Finite Group Databasebeta

← #391757

Subgroups — #391757

Order 512.

1135 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35530C211yesmaximal, characteristic
2#20640C211yesmaximal, characteristic, complemented
3#20634C211yesmaximal, characteristic, complemented
4#11225C212yesmaximal, complemented
5#11210C212yesmaximal, complemented
6#11148C212yesmaximal, complemented
7#11072C212yesmaximal, complemented
8#11056C212yesmaximal, complemented
9#10986C212yesmaximal
10C2×#537C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2905C2212yes—
13C2×#421C2214yes—
14C2×#413C2212yes—
15C2×#410C2212yes—
16#2502C412yescomplemented
17#2501C412yescomplemented
18#2500C412yescomplemented
19#2499C412yescomplemented
20#2443—24no—
21#2423—22no—
22#2422—22no—
23#2317C2211yescharacteristic
24#2316C2212yes—
25#2311C2211yescharacteristic
26#2297C2211yescharacteristic
27#2297C2211yescharacteristic
28C2×#384C2211yescharacteristic
29#1801—22no—
30#1800—22no—
31#1799—24no—
32C2×#343C2212yes—
33#1740C2212yes—
34#1736C2214yes—
35C2×#339C2214yes—
36C2×H(3,ℤ/4)C2214yes—
37C22×#41C2311yescharacteristic
38C22×#41C2311yescharacteristic
39C22×M16C2314yes—
40#537—44no—
41C2×#123—24no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8C2×C412yes—
44C2×#121—21no—
45C2×#121—22no—
46C2×#121C2×C412yes—
47#520—44no—
48C2×C4×Q8D412yes—
49C2×C4×D4D414yes—
50C2×C4×D4D412yes—
51C2×#117C2×C412yes—
52C2×#117C2314yes—
53C2×#117C2311yescharacteristic
54C2×#117C2311yescharacteristic
55C22×C4⋊C4—22no—
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C42C2311yescharacteristic
60#424—44no—
61#423—24no—
62#421—44no—
63#421—44no—
64C2×#104—44no—
65#413—44no—
66C2×#101—22no—
67C2×#100—22no—
68#410—44no—
69C2×#99—24no—
70#399—24no—
71#398—24no—
72#397—24no—
73#394—24no—
74#391—21no—
75#391C2×C412yes—
76#391C2×C411yescharacteristic
77#391C2×C411yescharacteristic
78#390—21no—
79#390C2×C412yes—
80#390C2×C411yescharacteristic
81#390C2×C411yescharacteristic
82#389—22no—
83#389—44no—
84#388—24no—
85#387—24no—
86#386—22no—
87#386—22no—
88#386—22no—
89#385—24no—
90#385—22no—
91#384—22no—
92#384C2×C412yes—
93#384C2×C412yes—
94#384C2×C412yes—
95#382C2×C412yes—
96#380—22no—
97#380—24no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4C2×C412yes—
100C4×C22⋊C4C2×C412yes—
101#343—24no—
102#342—24no—
103#340—44no—
104#339—44no—
105H(3,ℤ/4)—44no—
106C2×#41—24no—
107C2×#41—22no—
108C2×#41—22no—
109C2×#41—22no—
110C2×#41—22no—
111C2×#41—24no—
112C2×#41—22no—
113C2×#41—22no—
114C2×#41—22no—
115C2×#41—22no—
116C22×Q8C22×C411yescharacteristic
117C22×Q8C2×D411yescharacteristic
118C22×D4—21no—
119C22×D4—22no—
120C22×D4C22×C411yescharacteristic
121C22×D4C2×D412yes—
122C22×D4C2×D411yescharacteristic
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4C22⋊C414yes—
127C23×C4C22×C412yes—
128C23×C4C2×D412yes—
129C23×C4C2411yescharacteristic
130C2×M16—24no—
131C2×M16—24no—
132C2×M16—24no—
133C2×M16—24no—
134C2×M16—24no—
135C2×M16—24no—
136C22×C8—24no—
137#123—88no—
138C22⋊Q8—44no—
139C22⋊Q8—44no—
140#121—44no—
141#121—48no—
142#121—44no—
143C4×Q8—44no—
144C4×D4—44no—
145C4×D4—48no—
146#117—24no—
147#117—22no—
148#117—22no—
149#117—24no—
150#117—24no—
151#117—24no—
152#117—24no—
153#117—44no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—84no—
166C2×C4⋊C4C22⋊C414yes—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C411yescharacteristic
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C411yescharacteristic
171C2×C4⋊C4C2×D412yes—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4C22⋊C414yes—
182C2×C22⋊C4C2×D412yes—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—24no—
187C2×C42C22⋊C414yes—
188C2×C42C22×C412yes—
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C2×D412yes—
192C4⋊C8—48no—
