Finite Group Databasebeta

← #108616

Subgroups — #108616

Order 512.

852 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#407C211yesmaximal, characteristic
2#10719C212yesmaximal
3#8104C212yesmaximal
4#8055C211yesmaximal, characteristic
5#8049C211yesmaximal, characteristic
6C23×C42C2211yescharacteristic
7C2×#407C2212yes—
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×C4×M16C411yescharacteristic, complemented
11C2×C4×M16C411yescharacteristic, complemented
12#2280C2212yes—
13#1749—22no—
14C4×C22⋊C8—22no—
15C2×#339—22no—
16C2×#339—22no—
17#1728C411yescharacteristic, complemented
18#1728C411yescharacteristic, complemented
19#1305C418yescomplemented
20#1298—22no—
21#1297—22no—
22C24×C4—21no—
23C24×C4C2311yescharacteristic
24C22×M16—22no—
25C22×M16C2×C411yescharacteristic
26C22×M16C2×C411yescharacteristic
27C22×M16C2×C411yescharacteristic
28C22×M16C2×C411yescharacteristic
29C2×#117—22no—
30C22×C42—22no—
31C22×C42—21no—
32C22×C42—42no—
33C22×C42—42no—
34C22×C42C2×C411yescharacteristic
35C22×C42C2×C411yescharacteristic
36C22×C42D412yes—
37C22×C42D412yes—
38#407—21no—
39#407—21no—
40#407—22no—
41#407—22no—
42#407—22no—
43#407—22no—
44#407C2×C414yes—
45#407C2×C414yes—
46#407D412yes—
47#407Q812yes—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C4×M16—42no—
53C4×M16—42no—
54C2×#97—21no—
55C2×#97—21no—
56C2×C4×C8—21no—
57C2×C4×C8—21no—
58#381—22no—
59#380—22no—
60#380C2×C414yes—
61#379—22no—
62C4×C22⋊C4—22no—
63C4×C22⋊C4—22no—
64#339—44no—
65#339—44no—
66#336—22no—
67#336—22no—
68C25#4111yescharacteristic
69C23×C4—21no—
70C23×C4—22no—
71C23×C4—21no—
72C23×C4—22no—
73C23×C4—41no—
74C23×C4—41no—
75C23×C4—41no—
76C23×C4—41no—
77C23×C4—48no—
78C23×C4—44no—
79C23×C4C22×C411yescharacteristic
80C23×C4C22×C412yes—
81C23×C4C2×D412yes—
82C23×C4C2×D411yescharacteristic
83C23×C4C2×Q811yescharacteristic
84C23×C4#4112yes—
85C23×C4#4111yescharacteristic
86C23×C4#4111yescharacteristic
87C23×C4#4111yescharacteristic
88C23×C4#4112yes—
89C2×M16—21no—
90C2×M16—21no—
91C2×M16—21no—
92C2×M16—21no—
93C2×M16—24no—
94C2×M16—42no—
95C2×M16—41no—
96C2×M16—41no—
97C2×M16—41no—
98C2×M16—41no—
99C2×M16—41no—
100C2×M16—41no—
101C2×M16—41no—
102C2×M16—41no—
103C2×M16—42no—
104C2×M16—42no—
105C2×M16—42no—
106C2×M16—42no—
107C2×M16C4212yes—
108C2×M16C4212yes—
109C2×M16C4⋊C412yes—
110C2×M16C4⋊C412yes—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—42no—
116#117—42no—
117#117—42no—
118#117—42no—
119#117—42no—
120C2×C4⋊C4—24no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—42no—
124C2×C42—21no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—24no—
128C2×C42—24no—
129C2×C42—21no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—41no—
133C2×C42—41no—
134C2×C42—42no—
135C2×C42—42no—
136C2×C42—48no—
137C2×C42—44no—
138C2×C42—42no—
139C2×C42C22⋊C412yes—
140C2×C42C22⋊C411yescharacteristic
141C2×C42C22⋊C411yescharacteristic
142C22⋊C8—24no—
143C22⋊C8—24no—
144C22⋊C8—24no—
145C22⋊C8—24no—
146C22⋊C8—24no—
147C22⋊C8—24no—
148C22⋊C8—42no—
149C22⋊C8—42no—
150#97—42no—
151#97—42no—
152C4×C8—42no—
153C4×C8—42no—
154#95—24no—
155#95—42no—
156C24—41no—
157C24—41no—
158C24—44no—
159C24#11711yescharacteristic
160C24#11712yes—
161C24#12312yes—
162C24#12411yescharacteristic
163C24#12511yescharacteristic
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—24no—
