Finite Group Databasebeta

← #127620

Subgroups — #127620

Order 512.

720 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1801C211yesmaximal, characteristic
2C2×#1461C211yesmaximal, characteristic
3#13942C211yesmaximal, characteristic
4#7251C214yesmaximal
5C22×C4×Q8C2211yescharacteristic
6C2×#431—21no—
7C2×#423—21no—
8C22×#101—21no—
9C2×#391—21no—
10C2×#382C411yescharacteristic
11C2×C4×C4⋊C4C411yescharacteristic
12#1801—22no—
13#1801C414yescomplemented
14C2×#343C2212yes—
15#1461—22no—
16#1461C414yescomplemented
17C2×#324—21no—
18C2×#324C2212yes—
19#1441C412yes—
20#1441C2212yes—
21C4⋊#95—21no—
22#1434—21no—
23#1434—21no—
24C23×Q8D411yescharacteristic
25C22×M16—22no—
26C2×C4×Q8—41no—
27C2×C4×Q8—41no—
28C2×C4×Q8—42no—
29C2×C4×Q8C2×C412yes—
30C2×C4×Q8C2×C412yes—
31C2×C4×Q8D412yes—
32C2×C4×Q8Q812yes—
33C22×C4⋊C4—21no—
34C22×C4⋊C4C2×C412yes—
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4D411yescharacteristic
37C22×C4⋊C4C2311yescharacteristic
38C22×C42—21no—
39C22×C42C2×C411yescharacteristic
40C22×C42D411yescharacteristic
41C22×C42D411yescharacteristic
42#431—24no—
43#431—42no—
44#423—24no—
45#423—42no—
46C2×C4⋊C8—41no—
47C2×C4⋊C8—41no—
48C2×C4⋊C8—41no—
49C2×#101—24no—
50C2×#101—42no—
51C2×#101—42no—
52C2×C22⋊C8—22no—
53C2×C4×C8—41no—
54#391—42no—
55#391—42no—
56#382—42no—
57C4×C4⋊C4—41no—
58C4×C4⋊C4—41no—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95C2×C412yes—
66#343—22no—
67#343C2×C414yes—
68#324—24no—
69#324—22no—
70#324C2×C414yes—
71C22×Q8—41no—
72C22×Q8—41no—
73C22×Q8—41no—
74C22×Q8—41no—
75C22×Q8C22⋊C412yes—
76C22×Q8C4⋊C412yes—
77C22×Q8D812yes—
78C22×Q8SD1612yes—
79C22×Q8SD1612yes—
80C22×Q8Q1612yes—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22×C411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C2×M16—24no—
91C2×M16—42no—
92C2×M16—42no—
93C2×M16—42no—
94C2×M16—42no—
95C2×M16—42no—
96C22×C8—42no—
97C22×C8—42no—
98C4×Q8—42no—
99C4×Q8—42no—
100C4×Q8—42no—
101C4×Q8—42no—
102C4×Q8—42no—
103C4×Q8—42no—
104C4×Q8—42no—
105C4×Q8—42no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—44no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4C4212yes—
118C2×C4⋊C4C22⋊C412yes—
119C2×C4⋊C4C4⋊C412yes—
120C2×C4⋊C4C4⋊C412yes—
121C2×C4⋊C4C22×C411yescharacteristic
122C2×C4⋊C4C22×C411yescharacteristic
123C2×C4⋊C4C2×D411yescharacteristic
124C2×C4⋊C4C2×Q811yescharacteristic
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—22no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—22no—
133C2×C42—22no—
134C2×C42—42no—
135C2×C42—42no—
136C2×C42—42no—
137C2×C42C4212yes—
138C2×C42C22⋊C412yes—
139C2×C42C22⋊C412yes—
140C2×C42C4⋊C412yes—
141C2×C42C4⋊C412yes—
142C2×C42C4⋊C412yes—
143C4⋊C8—44no—
144C4⋊C8—44no—
145C4⋊C8—44no—
146#101—48no—
147C22⋊C8—24no—
148C22⋊C8—42no—
149C4×C8—44no—
150#95—42no—
151#95—44no—
152#95—44no—
153#95—41no—
154#95—41no—
155#95—42no—
156#95—44no—
157#95—42no—
158#95—42no—
159C24C2×C22⋊C411yescharacteristic
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8#10214yes—
173C2×Q8#10314yes—
174C2×Q8#10614yes—
175C2×Q8#10714yes—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
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—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—24no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4#9512yes—
213C22×C4#9512yes—
214C22×C4#9512yes—
215C22×C4#9512yes—
216C22×C4#9911yescharacteristic
217C22×C4#9911yescharacteristic
218C22×C4#9912yes—
219C22×C4#10212yes—
220C22×C4#10212yes—
221C22×C4#10312yes—
222C22×C4#10312yes—
223C22×C4C2×C4211yescharacteristic
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C4⋊C411yescharacteristic
227C22×C4C2×C4⋊C411yescharacteristic
228C22×C4C2×C4⋊C411yescharacteristic
229C22×C4C2×D811yescharacteristic
230C22×C4C2×SD1611yescharacteristic
231C22×C4C2×SD1611yescharacteristic
232C22×C4C2×Q1611yescharacteristic
233M16—44no—
234M16—44no—
235M16—82no—
236M16—82no—
237C2×C8—44no—
238C2×C8—44no—
239C2×C8—82no—
240C2×C8—82no—
241C2×C8—84no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—84no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—22no—
260C42—22no—
261C42—22no—
262C42—22no—
263C42—22no—
264C42—22no—
265C42—84no—
266C23—22no—
267C23—22no—
268C23C2×#9511yescharacteristic
269C23C2×#9911yescharacteristic
270C23C2×#9911yescharacteristic
271C23C2×#10211yescharacteristic
272C23C2×#10311yescharacteristic
273C23#42111yescharacteristic
274C23#42111yescharacteristic
275Q8—44no—
276Q8—44no—
277Q8—44no—
278Q8—44no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4#34014yes—
338C2×C4#34014yes—
339C2×C4#34212yes—
340C2×C4#34212yes—
341C2×C4C2×#10211yescharacteristic
342C2×C4C2×#10311yescharacteristic
343C2×C4C2×#10611yescharacteristic
344C2×C4C2×#10711yescharacteristic
345C8—84no—
346C8—84no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22#151911yescharacteristic
356C22#151911yescharacteristic
357C22#152011yescharacteristic
358C22#152011yescharacteristic
359C22#173311yescharacteristic
360C22C2×#34011yescharacteristic
361C22C2×#34211yescharacteristic
362C22#212012yes—
363C22#213312yes—
364C4—22no—
365C4—22no—
366C4—21no—
367C4—24no—
368C4—24no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—42no—
376C4—41no—
377C4—84no—
378C4—162no—
379C2—21no—
380C2—21no—
381C2—21no—
382C2—21no—
383C2#802311yescharacteristic
384C2#1187912yes—
385C2#1190512yes—
386C2#1257311yescharacteristic
387C2#1258611yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #1801C4semidirect
16. #1461C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18012
C2×#14612
#139422
#72512