Finite Group Databasebeta

← #375505

Subgroups — #375505

Order 512.

1345 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35529C211yesmaximal, characteristic
2#20398C211yesmaximal, characteristic
3#20354C211yesmaximal, characteristic
4#11388C212yesmaximal, complemented
5#11347C212yesmaximal, complemented
6#10893C212yesmaximal, complemented
7#10889C212yesmaximal, complemented
8#10749C212yesmaximal
9#10740C212yesmaximal
10C2×#544C2211yescharacteristic
11C2×#518C2212yes—
12C2×#518C2211yescharacteristic
13C22×#100C2212yes—
14#2873C2212yes—
15#2566C412yescomplemented
16#2566C412yescomplemented
17#2563C412yescomplemented
18#2542C412yescomplemented
19#2454—22no—
20#2452—22no—
21#2437—24no—
22#2317C2211yescharacteristic
23#2309C2211yescharacteristic
24#2296C2211yescharacteristic
25C2×#390C2211yescharacteristic
26#2285C2211yescharacteristic
27C4×#100—22no—
28C4×#100—22no—
29#1742C2212yes—
30#1740C2212yes—
31#1740C2212yes—
32C2×#342—24no—
33#1736C2214yes—
34#1733C2214yes—
35C2×#339C2214yes—
36C2×H(3,ℤ/4)C2214yes—
37C23×D4C2311yescharacteristic
38C22×M16C2314yes—
39#544—44no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8C2×C412yescomplemented
42C2×#125—21no—
43C2×#125C2×C412yescomplemented
44C2×C22⋊Q8—24no—
45C2×C22⋊Q8—22no—
46#518—28no—
47#518—24no—
48#518—24no—
49#518—44no—
50C2×C4×D4—44no—
51C2×C4×D4C2×C412yes—
52C2×C4×D4C2×C412yes—
53C2×C4×D4D412yes—
54C2×C4×D4D412yes—
55C2×#117C2314yes—
56C2×#117C2311yescharacteristic
57C2×#117C2311yescharacteristic
58C2×#117C2311yescharacteristic
59C2×#117C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4—24no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4D414yes—
64C22×C42C2311yescharacteristic
65C2×#100—24no—
66C2×#100—24no—
67C2×#100Q818yescomplemented
68#405—44no—
69C4×M16—24no—
70#399—24no—
71#396—24no—
72#396—24no—
73#393—24no—
74#390—22no—
75#390—22no—
76#390C2×C412yes—
77#390C2×C412yes—
78#390C2×C412yes—
79#390C2×C412yes—
80#389—21no—
81#389—21no—
82#389—22no—
83#385—24no—
84#385—24no—
85#385—22no—
86#385—22no—
87#384C2×C411yescharacteristic, complemented
88#384C2×C412yescomplemented
89#384C2×C411yescharacteristic, complemented
90#382C2×C411yescharacteristic, complemented
91#382C2×C412yescomplemented
92#382C2×C411yescharacteristic, complemented
93#380—24no—
94#380—22no—
95#379—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4C2×C412yescomplemented
98C4×C4⋊C4C2×C412yescomplemented
99#343—22no—
100#343—22no—
101#342—24no—
102#342—24no—
103#340—44no—
104#339—28no—
105#339—44no—
106H(3,ℤ/4)—28no—
107C25C2×D412yes—
108C22×Q8—22no—
109C22×D4—24no—
110C22×D4—24no—
111C22×D4C22×C412yes—
112C22×D4C2×D412yes—
113C22×D4C2×Q814yes—
114C22×D4#4114yes—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—44no—
118C23×C4C22⋊C414yes—
119C23×C4C22⋊C414yes—
120C23×C4C2411yescharacteristic
121C2×M16—24no—
122C2×M16—24no—
123C2×M16—24no—
124C2×M16—24no—
125C2×M16—24no—
126C2×M16C2×Q818yes—
127C22×C8—24no—
128C4⋊Q8—44no—
129C4⋊Q8—44no—
130#125—44no—
131#125—44no—
132C22⋊Q8—48no—
133C22⋊Q8—88no—
134C4×D4—44no—
135C4×D4—44no—
136C4×D4—44no—
137C4×D4—44no—
138C4×D4—416no—
139C4×D4—416no—
140#117—24no—
141#117—24no—
142#117—24no—
143#117—24no—
144#117—24no—
145#117—24no—
146#117—24no—
147#117#4118yes—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C22×C411yescharacteristic
162C2×C4⋊C4C22×C411yescharacteristic
163C2×C4⋊C4C22×C411yescharacteristic
164C2×C4⋊C4C22×C411yescharacteristic
165C2×C4⋊C4C2×D411yescharacteristic
166C2×C4⋊C4C2×D411yescharacteristic
167C2×C22⋊C4—216no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—48no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—48no—
178C2×C22⋊C4—84no—
179C2×C22⋊C4C22⋊C414yes—
180C2×C22⋊C4C22⋊C414yes—
181C2×C22⋊C4C2×D412yes—
