Finite Group Databasebeta

← #108593

Subgroups — #108593

Order 512.

748 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1753C211yesmaximal, characteristic
2#10729C212yesmaximal
3#8008C211yesmaximal, characteristic
4#7982C211yesmaximal, characteristic
5#7942C212yesmaximal
6C23×M16C2211yescharacteristic
7C2×#407C2212yes—
8C22×C22⋊C8C2211yescharacteristic
9C22×#97C411yescharacteristic, complemented
10C22×C4×C8C411yescharacteristic, complemented
11#2280C2212yes—
12C2×C4×C22⋊C4C2211yescharacteristic
13#1753—22no—
14#1753—22no—
15C2×#344—22no—
16C2×#344—22no—
17C2×#336C411yescharacteristic, complemented
18C2×#336C411yescharacteristic, complemented
19#1293—22no—
20#1289—22no—
21#1284C418yescomplemented
22C24×C4C2311yescharacteristic
23C22×M16—21no—
24C22×M16—21no—
25C22×M16—22no—
26C22×M16—22no—
27C22×M16—42no—
28C22×M16—42no—
29C22×M16D412yes—
30C22×M16D412yes—
31C23×C8C2×C411yescharacteristic
32C23×C8C2×C411yescharacteristic
33C23×C8C2×C411yescharacteristic
34C23×C8C2×C411yescharacteristic
35C2×#117—22no—
36C22×C22⋊C4—21no—
37C22×C42C2×C411yescharacteristic
38C22×C42C2×C411yescharacteristic
39#407—22no—
40#407—22no—
41#407—22no—
42#407C2×C414yes—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8D412yes—
48C2×C22⋊C8Q812yes—
49C2×#97—21no—
50C2×#97—21no—
51C2×#97—42no—
52C2×C4×C8—21no—
53C2×C4×C8—21no—
54C2×C4×C8—42no—
55#381C2×C414yes—
56#380—22no—
57#380—22no—
58#379—22no—
59C4×C22⋊C4—22no—
60C4×C22⋊C4—22no—
61C4×C22⋊C4—22no—
62C4×C22⋊C4C2×C414yes—
63C2×#95—21no—
64#344—44no—
65#344—44no—
66#336—22no—
67#336—22no—
68C25#4111yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—41no—
72C23×C4—41no—
73C23×C4—41no—
74C23×C4—41no—
75C23×C4C22×C411yescharacteristic
76C23×C4C22×C412yes—
77C23×C4C2×D412yes—
78C23×C4C2×D411yescharacteristic
79C23×C4C2×Q811yescharacteristic
80C23×C4#4112yes—
81C23×C4#4111yescharacteristic
82C23×C4#4111yescharacteristic
83C23×C4#4112yes—
84C23×C4#4111yescharacteristic
85C2×M16—22no—
86C2×M16—22no—
87C2×M16—22no—
88C2×M16—22no—
89C2×M16—24no—
90C2×M16—24no—
91C2×M16—24no—
92C2×M16—42no—
93C2×M16—42no—
94C2×M16—44no—
95C2×M16—48no—
96C2×M16—44no—
97C2×M16—42no—
98C2×M16—42no—
99C2×M16—42no—
100C2×M16—42no—
101C2×M16—42no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—41no—
113C22×C8—41no—
114C22×C8—41no—
115C22×C8—41no—
116C22×C8—41no—
117C22×C8—41no—
118C22×C8—42no—
119C22×C8C22⋊C412yes—
120C22×C8C22⋊C411yescharacteristic
121C22×C8C22⋊C411yescharacteristic
122C22×C8C4⋊C412yes—
123C22×C8C4⋊C412yes—
124#117—42no—
125#117—42no—
126#117—42no—
127#117—42no—
128C2×C4⋊C4—42no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—44no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—42no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—41no—
137C2×C42—41no—
138C2×C42—41no—
139C2×C42—41no—
140C2×C42—42no—
141C2×C42C4212yes—
142C2×C42C4212yes—
143C22⋊C8—24no—
144C22⋊C8—24no—
145C22⋊C8—24no—
146C22⋊C8—24no—
147C22⋊C8—42no—
148C22⋊C8—42no—
149#97—42no—
150#97—42no—
151C4×C8—42no—
152C4×C8—42no—
153#95—24no—
154#95—24no—
155#95—24no—
156C24—42no—
157C24—41no—
158C24—41no—
159C24—42no—
160C24#11711yescharacteristic
161C24C4×D412yes—
162C24#12411yescharacteristic
163C24#12511yescharacteristic
