Finite Group Databasebeta

← #380768

Subgroups — #380768

Order 512.

1030 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic
2C2×#1998C211yesmaximal, characteristic
3C2×#1997C211yesmaximal, characteristic
4#21314C211yesmaximal, characteristic
5#21313C211yesmaximal, characteristic
6C2×#1838C211yesmaximal, characteristic
7#20367C211yesmaximal, characteristic
8#11366C214yesmaximal
9#11365C214yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×#125C2211yescharacteristic
12C2×#501—24no—
13C2×#501—22no—
14C2×#501—22no—
15C2×#501C418yescomplemented
16C22×#107C2212yes—
17C22×#106C2212yes—
18C2×#423C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21C22×#97C2211yescharacteristic
22C4×#117C2211yescharacteristic
23#1998—24no—
24#1997—24no—
25#1989—22no—
26#1989—22no—
27#1988—22no—
28#1987—22no—
29#1838—24no—
30#1746—22no—
31#1745—22no—
32C2×#343C2214yes—
33C2×#343C2212yes—
34C2×#343C2212yes—
35C2×#341C2214yes—
36#1736C2214yes—
37#1736C2212yes—
38#1736C2212yes—
39#1729C2214yes—
40C23×Q8—21no—
41C22×M16C2312yes—
42C22×M16C2312yes—
43C22×M16C2312yes—
44C23×C8C2312yes—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8C2×C414yes—
53C2×#125—22no—
54C2×#125—21no—
55C2×#125—21no—
56C2×#125—22no—
57C2×#125—21no—
58C2×#125—21no—
59C2×#125—22no—
60C2×#125C2×C414yes—
61C2×#117C2312yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C42C2311yescharacteristic
69#501—216no—
70#501—216no—
71C2×#107—24no—
72C2×#107—24no—
73C2×#107C2×C418yes—
74C2×#106—24no—
75C2×#106—24no—
76C2×#106C2×C418yes—
77#423—22no—
78#423—22no—
79#423—22no—
80#423—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C4×M16—22no—
86C4×M16D414yes—
87C2×#97—21no—
88C2×#97C2×C414yes—
89C2×#97D414yes—
90C2×#97D412yes—
91C2×#97D412yes—
92C2×C4×C8—21no—
93C4×C4⋊C4D412yes—
94C4×C4⋊C4D412yes—
95C4×C22⋊C4—21no—
96C4×C22⋊C4—21no—
97#376—21no—
98C43—21no—
99#343—28no—
100#343—24no—
101#343—24no—
102#341—28no—
103#340—28no—
104#340—24no—
105#340—24no—
106#336—28no—
107C22×Q8—24no—
108C22×Q8—41no—
109C22×Q8—41no—
110C22×Q8—42no—
111C22×Q8—84no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4#4111yescharacteristic
116C23×C4#4111yescharacteristic
117C23×C4C2411yescharacteristic
118C2×M16—22no—
119C2×M16—24no—
120C2×M16—24no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—28no—
126C2×M16—28no—
127C2×M16C2×D414yes—
128C22×C8—24no—
129C22×C8—24no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8C22×C414yes—
134C22×C8C2×D414yes—
135C22×C8C2×D414yes—
136C4⋊Q8—24no—
137C4⋊Q8—24no—
138C4⋊Q8—24no—
139C4⋊Q8—24no—
140C4⋊Q8—42no—
141C4⋊Q8—42no—
142C4⋊Q8—42no—
143C4⋊Q8—42no—
144#125—24no—
145#125—24no—
146#125—24no—
147#125—24no—
148#125—42no—
149#125—42no—
150#125—42no—
151#125—42no—
152#117—24no—
153#117—24no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C414yes—
184C2×C4⋊C4C2×D414yes—
185C2×C22⋊C4—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42C22×C412yes—
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D412yes—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D412yes—
196C2×C42#4112yes—
197#107—216no—
198#106—216no—
199C4⋊C8—42no—
200C4⋊C8—42no—
201C4⋊C8—42no—
202C4⋊C8—42no—
203#97—24no—
204#97C22⋊C418yes—
205#97C22⋊C418yes—
206C4×C8—24no—
207#95—22no—
208#95—22no—
209C24C2×#4111yescharacteristic
210C2×Q8—44no—
211C2×Q8—44no—
212C2×Q8—84no—
213C2×Q8—84no—
214C2×Q8—88no—
215C2×Q8—88no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4—84no—
231C22×C4C4×D412yes—
232C22×C4C4×D412yes—
233C22×C4#12111yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12412yes—
238C22×C4#12412yes—
239C22×C4C23×C411yescharacteristic
240C22×C4C22×D411yescharacteristic
241C22×C4C22×D411yescharacteristic
242C22×C4C22×D411yescharacteristic
243C22×C4C22×D411yescharacteristic
244C22×C4C2×#4111yescharacteristic
245C22×C42−1+414yes—
246C22×C42−1+414yes—
247M16—24no—
248M16—24no—
249M16—44no—
250M16—44no—
251M16—44no—
252M16—44no—
253M16—44no—
254C2×C8—28no—
255C2×C8—28no—
256C2×C8—24no—
257C2×C8—24no—
258C2×C8—24no—
259C2×C8—44no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—44no—
263C2×C8—44no—
264C2×C8C2×C22⋊C418yes—
265C4⋊C4—22no—
266C4⋊C4—24no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C22⋊C4—44no—
286C42—44no—
287C42—82no—
288C42—82no—
289C42C2×C22⋊C412yes—
290C42C2×C22⋊C412yes—
291C42C4×D412yes—
292C42C4×D412yes—
293C42#12112yes—
294C42#12112yes—
295C42#12712yes—
296C42#12712yes—
297C23—22no—
298C23—22no—
299C23C2×C4×D411yescharacteristic
300C23C2×#12111yescharacteristic
301C23C2×#12411yescharacteristic
302C23#54114yes—
303Q8—88no—
304Q8—88no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—84no—
322C2×C4—82no—
323C2×C4—84no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—84no—
328C2×C4#39012yes—
329C2×C4#39012yes—
330C2×C4#39012yes—
331C2×C4#39012yes—
332C2×C4C22×C22⋊C411yescharacteristic
333C2×C4C2×C4×D411yescharacteristic
334C2×C4#51914yes—
335C2×C4C2×#12111yescharacteristic
336C2×C4C2×#12711yescharacteristic
337C2×C4#53614yes—
338C2×C4#53612yes—
339C2×C4#53612yes—
340C2×C4#53612yes—
341C2×C4#53612yes—
342C2×C4#53612yes—
343C2×C4#53612yes—
344C2×C4#54114yes—
345C8—28no—
346C8—44no—
347C8—82no—
348C8—82no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—28no—
354C22C2×#39011yescharacteristic
355C22#232712yes—
356C22#262112yes—
357C22#264212yes—
358C22#306812yes—
359C22#306912yes—
360C4—22no—
361C4—22no—
362C4—22no—
363C4—22no—
364C4—82no—
365C4—84no—
366C4—82no—
367C4—84no—
368C4#231012yes—
369C4#232712yes—
370C4#262112yes—
371C4#265811yescharacteristic
372C4#265811yescharacteristic
373C2—22no—
374C2—22no—
375C2#1427411yescharacteristic
376C2#2124911yescharacteristic
377C2#2125011yescharacteristic
378C2#2988914yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
15. C2×#501C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
C2×#19982
C2×#19972
#213142
#213132
C2×#18382
#203672
#113662
#113652