Finite Group Databasebeta

← #362828

Subgroups — #362828

Order 512.

1290 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34183C211yesmaximal, characteristic
2#20634C211yesmaximal, characteristic
3#20397C211yesmaximal, characteristic
4#11224C212yesmaximal, complemented
5#11210C212yesmaximal, complemented
6#10893C212yesmaximal, complemented
7#10889C212yesmaximal, complemented
8#10765C212yesmaximal
9#10758C212yesmaximal
10C2×#520C2211yescharacteristic
11C2×#520C2211yescharacteristic
12C2×C4×#41C2211yescharacteristic
13C2×#413C2212yes—
14C2×#410C2212yes—
15#2873C2212yes—
16Q8×C4⋊C4—22no—
17D4×C4⋊C4—22no—
18D4×C4⋊C4—24no—
19#2317C412yescomplemented
20#2317C412yescomplemented
21#2317C2211yescharacteristic
22#2317C2211yescharacteristic
23#2316C2212yes—
24#2285C412yescomplemented
25#2285C2211yescharacteristic
26C2×#384C412yescomplemented
27C2×#384C2211yescharacteristic
28#1799—24no—
29C4×#101—22no—
30C4×#100—22no—
31C2×#343C2212yes—
32#1740C2212yes—
33#1736C2214yes—
34#1733C2214yes—
35C2×#339C2214yes—
36C2×H(3,ℤ/4)C2214yes—
37C22×#41C2311yescharacteristic
38C22×M16C2314yes—
39#520—44no—
40#520—44no—
41C4×#41—24no—
42C4×#41—24no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8D412yes—
46C2×C4×D4—22no—
47C2×C4×D4—21no—
48C2×C4×D4—22no—
49C2×C4×D4—21no—
50C2×C4×D4D414yes—
51C2×C4×D4D412yes—
52C2×#117C2×C412yes—
53C2×#117C2×C412yes—
54C2×#117C2×C412yes—
55C2×#117C2314yes—
56C2×#117C2311yescharacteristic
57C2×#117C2311yescharacteristic
58C2×#117C2311yescharacteristic
59C22×C4⋊C4—24no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2×C412yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C42C2×C412yes—
66C22×C42C2311yescharacteristic
67#413—24no—
68#413Q818yes—
69C2×#101—22no—
70C2×#100—22no—
71#410—24no—
72#410—24no—
73C2×#99—24no—
74#405—44no—
75C4×M16—24no—
76#389—24no—
77#389—22no—
78#389—24no—
79#389—22no—
80#389—44no—
81#385—24no—
82#385—22no—
83#385—24no—
84#385—22no—
85#384—24no—
86#384—22no—
87#384C2×C412yes—
88#384C2×C412yes—
89#384C2×C412yes—
90#384C2×C411yescharacteristic
91#384C2×C412yes—
92#384C2×C411yescharacteristic
93#382C2×C411yescharacteristic
94#382C2×C411yescharacteristic
95#382C2×C412yes—
96#382C2×C412yes—
97#380—24no—
98#380—22no—
99#380—24no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—22no—
102#343—24no—
103#342—24no—
104#340—44no—
105#339—28no—
106#339—44no—
107H(3,ℤ/4)—28no—
108C2×#41—22no—
109C2×#41—22no—
110C2×#41—24no—
111C2×#41C2×Q814yes—
112C2×#41#4114yes—
113C22×Q8C2×D411yescharacteristic
114C22×D4C2×D412yes—
115C22×D4C2×D411yescharacteristic
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4C22⋊C414yes—
121C23×C4C22×C412yes—
122C23×C4C2×D412yes—
123C23×C4C2411yescharacteristic
124C2×M16—24no—
125C2×M16—24no—
126C2×M16—24no—
127C2×M16—24no—
128C2×M16—24no—
129C2×M16C2×Q818yes—
130C22×C8—24no—
131C4×Q8—28no—
132C4×Q8—44no—
133C4×Q8—44no—
134C4×D4—216no—
135C4×D4—28no—
136C4×D4—44no—
137C4×D4—48no—
138C4×D4—48no—
139C4×D4—44no—
140#117—22no—
141#117—24no—
142#117—24no—
143#117—24no—
144#117—24no—
145#117—24no—
146#117—24no—
147#117—24no—
148#117—24no—
149#117—44no—
150#117—44no—
151#117#4114yes—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—84no—
169C2×C4⋊C4—88no—
170C2×C4⋊C4C22⋊C414yes—
171C2×C4⋊C4C22×C411yescharacteristic
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C411yescharacteristic
174C2×C4⋊C4C22×C411yescharacteristic
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C411yescharacteristic
177C2×C4⋊C4C2×D412yes—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4C22⋊C414yes—
184C2×C22⋊C4C2×D412yes—
185C2×C42—22no—
186C2×C42—21no—
187C2×C42—22no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—22no—
