Finite Group Databasebeta

← #165673

Subgroups — #165673

Order 512.

941 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36419C211yesmaximal, characteristic
2#22787C212yesmaximal, complemented
3#7730C212yesmaximal, complemented
4#7699C212yesmaximal, complemented
5#7690C212yesmaximal, complemented
6#7639C211yesmaximal, characteristic, complemented
7#7612C212yesmaximal, complemented
8#7605C211yesmaximal, characteristic, complemented
9#7578C212yesmaximal, complemented
10C2×#556C411yescharacteristic
11C2×#550C411yescharacteristic
12#2858C2211yescharacteristic
13#2850C2212yes—
14#2568C411yescharacteristic
15#2559—22no—
16#2559—22no—
17#2548C411yescharacteristic
18#2544—22no—
19#2500C412yes—
20#2432C412yes—
21#2367—22no—
22C4×C4⋊Q8C2211yescharacteristic
23#2297C2212yes—
24C42×D4C2211yescharacteristic
25#1924—22no—
26#1917—22no—
27#1869—22no—
28#1839C2212yes—
29#1817C2212yes—
30#1797C2212yes—
31#1752—22no—
32#1749C2212yes—
33#1745C2212yes—
34C2×#336C2212yes—
35#1272C2211yescharacteristic
36#1272C2212yes—
37#1272C2211yescharacteristic
38#1272C2211yescharacteristic
39#1272C2212yes—
40#1272C2211yescharacteristic
41#1270C2212yes—
42#1270C2212yes—
43#1270C2212yes—
44#1270C2212yes—
45C22×#41C2×C412yes—
46C23×C8—22no—
47#556—44no—
48#550—44no—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#127—21no—
52C2×#127C2×C411yescharacteristic
53C2×#126—22no—
54C2×#124—21no—
55C2×#124—22no—
56C2×#124C2×C412yes—
57C2×C22⋊Q8C2×C412yes—
58C2×#121—22no—
59C2×#121C2×C412yes—
60C2×C4×Q8—21no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×D4—21no—
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4D412yes—
66C2×#117C2×C412yes—
67C22×C42C2312yes—
68C2×C4⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×#97—22no—
73C2×#97—22no—
74#400—21no—
75#400—21no—
76#394—22no—
77#394—22no—
78#394—22no—
79#394—22no—
80#394—21no—
81#394—21no—
82#391—22no—
83#391C2×C412yes—
84#391C2311yescharacteristic
85#391C2311yescharacteristic
86#390—22no—
87#390—22no—
88#390C2×C412yes—
89#390C2×C412yes—
90#390C2312yes—
91#389—22no—
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#386—22no—
97#386—22no—
98#386—22no—
99#386—22no—
100#385—22no—
101#385—22no—
102#384C2×C411yescharacteristic
103#382C2×C412yes—
104C4×C4⋊C4—22no—
105C4×C4⋊C4C2×C411yescharacteristic
106C4×C4⋊C4D412yes—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4C2×C412yes—
110C4×C22⋊C4D412yes—
111C4×C22⋊C4D412yes—
112C43C2311yescharacteristic
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#336C2312yes—
119#336C2312yes—
120#336C2312yes—
121#336C2312yes—
122C2×#41—24no—
123C2×#41—42no—
124C2×#41—44no—
125C2×#41—82no—
126C22×Q8—21no—
127C22×Q8C22×C411yescharacteristic
128C22×Q8C22×C411yescharacteristic
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4C22⋊C411yescharacteristic
134C22×D4C22×C412yes—
135C23×C4—22no—
136C23×C4C22×C412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—44no—
144C22×C8—44no—
145C22×C8—82no—
146C4⋊Q8—22no—
147C4⋊Q8—41no—
148C4⋊Q8—42no—
149#127—42no—
150#127—44no—
151#126—48no—
152#124—44no—
153#124—44no—
154#124—44no—
155#124—82no—
156C22⋊Q8—44no—
157#121—48no—
158#121—44no—
159C4×Q8—42no—
160C4×Q8—42no—
161C4×Q8—44no—
162C4×D4—24no—
163C4×D4—42no—
164C4×D4—42no—
165C4×D4—44no—
166#117—44no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4C22⋊C411yescharacteristic
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22×C411yescharacteristic
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C411yescharacteristic
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×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4C22⋊C412yes—
190C2×C22⋊C4C22⋊C412yes—
191C2×C22⋊C4C22⋊C412yes—
192C2×C42—22no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—82no—
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2411yescharacteristic
204C4⋊C8—82no—
205C22⋊C8—44no—
206C22⋊C8—44no—
207C22⋊C8—82no—
208#97—44no—
209#97—82no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95C22⋊C412yes—
217#95C22⋊C412yes—
218#95C22⋊C411yescharacteristic
219#95C22⋊C411yescharacteristic
220C24—22no—
221C24C2×C22⋊C412yes—
222#41—88no—
223#41—88no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—44no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—42no—
233C2×Q8#13112yes—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—42no—
237C2×D4—44no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—44no—
241C2×D4—42no—
242C2×D4—44no—
243C2×D4—44no—
244C2×D4—42no—
245C2×D4—82no—
246C2×D4—82no—
247C2×D4—82no—
248C2×D4—82no—
249C2×D4#10014yes—
250C22×C4—21no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—82no—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D412yes—
278C2×C8—24no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—88no—
291C4⋊C4—22no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—44no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—82no—
302C22⋊C4—24no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—44no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C42—21no—
319C42—21no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—42no—
324C42—42no—
325C42—42no—
326C42—41no—
327C42—42no—
328C42—44no—
329C42—42no—
330C42—42no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23C2×#10014yes—
345C23C22×C22⋊C411yescharacteristic
346Q8—42no—
347Q8—42no—
348Q8—81no—
349Q8—81no—
350Q8—84no—
351D4—44no—
352D4—82no—
353D4—82no—
354D4—84no—
355D4—88no—
356D4—162no—
357D4—164no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—88no—
397C2×C4C2×#10012yes—
398C2×C4C2×#13111yescharacteristic
399C2×C4#56811yescharacteristic
400C2×C4#57512yes—
401C2×C4#57512yes—
402C8—44no—
403C8—82no—
404C8—82no—
405C8—82no—
406C22—24no—
407C22—24no—
408C22—24no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—82no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22—82no—
418C22—82no—
419C22—82no—
420C22#288011yescharacteristic
421C22#288311yescharacteristic
422C22C22×#10011yescharacteristic
423C22#289612yes—
424C22#290112yes—
425C4—21no—
426C4—21no—
427C4—22no—
428C4—22no—
429C4—22no—
430C4—42no—
431C4—42no—
432C4—41no—
433C4—42no—
434C4—42no—
435C4—42no—
436C4—81no—
437C4—81no—
438C4—81no—
439C4—82no—
440C2—44no—
441C2—162no—
442C2#1742111yescharacteristic
443C2#1751612yes—
444C2#1762912yes—
445C2#1768511yescharacteristic
446C2#1775411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#550C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22787C2semidirect
3. #7730C2semidirect
4. #7699C2semidirect
5. #7690C2semidirect
6. #7639C2semidirect
7. #7612C2semidirect
8. #7605C2semidirect
9. #7578C2semidirect

Maximal subgroups

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

subgroupindex
#364192
#227872
#77302
#76992
#76902
#76392
#76122
#76052
#75782