Finite Group Databasebeta

← #382826

Subgroups — #382826

Order 512.

1284 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34207C211yesmaximal, characteristic
2#20695C211yesmaximal, characteristic, complemented
3#20634C211yesmaximal, characteristic, complemented
4#11210C212yesmaximal, complemented
5#11154C212yesmaximal, complemented
6#11072C212yesmaximal, complemented
7#10979C212yesmaximal
8#10824C212yesmaximal, complemented
9#10775C212yesmaximal, complemented
10C2×#532C2211yescharacteristic
11C2×#520C2211yescharacteristic
12C2×#520C2211yescharacteristic
13C2×#421C2214yes—
14C2×#410C2212yes—
15#2879C2212yes—
16#2361—24no—
17#2354—22no—
18D4×C4⋊C4—22no—
19#2326C412yescomplemented
20#2326C412yescomplemented
21#2316C2212yes—
22#2311C2211yescharacteristic
23#2311C2211yescharacteristic
24C4×C4⋊Q8C412yescomplemented
25C4×#127C412yescomplemented
26C2×#384C2211yescharacteristic
27C4×#117C2211yescharacteristic
28#1822—22no—
29#1821—22no—
30#1799—24no—
31#1742C2212yes—
32C2×#343C2212yes—
33#1740C2212yes—
34#1736C2214yes—
35C2×#339C2214yes—
36C2×H(3,ℤ/4)C2214yes—
37C22×#41C2311yescharacteristic
38C22×#41C2311yescharacteristic
39C22×M16C2314yes—
40#532—24no—
41#532—24no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#127C2×C411yescharacteristic
44C2×#124C2×C411yescharacteristic
45C2×#124C2×C411yescharacteristic
46C2×#121—22no—
47#520—44no—
48#520—44no—
49C2×C4×Q8C2×C412yes—
50C2×C4×Q8C2×C412yes—
51C2×C4×D4—22no—
52C2×C4×D4—24no—
53C2×C4×D4—22no—
54C2×C4×D4C2×C412yes—
55C2×C4×D4C2×C412yes—
56C2×C4×D4D414yes—
57C2×#117—21no—
58C2×#117D412yes—
59C2×#117C2314yes—
60C2×#117C2311yescharacteristic
61C2×#117C2311yescharacteristic
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4D412yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C42C2311yescharacteristic
68#421—28no—
69#421—44no—
70C2×#104—44no—
71#410—44no—
72C2×#99—24no—
73#408—44no—
74#407—24no—
75#391C2×C412yes—
76#391C2×C412yes—
77#390—24no—
78#390C2×C412yes—
79#390C2×C412yes—
80#389—24no—
81#389—44no—
82#388—24no—
83#388—24no—
84#386—22no—
85#386—22no—
86#385—24no—
87#385—24no—
88#384—22no—
89#384C2×C412yes—
90#384C2×C412yes—
91#380—22no—
92#380—22no—
93#380—24no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—24no—
97C4×C22⋊C4—22no—
98#376C2×C412yes—
99C43C2×C412yes—
100#343—22no—
101#343—24no—
102#343—22no—
103#342—24no—
104#340—44no—
105#339—28no—
106H(3,ℤ/4)—44no—
107C2×#41—22no—
108C2×#41—22no—
109C2×#41—22no—
110C2×#41—24no—
111C2×#41—22no—
112C2×#41—22no—
113C2×#41—24no—
114C2×#41—22no—
115C2×#41—22no—
116C2×#41#4114yes—
117C22×Q8C22×C411yescharacteristic
118C22×Q8C22×C411yescharacteristic
119C22×D4—22no—
120C22×D4—21no—
121C22×D4C22×C411yescharacteristic
122C22×D4C22×C411yescharacteristic
123C22×D4C2×D412yes—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4C22⋊C414yes—
129C23×C4C2×D412yes—
130C23×C4C2×D412yes—
131C23×C4C2411yescharacteristic
132C2×M16—24no—
133C2×M16—24no—
134C2×M16—24no—
135C2×M16—24no—
136C2×M16—24no—
137C2×M16—24no—
138C22×C8—24no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141#127—22no—
142#127—22no—
143#124—22no—
144#124—22no—
145#124—22no—
146#124—22no—
147#121—216no—
148C4×Q8—44no—
149C4×Q8—44no—
150C4×D4—216no—
151C4×D4—44no—
152C4×D4—44no—
153C4×D4—48no—
154C4×D4—48no—
155C4×D4—88no—
156#117—24no—
157#117—22no—
158#117—22no—
159#117—24no—
160#117—24no—
161#117—24no—
162#117—24no—
163#117—44no—
164#117—44no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—84no—
178C2×C4⋊C4C22⋊C414yes—
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C2×D411yescharacteristic
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4C22⋊C414yes—
192C2×C22⋊C4C2×D412yes—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—24no—
199C2×C42C22⋊C414yes—
200C2×C42C22×C412yes—
201C2×C42C22×C412yes—
