Finite Group Databasebeta

← #379760

Subgroups — #379760

Order 512.

2177 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35242C211yesmaximal, characteristic
2C2×#2310C211yesmaximal, characteristic, complemented
3#20979C212yesmaximal
4C2×#1784C211yesmaximal, characteristic, complemented
5#11156C214yesmaximal, complemented
6#10990C214yesmaximal, complemented
7#10716C212yesmaximal, complemented
8C24×Q8C2211yescharacteristic
9C2×#536C2212yes—
10C2×#525C2211yescharacteristic
11C22×#117C2212yes—
12C2×#421C2214yes—
13C22×#101C2214yes—
14C2×#407C2212yes—
15#2849—22no—
16#2614—22no—
17#2436C414yescomplemented
18#2434C414yescomplemented
19#2409—24no—
20#2310—22no—
21#2310—24no—
22#2310C2214yes—
23#2310C2212yes—
24C2×#391—24no—
25#2280C2211yescharacteristic
26C2×#381—22no—
27#1897—24no—
28#1896—24no—
29#1784—24no—
30#1733C2214yes—
31C2×H(3,ℤ/4)C2218yes—
32C22×#41C2311yescharacteristic
33C23×Q8—22no—
34C23×Q8—22no—
35C23×Q8—24no—
36C23×Q8—48no—
37C23×Q8C2×C414yes—
38C23×Q8D414yes—
39C23×Q8D412yes—
40C23×Q8C2312yes—
41C24×C4D412yes—
42C24×C4C2311yescharacteristic
43C22×M16C2314yes—
44#536—48no—
45C2×C4⋊Q8C2×C414yescomplemented
46C2×#124C2×C414yescomplemented
47#525—22no—
48#525—22no—
49#525—24no—
50C2×#123—24no—
51C2×#123—21no—
52C2×C22⋊Q8—24no—
53C2×C22⋊Q8C2×C412yes—
54C2×#121C2×C412yes—
55C2×#120—21no—
56C2×C4×D4—22no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—48no—
60C2×#117D414yes—
61C2×#117C2311yescharacteristic
62C2×#117C2312yes—
63C2×#117C2314yes—
64C22×C4⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C42D414yes—
67#421—48no—
68C2×#104—28no—
69C2×#101—28no—
70C2×#101—48no—
71#407—22no—
72#407—22no—
73#407—24no—
74C2×C22⋊C8—24no—
75#395—28no—
76#393—28no—
77#391—216no—
78#391—24no—
79#391—216no—
80#391C2×C418yescomplemented
81#386—28no—
82#385—28no—
83#381—28no—
84#381—28no—
85#381—24no—
86#381—21no—
87#380—22no—
88#379—21no—
89C4×C22⋊C4C2×C414yes—
90C2×#95—28no—
91#339—28no—
92H(3,ℤ/4)—48no—
93C25C2×D411yescharacteristic
94C2×#41—22no—
95C2×#41—22no—
96C2×#41—28no—
97C22×Q8—28no—
98C22×Q8—28no—
99C22×Q8—24no—
100C22×Q8—28no—
101C22×Q8—28no—
102C22×Q8—24no—
103C22×Q8—24no—
104C22×Q8—24no—
105C22×Q8—28no—
106C22×Q8—28no—
107C22×Q8—48no—
108C22×Q8—416no—
109C22×Q8—416no—
110C22×Q8—48no—
111C22×Q8—48no—
112C22×Q8—48no—
113C22×Q8—48no—
114C22×Q8—432no—
115C22×Q8C22⋊C418yes—
116C22×Q8C22×C411yescharacteristic
117C22×Q8C22×C414yes—
118C22×Q8C2×D414yes—
119C22×D4—22no—
120C22×D4—21no—
121C22×D4C22×C411yescharacteristic
122C23×C4—28no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—21no—
128C23×C4—48no—
129C23×C4C22⋊C418yes—
130C23×C4C22×C412yes—
131C23×C4C2×D414yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4#4114yes—
137C23×C4C2411yescharacteristic
138C2×M16—24no—
139C2×M16—24no—
140C2×M16—28no—
141C2×M16—28no—
142C22×C8—24no—
143C4⋊Q8—48no—
144#124—48no—
145#123—416no—
146#123—42no—
147#123—42no—
148C22⋊Q8—24no—
149C22⋊Q8—24no—
150C22⋊Q8—416no—
151#121—24no—
152#121—24no—
153#120—41no—
154#120—41no—
155#120—42no—
156C4×D4—44no—
157C4×D4—44no—
158#117—28no—
159#117—28no—
160#117—28no—
161#117—22no—
162#117—22no—
163#117—48no—
164#117—48no—
165#117—48no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—48no—
172C2×C4⋊C4C22×C412yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—48no—
182C2×C42—24no—
183C2×C42—21no—
184C2×C42—48no—
185C2×C42C22×C414yes—
186C2×C42C2×D418yes—
187#104—416no—
188#101—416no—
189C22⋊C8—44no—
190C22⋊C8—44no—
191#95—232no—
192#95—24no—
193#95—216no—
194C24—24no—
195C24—21no—
196C24—44no—
197C24C2×C22⋊C412yes—
198C24#12011yescharacteristic
199C24#12011yescharacteristic
200C24#12312yes—
201C24C22×D411yescharacteristic
