Finite Group Databasebeta

← #119173

Subgroups — #119173

Order 512.

536 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17715C211yesmaximal, characteristic
2C2×#1506C211yesmaximal, characteristic
3#17464C211yesmaximal, characteristic
4#8072C211yesmaximal, characteristic
5#8058C211yesmaximal, characteristic
6#7933C211yesmaximal, characteristic
7C2×#1274C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#376C2211yescharacteristic
15#1552—22no—
16#1549C412yescomplemented
17#1546C412yescomplemented
18#1544—22no—
19#1506—22no—
20#1506—22no—
21C2×#328—22no—
22#1485—22no—
23#1481—22no—
24#1479—22no—
25#1300—22no—
26#1298—22no—
27#1283—22no—
28#1274—22no—
29#533—21no—
30#533—21no—
31#533—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533D412yes—
36#533D412yes—
37C2×C4⋊Q8C2×C411yescharacteristic
38C2×C4⋊Q8C2×C411yescharacteristic
39C2×#125—21no—
40C2×C22⋊Q8—21no—
41C2×C22⋊Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44C2×#117—21no—
45C22×C42—21no—
46C22×C42C2311yescharacteristic
47#432—21no—
48#432—21no—
49#432—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431C2×C412yes—
66#431D412yes—
67#423—21no—
68#423C2×C412yes—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72#407—21no—
73#407—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8D412yes—
77C4×M16—21no—
78C4×M16D412yes—
79C2×#97—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82#376—21no—
83#376—21no—
84#376—21no—
85#376D412yes—
86C2×#95—21no—
87C2×#95—21no—
88#332—24no—
89#332—24no—
90#328—28no—
91Q8⋊C8—28no—
92C22×Q8—21no—
93C22×Q8—21no—
94C23×C4—41no—
95C23×C4C2×D411yescharacteristic
96C23×C4#4111yescharacteristic
97C23×C4#4111yescharacteristic
98C2×M16—22no—
99C2×M16—22no—
100C22×C8—22no—
101C22×C8—41no—
102C22×C8—41no—
103C4⋊Q8—21no—
104C4⋊Q8—21no—
105C4⋊Q8—21no—
106C4⋊Q8—21no—
107C4⋊Q8—21no—
108C4⋊Q8—21no—
109C4⋊Q8C22⋊C412yes—
110C4⋊Q8C22⋊C412yes—
111#125—22no—
112#125—22no—
113#125—41no—
114#125—41no—
115C22⋊Q8—22no—
116C22⋊Q8—22no—
117C22⋊Q8—22no—
118C22⋊Q8—22no—
119C22⋊Q8—41no—
120C22⋊Q8—41no—
121C22⋊Q8—41no—
122C22⋊Q8—41no—
123C4×Q8—22no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—41no—
128C4×Q8—41no—
129C4×Q8—41no—
130C4×Q8—41no—
131#117—22no—
132#117—22no—
133#117—41no—
134#117—41no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C22⋊C4—41no—
142C2×C42—24no—
143C2×C42—22no—
144C2×C42—42no—
145C2×C42—41no—
146C2×C42—41no—
147C2×C42C22×C411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42#4111yescharacteristic
154C2×C42#4111yescharacteristic
155C2×C42#4111yescharacteristic
156C2×C42#4111yescharacteristic
157C2×C42#4111yescharacteristic
158C2×C42#4111yescharacteristic
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—41no—
164C4⋊C8—41no—
165C4⋊C8—41no—
166C4⋊C8—41no—
167C4⋊C8—41no—
168C4⋊C8—41no—
169C22⋊C8—22no—
170C22⋊C8—22no—
171C22⋊C8—22no—
172C22⋊C8—22no—
173C22⋊C8—41no—
174C22⋊C8—41no—
175C22⋊C8—41no—
176C22⋊C8—41no—
177C22⋊C8—41no—
178C22⋊C8—41no—
179#97—22no—
180#97—41no—
181#97—41no—
182#97—41no—
183C4×C8—22no—
184C4×C8—22no—
185C4×C8—22no—
186C4×C8—41no—
187C4×C8—41no—
188C4×C8—41no—
189#95—22no—
190#95—22no—
191#95—41no—
192#95—41no—
193C24#12411yescharacteristic
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—82no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—42no—
204C22×C4—44no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4—81no—
209C22×C4—82no—
210C22×C4—82no—
211C22×C4#11711yescharacteristic
212C22×C4C4×D411yescharacteristic
213C22×C4C4×D411yescharacteristic
214C22×C4#12011yescharacteristic
215C22×C4#12111yescharacteristic
216C22×C4#12111yescharacteristic
217C22×C4#12111yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4#12311yescharacteristic
221C22×C4#12311yescharacteristic
222C22×C4#12311yescharacteristic
223C22×C4#12311yescharacteristic
224C22×C4#12311yescharacteristic
225C22×C4#12311yescharacteristic
226C22×C4#12611yescharacteristic
227C22×C4#12611yescharacteristic
228C22×C4#12611yescharacteristic
229M16—42no—
230M16—42no—
231M16—82no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—82no—
238C2×C8—82no—
239C2×C8—82no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C42—24no—
255C42—44no—
256C42—42no—
257C42—42no—
258C42—42no—
259C42C2×C22⋊C411yescharacteristic
260C42#11711yescharacteristic
261C42C4×D411yescharacteristic
262C42C4×D411yescharacteristic
263C42#12011yescharacteristic
264C42#12111yescharacteristic
265C42#12111yescharacteristic
266C42#12111yescharacteristic
267C42#12311yescharacteristic
268C42#12311yescharacteristic
269C42#12311yescharacteristic
270C42#12311yescharacteristic
271C42#12411yescharacteristic
272C42#12411yescharacteristic
273C42#12411yescharacteristic
274C42#12611yescharacteristic
275C23—22no—
276C23—22no—
277C23—41no—
278C23#38811yescharacteristic
279C23#39411yescharacteristic
280C23#39711yescharacteristic
281Q8—44no—
282Q8—82no—
283Q8—82no—
284Q8—81no—
285Q8—81no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—88no—
312C2×C4—82no—
313C2×C4#38111yescharacteristic
314C2×C4#38611yescharacteristic
315C2×C4#38611yescharacteristic
316C2×C4#38811yescharacteristic
317C2×C4#38811yescharacteristic
318C2×C4#38811yescharacteristic
319C2×C4#39411yescharacteristic
320C2×C4#39411yescharacteristic
321C2×C4#39411yescharacteristic
322C2×C4#39711yescharacteristic
323C2×C4#39711yescharacteristic
324C2×C4#39711yescharacteristic
325C2×C4#44312yes—
326C2×C4#44412yes—
327C8—82no—
328C8—82no—
329C8—82no—
330C8—82no—
331C8—82no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—44no—
340C22#144011yescharacteristic
341C22#192912yes—
342C22#208212yes—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—84no—
354C4—82no—
355C4—82no—
356C4#187612yes—
357C4#193711yescharacteristic
358C4#193811yescharacteristic
359C2—22no—
360C2—22no—
361C2—41no—
362C2#1115811yescharacteristic
363C2#1118011yescharacteristic
364C2#1119111yescharacteristic
subgroupG/H
center, socleC22#1440
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#124
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
16. #1549C4semidirect
17. #1546C4semidirect

Maximal subgroups

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

subgroupindex
#177152
C2×#15062
#174642
#80722
#80582
#79332
C2×#12742