Finite Group Databasebeta

← #165575

Subgroups — #165575

Order 512.

830 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36534C211yesmaximal, characteristic
2#22832C212yesmaximal, complemented
3#7716C212yesmaximal, complemented
4#7702C212yesmaximal, complemented
5#7676C212yesmaximal, complemented
6#7637C211yesmaximal, characteristic
7#7618C212yesmaximal, complemented
8#7603C211yesmaximal, characteristic
9#7584C212yesmaximal, complemented
10C2×#556C411yescharacteristic
11C2×#554C411yescharacteristic
12#2862C2211yescharacteristic
13#2861C2212yes—
14#2565C412yes—
15#2563C411yescharacteristic
16#2560—22no—
17#2560—22no—
18#2547C411yescharacteristic
19#2544C412yes—
20#2536—22no—
21#2352—22no—
22C4×#127C2211yescharacteristic
23C4×#124C2212yes—
24C42×Q8C2211yescharacteristic
25#1984—22no—
26#1917—22no—
27#1917—22no—
28#1839C2212yes—
29#1839C2212yes—
30#1831C2212yes—
31#1770—22no—
32C4×C4⋊C8C2212yes—
33#1745C2212yes—
34#1745C2212yes—
35#1272C2212yes—
36#1272C2212yes—
37#1272C2212yes—
38#1272C2212yes—
39#1270C2211yescharacteristic
40#1270C2212yes—
41#1270C2211yescharacteristic
42#1270C2212yes—
43#1270C2211yescharacteristic
44#1270C2211yescharacteristic
45#556—44no—
46#554—44no—
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×#127C2×C411yescharacteristic
50C2×#126—22no—
51C2×#125C2×C412yes—
52C2×#124C2×C412yes—
53C2×#124C2×C412yes—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—22no—
57C2×C4×Q8C2×C412yes—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8D412yes—
60C2×C4×D4—22no—
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4C2×C412yes—
64C2×#117—21no—
65C22×C4⋊C4—21no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×#97—22no—
69C2×#97—22no—
70C2×C4×C8—22no—
71#398—22no—
72#398—22no—
73#397—21no—
74#397—21no—
75#394—22no—
76#394—22no—
77#393—21no—
78#393—21no—
79#391—22no—
80#391C2×C412yes—
81#391C2×C412yes—
82#391C2312yes—
83#390—22no—
84#390C2×C412yes—
85#390C2311yescharacteristic
86#390C2311yescharacteristic
87#388—22no—
88#388—22no—
89#388—22no—
90#388—22no—
91#388—22no—
92#388—22no—
93#387—22no—
94#387—22no—
95#386—22no—
96#386—22no—
97#384C2×C411yescharacteristic
98#382—22no—
99#382C2×C412yes—
100#380—22no—
101#380—22no—
102C4×C4⋊C4C2×C411yescharacteristic
103C4×C4⋊C4C2×C412yes—
104C4×C4⋊C4D412yes—
105C4×C4⋊C4D412yes—
106C4×C4⋊C4D412yes—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C43C2311yescharacteristic
110C43C2312yes—
111#336—22no—
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#336C2312yes—
119#336C2312yes—
120#336C2312yes—
121#336C2312yes—
122C22×Q8—21no—
123C22×Q8C22⋊C411yescharacteristic
124C22×Q8C22×C412yes—
125C22×D4—22no—
126C22×D4C22×C411yescharacteristic
127C22×D4C22×C411yescharacteristic
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C4⋊Q8—42no—
136C4⋊Q8—42no—
137#127—22no—
138#127—41no—
139#126—48no—
140#125—44no—
141#124—24no—
142#124—42no—
143#124—44no—
144C22⋊Q8—48no—
145C22⋊Q8—44no—
146C22⋊Q8—44no—
147C4×Q8—24no—
148C4×Q8—42no—
149C4×Q8—42no—
150C4×Q8—44no—
151C4×D4—44no—
152C4×D4—44no—
153C4×D4—42no—
154C4×D4—42no—
155C4×D4—44no—
156#117—44no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22×C411yescharacteristic
171C2×C4⋊C4C22×C411yescharacteristic
172C2×C4⋊C4C22×C412yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C42—22no—
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C412yes—
187C2×C42C22×C412yes—
188C2×C42C22×C411yescharacteristic
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2411yescharacteristic
196C4⋊C8—44no—
197C4⋊C8—44no—
198#97—44no—
199#97—44no—
200C4×C8—44no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95C22⋊C412yes—
210#95C22⋊C412yes—
211#95C22⋊C411yescharacteristic
212#95C22⋊C411yescharacteristic
213C24—21no—
214C24—21no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—82no—
222C2×Q8#10014yes—
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—44no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—44no—
232C2×D4—82no—
233C2×D4#13112yes—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C23×C411yescharacteristic
253C22×C4C22×D412yes—
254C2×C8—24no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C22⋊C4—44no—
282C22⋊C4—44no—
283C22⋊C4—44no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—44no—
291C22⋊C4—44no—
292C22⋊C4—82no—
293C22⋊C4—82no—
294C42—22no—
295C42—21no—
296C42—22no—
297C42—21no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—24no—
306C42—42no—
307C42—42no—
308C42—42no—
309C42—41no—
310C42—44no—
311C23—22no—
312C23—41no—
313C23—41no—
314C23—42no—
315C23—42no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—42no—
320C23—42no—
321C23C22×C22⋊C411yescharacteristic
322Q8—44no—
323Q8—82no—
324Q8—82no—
325Q8—162no—
326D4—42no—
327D4—42no—
328D4—81no—
329D4—81no—
330D4—88no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—164no—
365C2×C4C2×#10014yes—
366C2×C4C2×#10012yes—
367C2×C4C2×#13111yescharacteristic
368C2×C4#56811yescharacteristic
369C2×C4#57512yes—
370C2×C4#57512yes—
371C8—44no—
372C8—82no—
373C8—82no—
374C8—82no—
375C22—42no—
376C22—44no—
377C22—81no—
378C22—81no—
379C22—82no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—82no—
384C22—82no—
385C22#288011yescharacteristic
386C22#288411yescharacteristic
387C22C22×#10011yescharacteristic
388C22#289612yes—
389C22#290112yes—
390C4—21no—
391C4—21no—
392C4—22no—
393C4—24no—
394C4—22no—
395C4—22no—
396C4—42no—
397C4—41no—
398C4—42no—
399C4—42no—
400C4—42no—
401C4—82no—
402C4—82no—
403C4—81no—
404C4—81no—
405C2—82no—
406C2—161no—
407C2#1742511yescharacteristic
408C2#1751312yes—
409C2#1762412yes—
410C2#1768811yescharacteristic
411C2#1775411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22832C2semidirect
3. #7716C2semidirect
4. #7702C2semidirect
5. #7676C2semidirect
7. #7618C2semidirect
9. #7584C2semidirect

Maximal subgroups

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

subgroupindex
#365342
#228322
#77162
#77022
#76762
#76372
#76182
#76032
#75842