202#41—44no—
203#41—44no—
204#41—48no—
205C2×Q8—28no—
206C2×Q8—28no—
207C2×Q8—28no—
208C2×Q8—28no—
209C2×Q8—28no—
210C2×Q8—28no—
211C2×Q8—216no—
212C2×Q8—22no—
213C2×Q8—24no—
214C2×Q8—416no—
215C2×Q8—48no—
216C2×Q8—48no—
217C2×Q8—416no—
218C2×Q8—432no—
219C2×Q8—432no—
220C2×Q8—432no—
221C2×Q8—432no—
222C2×Q8—48no—
223C2×Q8—48no—
224C2×Q8—416no—
225C2×D4—22no—
226C2×D4—24no—
227C2×D4—48no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—44no—
231C2×D4—42no—
232C2×D4—42no—
233C22×C4—24no—
234C22×C4—28no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—28no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—216no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—28no—
244C22×C4—28no—
245C22×C4—28no—
246C22×C4—28no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—21no—
250C22×C4—28no—
251C22×C4—416no—
252C22×C4—48no—
253C22×C4—48no—
254C22×C4—416no—
255C22×C4—48no—
256C22×C4—416no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4C2×C22⋊C418yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C4×D414yes—
262C22×C4C4×D414yes—
263C22×C4#12012yes—
264C22×C4#12114yes—
265C22×C4#12114yes—
266C22×C4#12312yes—
267C22×C4C23×C411yescharacteristic
268C22×C4C22×D412yes—
269C22×C4C22×D411yescharacteristic
270C22×C4C2×#4112yes—
271C22×C4H(5,𝔽2)18yes—
272M16—48no—
273M16—44no—
274M16—44no—
275C2×C8—44no—
276C2×C8—48no—
277C4⋊C4—24no—
278C4⋊C4—48no—
279C4⋊C4—416no—
280C4⋊C4—44no—
281C4⋊C4—48no—
282C4⋊C4—44no—
283C22⋊C4—416no—
284C22⋊C4—48no—
285C22⋊C4—48no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—48no—
289C22⋊C4—44no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—44no—
293C42—28no—
294C42—28no—
295C42—28no—
296C42—48no—
297C42—42no—
298C23—22no—
299C23—24no—
300C23—24no—
301C23—22no—
302C23—24no—
303C23—24no—
304C23—24no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—42no—
309C23—416no—
310C23—44no—
311C23—42no—
312C23—41no—
313C23#38612yes—
314C23#38612yes—
315C23C22×C22⋊C411yescharacteristic
316C23C2×#12011yescharacteristic
317C23C2×#12311yescharacteristic
318C23#53414yes—
319Q8—432no—
320Q8—416no—
321Q8—416no—
322Q8—44no—
323D4—44no—
324D4—84no—
325D4—82no—
326D4—82no—
327D4—84no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—28no—
333C2×C4—28no—
334C2×C4—28no—
335C2×C4—28no—
336C2×C4—28no—
337C2×C4—28no—
338C2×C4—28no—
339C2×C4—416no—
340C2×C4—416no—
341C2×C4—48no—
342C2×C4—48no—
343C2×C4—48no—
344C2×C4—416no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—48no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—816no—
352C2×C4—84no—
353C2×C4#38614yes—
354C2×C4C2×C4×D412yes—
355C2×C4#51814yes—
356C2×C4C2×#12112yes—
357C2×C4#53414yes—
358C2×C4#53414yes—
359C2×C4#53514yes—
360C2×C4#53818yes—
361C8—84no—
362C22—28no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—44no—
372C22—48no—
373C22—48no—
374C22—44no—
375C22—84no—
376C22—82no—
377C22—81no—
378C22#191214yes—
379C22C2×#38611yescharacteristic
380C22#230812yes—
381C22#261312yes—
382C22#262212yes—
383C4—24no—
384C4—48no—
385C4—84no—
386C4—88no—
387C4—82no—
388C4—81no—
389C4#232114yes—
390C4#261714yes—
391C2—22no—
392C2—22no—
393C2—24no—
394C2—22no—
395C2—42no—
396C2—161no—
397C2#1425511yescharacteristic
398C2C2×#191212yes—
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2310C2semidirect
4. C2×#1784C2semidirect
5. #11156C2semidirect
6. #10990C2semidirect
7. #10716C2semidirect
17. #2436C4semidirect
18. #2434C4semidirect
45. C2×C4⋊Q8C2×C4semidirect
46. C2×#124C2×C4semidirect
80. #391C2×C4semidirect

Maximal subgroups

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

subgroupindex
#352422
C2×#23102
#209792
C2×#17842
#111562
#109902
#107162