174C22×C4—28no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—432no—
202C22×C4—44no—
203C22×C4C2×C4211yescharacteristic
204C22×C4C2×C22⋊C411yescharacteristic
205C22×C4C2×C4⋊C411yescharacteristic
206C22×C4#11711yescharacteristic
207C22×C4#11711yescharacteristic
208C22×C4#11711yescharacteristic
209C22×C4#11712yes—
210C22×C4C4×D412yes—
211C22×C4C4×D412yes—
212C22×C4C4×D412yes—
213C22×C4C4×Q812yes—
214C22×C4#12112yes—
215C22×C4C22⋊Q812yes—
216C22×C4#12312yes—
217C22×C4#12411yescharacteristic
218C22×C4#12511yescharacteristic
219C22×C4#12612yes—
220C22×C4#12612yes—
221M16—42no—
222M16—42no—
223M16—42no—
224M16—42no—
225M16—42no—
226M16—42no—
227M16—42no—
228M16—42no—
229M16—44no—
230M16—44no—
231M16—82no—
232M16—82no—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—44no—
246C2×C8—82no—
247C2×C8—82no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C22⋊C4—84no—
251C22⋊C4—82no—
252C22⋊C4—82no—
253C22⋊C4—84no—
254C42—24no—
255C42—21no—
256C42—21no—
257C42—22no—
258C42—22no—
259C42—44no—
260C42—44no—
261C42—42no—
262C42—82no—
263C42—82no—
264C42#9912yes—
265C42#9912yes—
266C23—22no—
267C23—21no—
268C23—22no—
269C23—21no—
270C23—24no—
271C23—24no—
272C23—24no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—41no—
277C23—41no—
278C23—42no—
279C23—42no—
280C23—48no—
281C23—48no—
282C23#37611yescharacteristic
283C23#38111yescharacteristic
284C23#38211yescharacteristic
285C23#38712yes—
286C23#38812yes—
287C2×C4—24no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
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—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—22no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—416no—
319C2×C4—48no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—416no—
323C2×C4—48no—
324C2×C4—42no—
325C2×C4—84no—
326C2×C4—84no—
327C2×C4—84no—
328C2×C4C4×C22⋊C411yescharacteristic
329C2×C4C4×C4⋊C411yescharacteristic
330C2×C4#38011yescharacteristic
331C2×C4#38311yescharacteristic
332C2×C4#38712yes—
333C2×C4#38812yes—
334C2×C4C2×#9911yescharacteristic
335C2×C4C2×#9911yescharacteristic
336C2×C4#41011yescharacteristic
337C2×C4#41011yescharacteristic
338C8—42no—
339C8—42no—
340C8—42no—
341C8—42no—
342C8—84no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—21no—
347C22—21no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—44no—
354C22—44no—
355C22—44no—
356C22—42no—
357C22—48no—
358C22—42no—
359C22—44no—
360C22#143411yescharacteristic
361C22#181911yescharacteristic
362C22#182011yescharacteristic
363C22#193214yes—
364C4—24no—
365C4—22no—
366C4—21no—
367C4—21no—
368C4—22no—
369C4—44no—
370C4—42no—
371C4—164no—
372C4C4×#9911yescharacteristic
373C4C4×#9911yescharacteristic
374C4#179911yescharacteristic
375C4#179911yescharacteristic
376C2—21no—
377C2—24no—
378C2—21no—
379C2—41no—
380C2—41no—
381C2—42no—
382C2#1080111yescharacteristic
383C2#1082612yes—
subgroupG/H
centerC2×C4#380
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
socleC22#1434
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
10. C2×C4×M16C4semidirect
11. C2×C4×M16C4semidirect
17. #1728C4semidirect
18. #1728C4semidirect
19. #1305C4semidirect

Maximal subgroups

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

subgroupindex
C4×#4072
#107192
#81042
#80552
#80492