193#104—416no—
194#101—44no—
195#100—44no—
196#99—48no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—24no—
202#95—24no—
203#95—44no—
204C24—21no—
205C24C2×C22⋊C412yes—
206C24C22×D411yescharacteristic
207#41—44no—
208#41—44no—
209#41—44no—
210#41—44no—
211#41—44no—
212#41—44no—
213#41—44no—
214#41—44no—
215C2×Q8—22no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×D4—24no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—24no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—22no—
233C2×D4—42no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—42no—
237C2×D4—44no—
238C2×D4—84no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—44no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—84no—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C414yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C23×C411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4H(5,𝔽2)14yes—
268C22×C4H(5,𝔽2)12yes—
269C22×C4H(5,𝔽2)12yes—
270C22×C42−1+412yes—
271C22×C42−1+412yes—
272M16—44no—
273M16—44no—
274M16—44no—
275M16—44no—
276C2×C8—44no—
277C2×C8—44no—
278C2×C8—44no—
279C4⋊C4—24no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—44no—
285C4⋊C4—48no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—84no—
292C4⋊C4—84no—
293C4⋊C4—164no—
294C22⋊C4—24no—
295C22⋊C4—44no—
296C22⋊C4—44no—
297C22⋊C4—44no—
298C22⋊C4—44no—
299C22⋊C4—44no—
300C22⋊C4—44no—
301C22⋊C4—44no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—44no—
305C22⋊C4—44no—
306C22⋊C4—84no—
307C42—24no—
308C42—22no—
309C42—22no—
310C42—24no—
311C42—44no—
312C42—41no—
313C42—41no—
314C42—44no—
315C42—44no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—41no—
320C23—41no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23C22×C22⋊C411yescharacteristic
329C23#53412yes—
330C23#53511yescharacteristic
331C23#53611yescharacteristic
332C23#53712yes—
333Q8—42no—
334Q8—42no—
335Q8—44no—
336D4—44no—
337D4—44no—
338D4—44no—
339D4—42no—
340D4—42no—
341D4—82no—
342D4—82no—
343D4—164no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—48no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—48no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—84no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4—164no—
376C2×C4—162no—
377C2×C4#51812yes—
378C2×C4#52012yes—
379C2×C4#52012yes—
380C2×C4#53412yes—
381C2×C4#53511yescharacteristic
382C2×C4#53611yescharacteristic
383C2×C4#53712yes—
384C2×C4#56014yes—
385C2×C4#56214yes—
386C2×C4#56214yes—
387C2×C4#56314yes—
388C8—84no—
389C22—21no—
390C22—22no—
391C22—21no—
392C22—22no—
393C22—42no—
394C22—41no—
395C22—41no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—82no—
400C22—82no—
401C22—82no—
402C22—82no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22#230811yescharacteristic
407C22#231111yescharacteristic
408C22#231111yescharacteristic
409C22#281312yes—
410C22#281412yes—
411C4—42no—
412C4—42no—
413C4—44no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4—82no—
418C4—82no—
419C4—81no—
420C4—164no—
421C4#237614yes—
422C4#281312yes—
423C4#281412yes—
424C2—21no—
425C2—21no—
426C2—22no—
427C2—22no—
428C2—82no—
429C2—161no—
430C2#1465811yescharacteristic
431C2#2178211yescharacteristic
432C2#2178311yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#537C22
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. #20640C2semidirect
3. #20634C2semidirect
4. #11225C2semidirect
5. #11210C2semidirect
6. #11148C2semidirect
7. #11072C2semidirect
8. #11056C2semidirect
16. #2502C4semidirect
17. #2501C4semidirect
18. #2500C4semidirect
19. #2499C4semidirect

Maximal subgroups

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

subgroupindex
#355302
#206402
#206342
#112252
#112102
#111482
#110722
#110562
#109862