164C24#12612yes—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—44no—
198C22×C4C2×C4211yescharacteristic
199C22×C4C2×C22⋊C411yescharacteristic
200C22×C4C2×C4⋊C411yescharacteristic
201C22×C4#11711yescharacteristic
202C22×C4#11711yescharacteristic
203C22×C4#11711yescharacteristic
204C22×C4#11712yes—
205C22×C4#11712yes—
206C22×C4C4×D412yes—
207C22×C4C4×D412yes—
208C22×C4C4×Q812yes—
209C22×C4#12112yes—
210C22×C4C22⋊Q812yes—
211C22×C4#12312yes—
212C22×C4#12312yes—
213C22×C4#12411yescharacteristic
214C22×C4#12511yescharacteristic
215C22×C4#12612yes—
216M16—44no—
217M16—44no—
218M16—44no—
219M16—44no—
220M16—44no—
221M16—44no—
222M16—82no—
223M16—82no—
224C2×C8—24no—
225C2×C8—24no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—22no—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—44no—
246C2×C8—82no—
247C2×C8—82no—
248C2×C8#9912yes—
249C2×C8#9912yes—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C22⋊C4—48no—
253C22⋊C4—44no—
254C22⋊C4—44no—
255C22⋊C4—84no—
256C22⋊C4—84no—
257C42—42no—
258C42—42no—
259C42—41no—
260C42—41no—
261C42—41no—
262C42—41no—
263C42—82no—
264C42—82no—
265C23—22no—
266C23—22no—
267C23—21no—
268C23—21no—
269C23—22no—
270C23—22no—
271C23—24no—
272C23—22no—
273C23—22no—
274C23—42no—
275C23—41no—
276C23—41no—
277C23—42no—
278C23—42no—
279C23—41no—
280C23—41no—
281C23—42no—
282C23—84no—
283C23—84no—
284C23C4×C22⋊C411yescharacteristic
285C23#38211yescharacteristic
286C23#38311yescharacteristic
287C23#38712yes—
288C23#38812yes—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
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—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—24no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—48no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—84no—
324C2×C4—84no—
325C2×C4—84no—
326C2×C4—84no—
327C2×C4#37611yescharacteristic
328C2×C4C4×C4⋊C411yescharacteristic
329C2×C4#38011yescharacteristic
330C2×C4#38111yescharacteristic
331C2×C4#38712yes—
332C2×C4#38812yes—
333C2×C4C2×#9911yescharacteristic
334C2×C4C2×#9911yescharacteristic
335C2×C4#41011yescharacteristic
336C2×C4#41011yescharacteristic
337C8—22no—
338C8—22no—
339C8—42no—
340C8—42no—
341C8—42no—
342C8—84no—
343C22—24no—
344C22—24no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—82no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22—84no—
362C22#143411yescharacteristic
363C22C4×#9911yescharacteristic
364C22C4×#9911yescharacteristic
365C22#183312yes—
366C22#184112yes—
367C4—22no—
368C4—21no—
369C4—21no—
370C4—22no—
371C4—44no—
372C4—82no—
373C4—164no—
374C4#179911yescharacteristic
375C4#179911yescharacteristic
376C4#181911yescharacteristic
377C4#182011yescharacteristic
378C2—21no—
379C2—21no—
380C2—22no—
381C2—22no—
382C2—42no—
383C2—42no—
384C2#1080111yescharacteristic
385C2#1081711yescharacteristic
386C2#1082511yescharacteristic
subgroupG/H
centerC2×C4#381
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
9. C22×#97C4semidirect
10. C22×C4×C8C4semidirect
17. C2×#336C4semidirect
18. C2×#336C4semidirect
21. #1284C4semidirect

Maximal subgroups

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

subgroupindex
C2×#17532
#107292
#80082
#79822
#79422