191C2×C42—21no—
192C2×C42—24no—
193C2×C42C22⋊C414yes—
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C412yes—
196C2×C42C22×C411yescharacteristic
197C2×C42C2×D412yes—
198C2×C42#4114yes—
199#101—28no—
200#100—28no—
201#99—48no—
202#97—44no—
203C4×C8—44no—
204#95—24no—
205#95—22no—
206#95—24no—
207#95—22no—
208#95—44no—
209C24C2×C22⋊C412yes—
210C24C22×D411yescharacteristic
211#41—28no—
212#41—28no—
213#41—28no—
214#41—28no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8C22⋊Q814yes—
218C2×D4—24no—
219C2×D4—24no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4C22⋊Q814yes—
223C2×D4#12414yes—
224C2×D4C4⋊Q814yes—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—28no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—44no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4—84no—
248C22×C4—84no—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C414yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C4×Q818yes—
254C22×C4C23×C411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C22×Q812yes—
257C22×C4C2×#4112yes—
258C22×C4C2×#4112yes—
259C22×C4C2×#4112yes—
260C22×C4H(5,𝔽2)14yes—
261M16—28no—
262M16—28no—
263M16—44no—
264M16—44no—
265C2×C8—28no—
266C2×C8—44no—
267C2×C8—44no—
268C4⋊C4—28no—
269C4⋊C4—24no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—48no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—84no—
286C4⋊C4—84no—
287C4⋊C4—84no—
288C4⋊C4—164no—
289C22⋊C4—28no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C22⋊C4—42no—
295C22⋊C4—44no—
296C22⋊C4—44no—
297C42—28no—
298C42—21no—
299C42—21no—
300C42—22no—
301C42—24no—
302C42—24no—
303C42—44no—
304C42—44no—
305C42—42no—
306C42—44no—
307C42—44no—
308C42#11712yes—
309C42#11714yes—
310C42#11712yes—
311C23—22no—
312C23—22no—
313C23—41no—
314C23—41no—
315C23—42no—
316C23—42no—
317C23#39114yes—
318C23C22×C22⋊C411yescharacteristic
319C23#52512yes—
320C23C2×#12411yescharacteristic
321C23C2×C4⋊Q811yescharacteristic
322C23#53412yes—
323Q8—28no—
324D4—28no—
325D4—28no—
326D4—28no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—28no—
333C2×C4—28no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—48no—
340C2×C4—416no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—84no—
355C2×C4—84no—
356C2×C4—84no—
357C2×C4—84no—
358C2×C4—164no—
359C2×C4—162no—
360C2×C4—164no—
361C2×C4#39114yes—
362C2×C4C2×#11711yescharacteristic
363C2×C4C2×#11711yescharacteristic
364C2×C4C2×C4×Q812yes—
365C2×C4#51812yes—
366C2×C4C2×C22⋊Q812yes—
367C2×C4#52512yes—
368C2×C4#53412yes—
369C2×C4#55314yes—
370C2×C4#55314yes—
371C2×C4#55414yes—
372C2×C4#55414yes—
373C8—48no—
374C22—21no—
375C22—22no—
376C22—21no—
377C22—22no—
378C22—24no—
379C22—42no—
380C22—41no—
381C22—41no—
382C22—44no—
383C22—44no—
384C22—42no—
385C22—42no—
386C22#228011yescharacteristic
387C22C2×#39111yescharacteristic
388C22#230811yescharacteristic
389C22#258312yes—
390C22#259112yes—
391C4—24no—
392C4—24no—
393C4—44no—
394C4—42no—
395C4—44no—
396C4—42no—
397C4—42no—
398C4—82no—
399C4—81no—
400C4—81no—
401C4—164no—
402C4#234512yes—
403C4#234512yes—
404C4#257914yes—
405C2—21no—
406C2—22no—
407C2—21no—
408C2—22no—
409C2—44no—
410C2#1392311yescharacteristic
411C2#2178211yescharacteristic
412C2#2178311yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
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. #11224C2semidirect
5. #11210C2semidirect
6. #10893C2semidirect
7. #10889C2semidirect
19. #2317C4semidirect
20. #2317C4semidirect
24. #2285C4semidirect
26. C2×#384C4semidirect

Maximal subgroups

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

subgroupindex
#341832
#206342
#203972
#112242
#112102
#108932
#108892
#107652
#107582