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C411yescharacteristic
204C2×C42#4114yes—
205#104—416no—
206#99—48no—
207C22⋊C8—48no—
208#95—22no—
209#95—22no—
210#95—24no—
211#95—24no—
212#95—44no—
213C24—21no—
214C24C2×C22⋊C412yes—
215C24C22×D411yescharacteristic
216#41—28no—
217#41—28no—
218#41—44no—
219#41—44no—
220#41—44no—
221#41—44no—
222#41—44no—
223#41—44no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8#11712yes—
232C2×D4—28no—
233C2×D4—24no—
234C2×D4—24no—
235C2×D4—24no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—42no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4#11712yes—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—44no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—44no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—84no—
273C22×C4—84no—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C414yes—
278C22×C4C23×C411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C2×#4112yes—
281C22×C4C2×#4112yes—
282C22×C4H(5,𝔽2)14yes—
283C22×C4H(5,𝔽2)12yes—
284C22×C42−1+412yes—
285M16—44no—
286M16—44no—
287M16—44no—
288M16—44no—
289C2×C8—44no—
290C2×C8—44no—
291C2×C8—44no—
292C4⋊C4—28no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—24no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—48no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—84no—
306C4⋊C4—164no—
307C22⋊C4—28no—
308C22⋊C4—28no—
309C22⋊C4—24no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—84no—
317C22⋊C4—84no—
318C42—21no—
319C42—21no—
320C42—24no—
321C42—24no—
322C42—24no—
323C42—44no—
324C42—44no—
325C42—44no—
326C42—44no—
327C42—44no—
328C42—44no—
329C42#11712yes—
330C42#11712yes—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—41no—
336C23—41no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—41no—
342C23—41no—
343C23C22×C22⋊C411yescharacteristic
344C23#52512yes—
345C23#53412yes—
346C23#53511yescharacteristic
347C23#53611yescharacteristic
348Q8—24no—
349Q8—42no—
350Q8—42no—
351Q8—42no—
352D4—24no—
353D4—416no—
354D4—44no—
355D4—44no—
356D4—44no—
357D4—44no—
358D4—42no—
359D4—42no—
360D4—42no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—48no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—48no—
378C2×C4—48no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4—84no—
391C2×C4—164no—
392C2×C4—164no—
393C2×C4—162no—
394C2×C4C2×#11711yescharacteristic
395C2×C4C2×#11711yescharacteristic
396C2×C4#51812yes—
397C2×C4#52012yes—
398C2×C4#52512yes—
399C2×C4#53412yes—
400C2×C4#53712yes—
401C2×C4#53814yes—
402C2×C4#53914yes—
403C2×C4#54014yes—
404C2×C4#54214yes—
405C8—84no—
406C22—21no—
407C22—22no—
408C22—21no—
409C22—22no—
410C22—42no—
411C22—41no—
412C22—41no—
413C22—44no—
414C22—44no—
415C22—42no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—82no—
420C22—82no—
421C22—81no—
422C22—81no—
423C22#228011yescharacteristic
424C22#230811yescharacteristic
425C22#231111yescharacteristic
426C22#263512yes—
427C22#263812yes—
428C4—24no—
429C4—42no—
430C4—42no—
431C4—42no—
432C4—44no—
433C4—42no—
434C4—42no—
435C4—82no—
436C4—82no—
437C4—164no—
438C4#231812yes—
439C4#231912yes—
440C4#263014yes—
441C2—21no—
442C2—21no—
443C2—22no—
444C2—22no—
445C2—82no—
446C2—82no—
447C2#1434411yescharacteristic
448C2#2178211yescharacteristic
449C2#2178311yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
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
2. #20695C2semidirect
3. #20634C2semidirect
4. #11210C2semidirect
5. #11154C2semidirect
6. #11072C2semidirect
8. #10824C2semidirect
9. #10775C2semidirect
19. #2326C4semidirect
20. #2326C4semidirect
24. C4×C4⋊Q8C4semidirect
25. C4×#127C4semidirect

Maximal subgroups

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

subgroupindex
#342072
#206952
#206342
#112102
#111542
#110722
#109792
#108242
#107752