Finite Group Databasebeta

← #127622

Subgroups — #127622

Order 512.

742 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1801C211yesmaximal, characteristic
2C2×#1461C211yesmaximal, characteristic
3#13926C211yesmaximal, characteristic
4#7251C214yesmaximal
5C22×C4×Q8C2211yescharacteristic
6C2×#431—21no—
7C2×#423—21no—
8C22×#101—21no—
9C2×#391—21no—
10C2×#384C411yescharacteristic
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—
22C4⋊#95—21no—
23C4⋊#95—21no—
24C23×Q8D411yescharacteristic
25C22×M16—22no—
26C2×C4×Q8—42no—
27C2×C4×Q8—41no—
28C2×C4×Q8—41no—
29C2×C4×Q8C2×C412yes—
30C2×C4×Q8C2×C412yes—
31C2×C4×Q8D412yes—
32C2×C4×Q8Q812yes—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C4⋊C4C2×C412yes—
38C22×C4⋊C4D411yescharacteristic
39C22×C4⋊C4C2311yescharacteristic
40C22×C42—21no—
41C22×C42C2×C411yescharacteristic
42C22×C42D411yescharacteristic
43C22×C42D411yescharacteristic
44#431—24no—
45#431—42no—
46#423—24no—
47#423—42no—
48C2×C4⋊C8—41no—
49C2×C4⋊C8—41no—
50C2×C4⋊C8—41no—
51C2×#101—24no—
52C2×#101—42no—
53C2×#101—42no—
54C2×C22⋊C8—22no—
55C2×C4×C8—41no—
56#391—42no—
57#391—42no—
58#384—41no—
59#384—41no—
60C4×C4⋊C4—42no—
61C2×#95—22no—
62C2×#95—22no—
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—24no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—82no—
124C2×C4⋊C4C4212yes—
125C2×C4⋊C4C22⋊C412yes—
126C2×C4⋊C4C4⋊C412yes—
127C2×C4⋊C4C4⋊C412yes—
128C2×C4⋊C4C22×C411yescharacteristic
129C2×C4⋊C4C22×C411yescharacteristic
130C2×C4⋊C4C2×D411yescharacteristic
131C2×C4⋊C4C2×Q811yescharacteristic
132C2×C42—22no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—42no—
142C2×C42—42no—
143C2×C42—42no—
144C2×C42C4212yes—
145C2×C42C22⋊C412yes—
146C2×C42C22⋊C412yes—
147C2×C42C4⋊C412yes—
148C2×C42C4⋊C412yes—
149C2×C42C4⋊C412yes—
150C4⋊C8—44no—
151C4⋊C8—44no—
152C4⋊C8—44no—
153#101—48no—
154C22⋊C8—24no—
155C22⋊C8—42no—
156C4×C8—44no—
157#95—42no—
158#95—44no—
159#95—42no—
160#95—42no—
161#95—82no—
162#95—82no—
163C24C2×C22⋊C411yescharacteristic
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—42no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8#10214yes—
177C2×Q8#10314yes—
178C2×Q8#10614yes—
179C2×Q8#10714yes—
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—22no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—24no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4#9512yes—
221C22×C4#9512yes—
222C22×C4#9512yes—
223C22×C4#9512yes—
224C22×C4#9912yes—
225C22×C4#9911yescharacteristic
226C22×C4#9911yescharacteristic
227C22×C4#10212yes—
228C22×C4#10212yes—
229C22×C4#10312yes—
230C22×C4#10312yes—
231C22×C4C2×C4211yescharacteristic
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C4⋊C411yescharacteristic
235C22×C4C2×C4⋊C411yescharacteristic
236C22×C4C2×C4⋊C411yescharacteristic
237C22×C4C2×D811yescharacteristic
238C22×C4C2×SD1611yescharacteristic
239C22×C4C2×SD1611yescharacteristic
240C22×C4C2×Q1611yescharacteristic
241M16—44no—
242M16—44no—
243M16—82no—
244M16—82no—
245C2×C8—44no—
246C2×C8—44no—
247C2×C8—82no—
248C2×C8—82no—
249C2×C8—82no—
250C2×C8—82no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—48no—
260C4⋊C4—82no—
261C4⋊C4—84no—
262C4⋊C4—84no—
263C4⋊C4—82no—
264C42—22no—
265C42—22no—
266C42—22no—
267C42—22no—
268C42—22no—
269C42—22no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—84no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23C2×#9511yescharacteristic
282C23C2×#9911yescharacteristic
283C23C2×#9911yescharacteristic
284C23C2×#10211yescharacteristic
285C23C2×#10311yescharacteristic
286C23#42111yescharacteristic
287C23#42111yescharacteristic
288Q8—44no—
289Q8—44no—
290Q8—44no—
291Q8—44no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4#34014yes—
354C2×C4#34014yes—
355C2×C4#34212yes—
356C2×C4#34212yes—
357C2×C4#35512yes—
358C2×C4#35612yes—
359C2×C4C2×#10211yescharacteristic
360C2×C4C2×#10311yescharacteristic
361C2×C4C2×#10611yescharacteristic
362C2×C4C2×#10711yescharacteristic
363C8—84no—
364C8—84no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22#151911yescharacteristic
378C22#151911yescharacteristic
379C22#152011yescharacteristic
380C22#152011yescharacteristic
381C22#173311yescharacteristic
382C22C2×#34011yescharacteristic
383C22C2×#34211yescharacteristic
384C22#212211yescharacteristic
385C22#212311yescharacteristic
386C22C2×#35511yescharacteristic
387C22C2×#35611yescharacteristic
388C4—22no—
389C4—21no—
390C4—24no—
391C4—24no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—42no—
399C4—41no—
400C4—84no—
401C4—162no—
402C4#139414yes—
403C2—21no—
404C2—21no—
405C2—21no—
406C2—21no—
407C2#802311yescharacteristic
408C2#1188211yescharacteristic
409C2#1188311yescharacteristic
410C2#1190711yescharacteristic
411C2#1190811yescharacteristic
412C2#1257511yescharacteristic
413C2C2×#139411yescharacteristic
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
#139262
#72512