Finite Group Databasebeta

← #118778

Subgroups — #118778

Order 512.

561 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21841C211yesmaximal, characteristic
2#17746C211yesmaximal, characteristic
3C2×#1546C211yesmaximal, characteristic
4#8121C211yesmaximal, characteristic
5#8093C211yesmaximal, characteristic
6#8058C211yesmaximal, characteristic
7#7858C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14#2283C2211yescharacteristic
15#1993—22no—
16#1988—22no—
17#1824—22no—
18#1558—22no—
19#1548—22no—
20#1548C414yescomplemented
21#1546—22no—
22#1546—22no—
23C2×#332—22no—
24#1307—22no—
25#1303—22no—
26#1298—22no—
27#1274—22no—
28C23×Q8—21no—
29C22×M16—21no—
30C2×C4⋊Q8—21no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8C2×C412yes—
40C2×C4⋊Q8D412yes—
41C2×C4⋊Q8D412yes—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C42C2311yescharacteristic
45#432—21no—
46#432—21no—
47#432—21no—
48#432—21no—
49#432—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53#431—21no—
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431C2×C412yes—
60#431D412yes—
61#423—21no—
62#423—21no—
63#423—21no—
64#423—21no—
65#423C2×C412yes—
66#423D412yes—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71#407—21no—
72#407—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8D412yes—
75C4×M16—21no—
76C4×M16—21no—
77C2×#97—21no—
78C2×C4×C8—21no—
79#383—21no—
80#382—21no—
81#381—21no—
82#380—21no—
83C4×C4⋊C4D412yes—
84C4×C22⋊C4—21no—
85#376—21no—
86#340—28no—
87#333—28no—
88#332—28no—
89C22×Q8—22no—
90C22×Q8—22no—
91C22×Q8—22no—
92C22×Q8—41no—
93C22×Q8—41no—
94C22×Q8—41no—
95C22×Q8—82no—
96C22×Q8—82no—
97C23×C4—41no—
98C23×C4C2×D411yescharacteristic
99C23×C4#4111yescharacteristic
100C23×C4#4111yescharacteristic
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—41no—
104C2×M16—41no—
105C2×M16—82no—
106C22×C8—22no—
107C22×C8—41no—
108C22×C8—41no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—41no—
119C4⋊Q8—41no—
120C4⋊Q8—41no—
121C4⋊Q8—41no—
122C4⋊Q8—41no—
123C4⋊Q8—41no—
124C4⋊Q8C22⋊C414yes—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C22⋊C4—41no—
139C2×C42—22no—
140C2×C42C22×C411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42#4111yescharacteristic
147C2×C42#4111yescharacteristic
148C2×C42#4111yescharacteristic
149C2×C42#4111yescharacteristic
150C2×C42#4111yescharacteristic
151C2×C42#4111yescharacteristic
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—41no—
159C4⋊C8—41no—
160C4⋊C8—41no—
161C4⋊C8—41no—
162C4⋊C8—41no—
163C4⋊C8—41no—
164C22⋊C8—22no—
165C22⋊C8—22no—
166C22⋊C8—22no—
167C22⋊C8—22no—
168C22⋊C8—41no—
169C22⋊C8—41no—
170C22⋊C8—41no—
171C22⋊C8—41no—
172#97—41no—
173#97—41no—
174#97—41no—
175#97—41no—
176C4×C8—22no—
177C4×C8—22no—
178C4×C8—41no—
179C4×C8—41no—
180#95—22no—
181#95—41no—
182#95—41no—
183#95—41no—
184C24#12411yescharacteristic
185C2×Q8—24no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—84no—
190C2×Q8—82no—
191C2×Q8—82no—
192C2×Q8—84no—
193C2×Q8—82no—
194C2×Q8—82no—
195C2×Q8—84no—
196C2×Q8—82no—
197C2×Q8—82no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—84no—
207C22×C4#11711yescharacteristic
208C22×C4C4×D411yescharacteristic
209C22×C4C4×D411yescharacteristic
210C22×C4#12011yescharacteristic
211C22×C4#12111yescharacteristic
212C22×C4#12111yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4#12111yescharacteristic
216C22×C4#12311yescharacteristic
217C22×C4#12311yescharacteristic
218C22×C4#12311yescharacteristic
219C22×C4#12311yescharacteristic
220C22×C4#12311yescharacteristic
221C22×C4#12311yescharacteristic
222C22×C4#12611yescharacteristic
223C22×C4#12611yescharacteristic
224C22×C4#12611yescharacteristic
225M16—82no—
226M16—82no—
227M16—82no—
228M16—82no—
229C2×C8—44no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—82no—
236C2×C8—82no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—44no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C22⋊C4—82no—
247C42—42no—
248C42—42no—
249C42C2×C22⋊C411yescharacteristic
250C42#11711yescharacteristic
251C42C4×D411yescharacteristic
252C42C4×D411yescharacteristic
253C42#12011yescharacteristic
254C42#12111yescharacteristic
255C42#12111yescharacteristic
256C42#12111yescharacteristic
257C42#12311yescharacteristic
258C42#12311yescharacteristic
259C42#12311yescharacteristic
260C42#12311yescharacteristic
261C42#12411yescharacteristic
262C42#12411yescharacteristic
263C42#12411yescharacteristic
264C42#12611yescharacteristic
265C23—22no—
266C23—22no—
267C23—41no—
268C23#38811yescharacteristic
269C23#39411yescharacteristic
270C23#39711yescharacteristic
271Q8—84no—
272Q8—84no—
273Q8—84no—
274Q8—82no—
275Q8—82no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—44no—
296C2×C4—84no—
297C2×C4—84no—
298C2×C4—84no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4#38111yescharacteristic
303C2×C4#38611yescharacteristic
304C2×C4#38611yescharacteristic
305C2×C4#38811yescharacteristic
306C2×C4#38811yescharacteristic
307C2×C4#38811yescharacteristic
308C2×C4#39411yescharacteristic
309C2×C4#39411yescharacteristic
310C2×C4#39411yescharacteristic
311C2×C4#39711yescharacteristic
312C2×C4#39711yescharacteristic
313C2×C4#39711yescharacteristic
314C2×C4#47112yes—
315C2×C4#47212yes—
316C8—82no—
317C8—82no—
318C8—82no—
319C8—82no—
320C8—82no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—44no—
329C22#144011yescharacteristic
330C22#205112yes—
331C22#208212yes—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—84no—
342C4—162no—
343C4#190512yes—
344C4#193112yes—
345C4#204511yescharacteristic
346C4#204611yescharacteristic
347C2—22no—
348C2—22no—
349C2—41no—
350C2#1116311yescharacteristic
351C2#1118211yescharacteristic
352C2#1118411yescharacteristic
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
20. #1548C4semidirect

Maximal subgroups

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

subgroupindex
#218412
#177462
C2×#15462
#81212
#80932
#80582
#78582