Finite Group Databasebeta

← #391778

Subgroups — #391778

Order 512.

1672 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2#14685C211yesmaximal, characteristic, complemented
3#14684C211yesmaximal, characteristic, complemented
4#11353C212yesmaximal, complemented
5#11043C214yesmaximal, complemented
6#10980C212yesmaximal
7#10966C214yesmaximal, complemented
8C22×#123C2212yes—
9C22×#121C2211yescharacteristic
10C23×C42C2211yescharacteristic
11C22×#99C2218yes—
12#2854—22no—
13#2854—22no—
14#2854—22no—
15#2854—22no—
16C2×#390C412yescomplemented
17C2×#389C412yescomplemented
18C2×#388C414yescomplemented
19C2×#386C2212yes—
20C2×#381C2212yes—
21C2×#381C2212yes—
22C2×#380C2212yes—
23C2×#380C2211yescharacteristic
24C2×#342C2218yes—
25C4⋊#95—22no—
26#1434—22no—
27#1434—22no—
28#1434—22no—
29C22⋊#95C2212yes—
30C22⋊#95C2212yes—
31C22⋊#95C2212yes—
32C23×D4C2×C411yescharacteristic
33C23×D4C2312yes—
34C24×C4C2311yescharacteristic
35C24×C4C2311yescharacteristic
36C24×C4C2311yescharacteristic
37C2×#123—24no—
38C2×#123—22no—
39C2×#123—22no—
40C2×#123—48no—
41C2×#121—22no—
42C2×#121—22no—
43C2×#121—44no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C414yes—
47C22×C4⋊C4C2×C412yes—
48C22×C22⋊C4C2×C412yes—
49C22×C22⋊C4C2×C414yes—
50C22×C22⋊C4C2×C414yes—
51C22×C22⋊C4C2314yes—
52C22×C22⋊C4C2314yes—
53C22×C22⋊C4C2312yes—
54C22×C42—44no—
55C22×C42C2×C412yes—
56C22×C42C2×C412yes—
57C22×C42D412yes—
58C22×C42D412yes—
59C22×C42D412yes—
60C22×C42D412yes—
61C2×#99—48no—
62C2×#99—48no—
63C2×#99—48no—
64#390—22no—
65#390—22no—
66#390—22no—
67#390—22no—
68#389—22no—
69#389—22no—
70#389—22no—
71#389—22no—
72#388—24no—
73#388—24no—
74#388—24no—
75#388—24no—
76#386—22no—
77#386—24no—
78#386—22no—
79#381—44no—
80#381—44no—
81#380—22no—
82#380—22no—
83#380—44no—
84C2×#95—24no—
85C2×#95—24no—
86C2×#95—24no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—24no—
91C2×#95C2×C412yes—
92C2×#95C2×C414yes—
93#342—48no—
94C25C22×C412yes—
95C25C2411yescharacteristic
96C22×D4—22no—
97C22×D4—22no—
98C22×D4—28no—
99C22×D4—28no—
100C22×D4—24no—
101C22×D4—24no—
102C22×D4—22no—
103C22×D4—42no—
104C22×D4—44no—
105C23×C4—24no—
106C23×C4—24no—
107C23×C4—24no—
108C23×C4—22no—
109C23×C4—24no—
110C23×C4—22no—
111C23×C4—24no—
112C23×C4—24no—
113C23×C4—44no—
114C23×C4—44no—
115C23×C4C22×C412yes—
116C23×C4C22×C412yes—
117C23×C4C22×C411yescharacteristic
118C23×C4C22×C412yes—
119C23×C4C22×C414yes—
120C23×C4C22×C411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129#123—416no—
130#123—48no—
131#123—48no—
132#121—48no—
133#121—48no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—48no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—48no—
143C2×C4⋊C4—44no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—28no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—28no—
154C2×C22⋊C4—28no—
155C2×C22⋊C4—28no—
156C2×C22⋊C4—28no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—44no—
161C2×C22⋊C4—48no—
162C2×C22⋊C4—48no—
163C2×C42—44no—
164C2×C42—44no—
165C2×C42—44no—
166C2×C42—44no—
167C2×C42—44no—
168C2×C42—44no—
169C2×C42C22⋊C412yes—
170C2×C42C22⋊C412yes—
171C2×C42C22⋊C412yes—
172C2×C42C22⋊C412yes—
173C2×C42C22⋊C412yes—
174C2×C42C22⋊C412yes—
175C2×C42C22⋊C412yes—
176C2×C42C22⋊C412yes—
177#99—432no—
178#95—24no—
179#95—24no—
180#95—24no—
181#95—48no—
182#95—48no—
