—
55#423—21no—
56#423—21no—
57#423—21no—
58#423—21no—
59#423—21no—
60#423—21no—
61#423—21no—
62#423—21no—
63#423C2×C412yes—
64#423D411yescharacteristic
65#423D411yescharacteristic
66#423D411yescharacteristic
67#423D411yescharacteristic
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8D411yescharacteristic
74C2×C4⋊C8D411yescharacteristic
75C2×C4⋊C8Q812yes—
76#407—21no—
77#407—21no—
78C2×C22⋊C8—21no—
79C4×M16—21no—
80C4×M16—21no—
81C2×C4×C8—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C2×#95Q812yes—
87#343—44no—
88#341—22no—
89#341—22no—
90#335—24no—
91#334—24no—
92C23×C4—22no—
93C23×C4C2×D411yescharacteristic
94C23×C4C2×Q811yescharacteristic
95C23×C4#4111yescharacteristic
96C2×M16—21no—
97C2×M16—21no—
98C2×M16—22no—
99C2×M16—22no—
100C2×M16—21no—
101C2×M16—21no—
102C2×M16—22no—
103C2×M16—41no—
104C2×M16—41no—
105C2×M16—41no—
106C2×M16—42no—
107C2×M16—42no—
108C2×M16—42no—
109C2×M16—42no—
110C2×M16—81no—
111C2×M16—81no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—41no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C42—41no—
124C2×C42—41no—
125C2×C42—81no—
126C2×C42—81no—
127C2×C42C22×C411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42C2×Q811yescharacteristic
133C2×C42C2×Q811yescharacteristic
134C2×C42#4111yescharacteristic
135C2×C42#4111yescharacteristic
136C2×C42#4111yescharacteristic
137C2×C42#4111yescharacteristic
138C2×C42#4111yescharacteristic
139#108—82no—
140#108—82no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8—21no—
144C4⋊C8—21no—
145C4⋊C8—21no—
146C4⋊C8—21no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—21no—
150C4⋊C8—21no—
151C4⋊C8—21no—
152C4⋊C8—21no—
153C4⋊C8—41no—
154C4⋊C8—41no—
155C4⋊C8—41no—
156C4⋊C8—41no—
157C4⋊C8—41no—
158C4⋊C8—41no—
159C4⋊C8—41no—
160C4⋊C8—41no—
161C4⋊C8C22⋊C412yes—
162C4⋊C8C22⋊C412yes—
163C4⋊C8C4⋊C412yes—
164C4⋊C8C4⋊C412yes—
165C22⋊C8—22no—
166C22⋊C8—22no—
167C22⋊C8—22no—
168C22⋊C8—22no—
169C22⋊C8—41no—
170C22⋊C8—41no—
171#97—22no—
172#97—41no—
173C4×C8—22no—
174C4×C8—22no—
175C4×C8—22no—
176C4×C8—41no—
177#95—22no—
178#95—22no—
179C24C22⋊Q811yescharacteristic
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—24no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—81no—
192C22×C4—82no—
193C22×C4—82no—
194C22×C4#11711yescharacteristic
195C22×C4C4×D411yescharacteristic
196C22×C4C4×Q811yescharacteristic
197C22×C4#12011yescharacteristic
198C22×C4#12111yescharacteristic
199C22×C4#12111yescharacteristic
200C22×C4#12111yescharacteristic
201C22×C4C22⋊Q811yescharacteristic
202C22×C4C22⋊Q811yescharacteristic
203C22×C4C22⋊Q811yescharacteristic
204C22×C4C22⋊Q811yescharacteristic
205C22×C4#12311yescharacteristic
206C22×C4#12511yescharacteristic
207C22×C4#12611yescharacteristic
208C22×C4#12611yescharacteristic
209C22×C4C4⋊Q811yescharacteristic
210C22×C4C4⋊Q811yescharacteristic
211C22×C4C4⋊Q811yescharacteristic
212M16—41no—
213M16—41no—
214M16—41no—
215M16—41no—
216M16—44no—
217M16—44no—
218M16—41no—
219M16—42no—
220M16—41no—
221M16—41no—
222M16—42no—
223M16—41no—
224M16—82no—
225M16—81no—
226M16—81no—
227M16—81no—
228M16—81no—
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—22no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—42no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—82no—
243C2×C8—82no—
244C2×C8—82no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C42—82no—
250C42—81no—
251C42—81no—
252C42C2×C22⋊C411yescharacteristic
253C42C2×C4⋊C411yescharacteristic
254C42C4×D411yescharacteristic
255C42C4×Q811yescharacteristic
256C42#12111yescharacteristic
257C42#12111yescharacteristic
258C42#12111yescharacteristic
259C42C22⋊Q811yescharacteristic
260C42C22⋊Q811yescharacteristic
261C42C22⋊Q811yescharacteristic
262C42#12311yescharacteristic
263C42#12311yescharacteristic
264C42#12411yescharacteristic
265C42#12511yescharacteristic
266C42#12511yescharacteristic
267C42#12511yescharacteristic
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—41no—
273C23#38711yescharacteristic
274C23#39511yescharacteristic
275C23#39611yescharacteristic
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—44no—
303C2×C4—82no—
304C2×C4—84no—
305C2×C4—82no—
306C2×C4—84no—
307C2×C4#38011yescharacteristic
308C2×C4#38611yescharacteristic
309C2×C4#38711yescharacteristic
310C2×C4#38911yescharacteristic
311C2×C4#38911yescharacteristic
312C2×C4#39111yescharacteristic
313C2×C4#39411yescharacteristic
314C2×C4#39611yescharacteristic
315C2×C4#39811yescharacteristic
316C2×C4#39811yescharacteristic
317C2×C4#39811yescharacteristic
318C2×C4#40011yescharacteristic
319C2×C4#45311yescharacteristic
320C2×C4#45411yescharacteristic
321C2×C4#45511yescharacteristic
322C2×C4#45611yescharacteristic
323C2×C4#47111yescharacteristic
324C2×C4#47111yescharacteristic
325C2×C4#47211yescharacteristic
326C2×C4#47311yescharacteristic
327C8—42no—
328C8—42no—
329C8—42no—
330C8—42no—
331C8—82no—
332C8—82no—
333C8—82no—
334C8—81no—
335C8—81no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22#144111yescharacteristic
348C22#206011yescharacteristic
349C22#206111yescharacteristic
350C22#206311yescharacteristic
351C22#206411yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—84no—
361C4#181511yescharacteristic
362C4#181611yescharacteristic
363C4#188211yescharacteristic
364C4#188311yescharacteristic
365C4#203711yescharacteristic
366C4#203811yescharacteristic
367C4#206311yescharacteristic
368C4#206411yescharacteristic
369C2—21no—
370C2—21no—
371C2—21no—
372C2—21no—
373C2—41no—
374C2#1121711yescharacteristic
375C2#1124211yescharacteristic
376C2#1124811yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C22⋊Q8
LCSG ▷ C2×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
21. #1574C4semidirect

Maximal subgroups

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

subgroupindex
#219932
#178092
#108472
#80982
C2×#13032
#80042