182C2×C22⋊C4C2×D412yes—
183C2×C42—24no—
184C2×C42—24no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42C22×C412yes—
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C412yes—
191C2×C42C2×D412yes—
192#100—216no—
193#97—44no—
194C4×C8—44no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—24no—
200C24—24no—
201C24—24no—
202C24—82no—
203C24—82no—
204C24C2×C22⋊C414yes—
205C24#12414yes—
206C24C4⋊Q814yes—
207C24C22×D411yescharacteristic
208C2×Q8—44no—
209C2×Q8—84no—
210C2×D4—24no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—28no—
215C2×D4—28no—
216C2×D4—28no—
217C2×D4—28no—
218C2×D4C4×Q818yes—
219C2×D4C22⋊Q818yes—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—416no—
232C22×C4—42no—
233C22×C4—81no—
234C22×C4—82no—
235C22×C4—84no—
236C22×C4—84no—
237C22×C4—164no—
238C22×C4—164no—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C412yes—
243C22×C4C23×C411yescharacteristic
244C22×C4C22×D411yescharacteristic
245C22×C4C22×Q812yes—
246C22×C4C2×#4112yes—
247C22×C4C2×#4114yes—
248C22×C4H(5,𝔽2)14yes—
249M16—28no—
250M16—28no—
251M16—44no—
252M16—44no—
253C2×C8—28no—
254C2×C8—44no—
255C2×C8—44no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—24no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—28no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—44no—
272C4⋊C4—82no—
273C4⋊C4—82no—
274C4⋊C4—84no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—28no—
278C22⋊C4—44no—
279C22⋊C4—44no—
280C22⋊C4—48no—
281C22⋊C4—416no—
282C22⋊C4—42no—
283C22⋊C4—44no—
284C22⋊C4—42no—
285C22⋊C4—84no—
286C22⋊C4—84no—
287C22⋊C4—84no—
288C42—24no—
289C42—24no—
290C42—22no—
291C42—22no—
292C42—24no—
293C42—28no—
294C42—42no—
295C42—42no—
296C42—44no—
297C23—24no—
298C23—28no—
299C23—28no—
300C23—24no—
301C23—41no—
302C23—41no—
303C23—44no—
304C23—44no—
305C23—84no—
306C23—84no—
307C23—84no—
308C23—84no—
309C23—164no—
310C23#39118yes—
311C23C22×C22⋊C411yescharacteristic
312C23#52512yes—
313C23C2×#12411yescharacteristic
314C23C2×C4⋊Q811yescharacteristic
315C23#53412yes—
316Q8—164no—
317D4—216no—
318D4—216no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—48no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—84no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4—164no—
347C2×C4—164no—
348C2×C4—162no—
349C2×C4—1616no—
350C2×C4C2×C4×Q812yes—
351C2×C4C4×#4112yes—
352C2×C4#51812yes—
353C2×C4C2×C22⋊Q812yes—
354C2×C4#53212yes—
355C2×C4#53412yes—
356C2×C4#55114yes—
357C2×C4#55114yes—
358C2×C4#55414yes—
359C2×C4#55414yes—
360C8—48no—
361C22—21no—
362C22—22no—
363C22—21no—
364C22—22no—
365C22—28no—
366C22—42no—
367C22—41no—
368C22—41no—
369C22—48no—
370C22—44no—
371C22—44no—
372C22—88no—
373C22—84no—
374C22—88no—
375C22C2×#39111yescharacteristic
376C22#229711yescharacteristic
377C22#230811yescharacteristic
378C22#256112yes—
379C22#258312yes—
380C4—42no—
381C4—42no—
382C4—42no—
383C4—42no—
384C4—42no—
385C4—82no—
386C4—81no—
387C4—88no—
388C4—82no—
389C4—81no—
390C4—81no—
391C4#236814yes—
392C4#256414yes—
393C2—21no—
394C2—22no—
395C2—21no—
396C2—22no—
397C2—48no—
398C2#1415911yescharacteristic
399C2#2178211yescharacteristic
400C2#2178411yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11388C2semidirect
5. #11347C2semidirect
6. #10893C2semidirect
7. #10889C2semidirect
15. #2566C4semidirect
16. #2566C4semidirect
17. #2563C4semidirect
18. #2542C4semidirect
41. C2×C4⋊Q8C2×C4semidirect
43. C2×#125C2×C4semidirect
67. C2×#100Q8semidirect
87. #384C2×C4semidirect
88. #384C2×C4semidirect
89. #384C2×C4semidirect
90. #382C2×C4semidirect
91. #382C2×C4semidirect
92. #382C2×C4semidirect
97. C4×C4⋊C4C2×C4semidirect
98. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#355292
#203982
#203542
#113882
#113472
#108932
#108892
#107492
#107402