183#95—48no—
184#95—44no—
185#95—44no—
186#95—44no—
187#95—48no—
188C24—22no—
189C24—24no—
190C24—24no—
191C24—24no—
192C24—42no—
193C24—42no—
194C24—44no—
195C24—44no—
196C24C23×C411yescharacteristic
197C24C22×D411yescharacteristic
198C24C22×D411yescharacteristic
199C24H(5,𝔽2)14yes—
200C24H(5,𝔽2)14yes—
201C24H(5,𝔽2)12yes—
202C242−1+412yes—
203C2×D4—44no—
204C2×D4—44no—
205C2×D4—44no—
206C2×D4—44no—
207C2×D4—44no—
208C2×D4—44no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—44no—
212C2×D4—48no—
213C2×D4—48no—
214C2×D4—416no—
215C2×D4—44no—
216C2×D4—88no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—28no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—416no—
257C22×C4—416no—
258C22×C4—48no—
259C22×C4—48no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—48no—
263C22×C4—48no—
264C22×C4—48no—
265C22×C4—48no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—48no—
281C4⋊C4—48no—
282C4⋊C4—44no—
283C4⋊C4—88no—
284C22⋊C4—48no—
285C22⋊C4—48no—
286C22⋊C4—48no—
287C22⋊C4—48no—
288C22⋊C4—44no—
289C22⋊C4—44no—
290C22⋊C4—48no—
291C22⋊C4—48no—
292C22⋊C4—48no—
293C22⋊C4—48no—
294C22⋊C4—48no—
295C42—42no—
296C42—42no—
297C42—42no—
298C42—42no—
299C42—42no—
300C42—42no—
301C42—42no—
302C42—42no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—44no—
310C23—44no—
311C23—48no—
312C23—44no—
313C23—44no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—44no—
318C23—44no—
319C23—48no—
320C23—84no—
321C23—88no—
322C23—88no—
323C23—84no—
324C23C22×C22⋊C411yescharacteristic
325C23#51812yes—
326C23#51812yes—
327C23#52012yes—
328C23#53412yes—
329C23#53412yes—
330C23#53512yes—
331C23#53512yes—
332C23#53511yescharacteristic
333C23#53611yescharacteristic
334C23#53712yes—
335C23#56018yes—
336C23#56218yes—
337D4—88no—
338D4—84no—
339D4—88no—
340D4—84no—
341C2×C4—28no—
342C2×C4—28no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—48no—
360C2×C4—48no—
361C2×C4—48no—
362C2×C4—48no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—48no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—48no—
377C2×C4—88no—
378C2×C4—84no—
379C2×C4—88no—
380C2×C4—88no—
381C2×C4—88no—
382C2×C4—88no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4—84no—
388C2×C4#41014yes—
389C2×C4#41014yes—
390C22—24no—
391C22—24no—
392C22—24no—
393C22—22no—
394C22—24no—
395C22—24no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—48no—
404C22—48no—
405C22—84no—
406C22—82no—
407C22—84no—
408C22—82no—
409C22—82no—
410C22—168no—
411C22#230811yescharacteristic
412C22#230911yescharacteristic
413C22#231111yescharacteristic
414C22#237714yes—
415C22#281314yes—
416C22#281512yes—
417C22#281612yes—
418C22C2×#41012yes—
419C22C2×#41012yes—
420C4—24no—
421C4—24no—
422C4—42no—
423C4—42no—
424C4—42no—
425C4—42no—
426C4—84no—
427C4—82no—
428C4—164no—
429C4—164no—
430C2—22no—
431C2—24no—
432C2—24no—
433C2—22no—
434C2—162no—
435C2#1465911yescharacteristic
436C2#2174711yescharacteristic
437C2#2174811yescharacteristic
438C2#2179412yes—
439C2#2187512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14685C2semidirect
3. #14684C2semidirect
4. #11353C2semidirect
5. #11043C2semidirect
7. #10966C2semidirect
16. C2×#390C4semidirect
17. C2×#389C4semidirect
18. C2×#388C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
#146852
#146842
#113532
#110432
#109802
#109662