Finite Group Databasebeta

← #391496

Subgroups — #391496

Order 512.

828 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×#2087C211yesmaximal, characteristic
2#14266C212yesmaximal
3#11315C212yesmaximal
4#11293C212yesmaximal
5#11256C212yesmaximal
6#11232C211yesmaximal, characteristic
7#11227C212yesmaximal
8#11219C212yesmaximal
9#10833C211yesmaximal, characteristic
10C22×#107C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C2×#389C2212yes—
15C2×#389C2212yes—
16C2×#376C2211yescharacteristic
17#2087C418yescomplemented
18C2×#340C2211yescharacteristic
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2211yescharacteristic
26C2×#336C2212yes—
27C2×#336C2212yes—
28#1445—22no—
29#1443—22no—
30#1442—22no—
31#1441—22no—
32C4⋊#95C2212yes—
33C4⋊#95C2212yes—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C23×C8C2312yes—
37C23×C8C2311yescharacteristic
38C23×C8C2311yescharacteristic
39C22×C4⋊C4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4C2312yes—
42C22×C4⋊C4C2312yes—
43C22×C4⋊C4C2312yes—
44C22×C4⋊C4C2312yes—
45C22×C42C2312yes—
46C22×C42C2311yescharacteristic
47C2×#107—41no—
48C2×#107—41no—
49C2×#107—42no—
50C2×#107—41no—
51C2×#107C2×C414yes—
52C2×C4⋊C8—22no—
53C2×C4⋊C8—41no—
54C2×C4⋊C8—41no—
55C2×C4⋊C8—41no—
56C2×C4⋊C8—41no—
57C2×C4⋊C8C2×C414yes—
58#389—24no—
59#389—42no—
60#389—42no—
61#389C2×C418yes—
62#376—41no—
63#376—41no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95—22no—
67C2×#95—22no—
68C2×#95—22no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95D411yescharacteristic
75C2×#95D412yes—
76C2×#95D411yescharacteristic
77#340—42no—
78#340—42no—
79#340—42no—
80#340—42no—
81#340—42no—
82#340—41no—
83#340C2×C414yes—
84#340C2×C418yes—
85#336—42no—
86#336—42no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4C2×D412yes—
92C23×C4C2×D412yes—
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C23×C4C2411yescharacteristic
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8C22×C412yes—
112C22×C8C22×C412yes—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4—82no—
138C2×C4⋊C4—82no—
139C2×C4⋊C4—82no—
140C2×C4⋊C4—82no—
141C2×C4⋊C4C22×C414yes—
142C2×C4⋊C4C22×C414yes—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42C22×C412yes—
154C2×C42#4112yes—
155#107—44no—
156#107—44no—
157C4⋊C8—44no—
158C4⋊C8—44no—
159#95—28no—
160#95—41no—
161#95—42no—
162#95—41no—
163#95—44no—
164#95—82no—
165#95—82no—
166#95—82no—
167#95—82no—
168#95—82no—
169#95—82no—
170#95—82no—
171#95—82no—
172C24C22×D411yescharacteristic
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
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—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4C4×D412yes—
226C22×C4C4×D412yes—
227C22×C4C23×C411yescharacteristic
228C22×C4C2×#4111yescharacteristic
229C22×C4H(5,𝔽2)11yescharacteristic
230C22×C4H(5,𝔽2)11yescharacteristic
231C22×C4H(5,𝔽2)11yescharacteristic
232C22×C4H(5,𝔽2)12yes—
233C22×C42−1+412yes—
234C22×C42−1+412yes—
235C22×C42−1+411yescharacteristic
236C22×C42−1+412yes—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—24no—
240C2×C8—24no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—88no—
271C4⋊C4—88no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—42no—
277C42—42no—
278C42—42no—
279C42—42no—
280C23C2×C4×D411yescharacteristic
281C23#53411yescharacteristic
282C23#53612yes—
283C23#53711yescharacteristic
284C23#53712yes—
285C23#57612yes—
286C23#57611yescharacteristic
287C23#57712yes—
288C23#57811yescharacteristic
289C23#57811yescharacteristic
290C23#57811yescharacteristic
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
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—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4#51811yescharacteristic
346C2×C4#51912yes—
347C2×C4#52012yes—
348C2×C4#52011yescharacteristic
349C2×C4#53811yescharacteristic
350C2×C4#53911yescharacteristic
351C2×C4#54112yes—
352C2×C4#54212yes—
353C2×C4#56011yescharacteristic
354C2×C4#56212yes—
355C2×C4#56212yes—
356C2×C4#56311yescharacteristic
357C2×C4#56311yescharacteristic
358C2×C4#56311yescharacteristic
359C2×C4#56312yes—
360C2×C4#56312yes—
361C2×C4#56312yes—
362C2×C4#56412yes—
363C8—42no—
364C8—42no—
365C8—82no—
366C22#232312yes—
367C22#232511yescharacteristic
368C22#281412yes—
369C22#282211yescharacteristic
370C22#282311yescharacteristic
371C22#294612yes—
372C22#295011yescharacteristic
373C22#295111yescharacteristic
374C22#319812yes—
375C22#320211yescharacteristic
376C22#320311yescharacteristic
377C22#324112yes—
378C22#324212yes—
379C22#324312yes—
380C22#324312yes—
381C22#325812yes—
382C22#325812yes—
383C22#325911yescharacteristic
384C22#325912yes—
385C22#326111yescharacteristic
386C22#326211yescharacteristic
387C22#326212yes—
388C22#326311yescharacteristic
389C4—42no—
390C4—42no—
391C4—84no—
392C4—84no—
393C4—82no—
394C4—82no—
395C4—162no—
396C4—162no—
397C4#237611yescharacteristic
398C4#238111yescharacteristic
399C4#238212yes—
400C4#283011yescharacteristic
401C4#283311yescharacteristic
402C4#283912yes—
403C2#1463411yescharacteristic
404C2#2212512yes—
405C2#2212612yes—
406C2#2219711yescharacteristic
407C2#2219911yescharacteristic
408C2#3275312yes—
409C2#3275412yes—
410C2#3285511yescharacteristic
411C2#3285611yescharacteristic
412C2#3286611yescharacteristic
413C2#3286711yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #2087C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20872
#142662
#113152
#112932
#112562
#112322
#112272
#112192
#108332