Finite Group Databasebeta

← #163797

Subgroups — #163797

Order 512.

776 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#35266C211yesmaximal, characteristic
2#22641C212yesmaximal, complemented
3#7726C212yesmaximal, complemented
4#7712C212yesmaximal, complemented
5#7686C212yesmaximal, complemented
6#7672C212yesmaximal, complemented
7#7640C211yesmaximal, characteristic
8#7626C211yesmaximal, characteristic
9#7613C212yesmaximal, complemented
10#2860C2212yes—
11#2860C2212yes—
12#2644C412yes—
13#2644C412yes—
14#2513—22no—
15#2512—22no—
16#2480—22no—
17#2480—22no—
18#2317C411yescharacteristic
19#2316C411yescharacteristic
20#2309C411yescharacteristic
21#2309C411yescharacteristic
22#2296C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C4×#117C2211yescharacteristic
25#1940—22no—
26#1924—22no—
27#1923—22no—
28#1839C2212yescomplemented
29#1839C2212yescomplemented
30#1797C2212yes—
31#1771—22no—
32#1767C2212yescomplemented
33C4×C4⋊C8C2212yescomplemented
34#1749C2212yes—
35#1271C2212yescomplemented
36#1271C2212yescomplemented
37#1271C2212yescomplemented
38#1271C2212yescomplemented
39#1270C2211yescharacteristic
40#1270C2212yes—
41#1270C2211yescharacteristic
42#1270C2212yes—
43#1270C2211yescharacteristic
44#1270C2211yescharacteristic
45C23×D4C2×C411yescharacteristic
46C2×#126—22no—
47C2×#126—22no—
48C2×#117C2×C411yescharacteristic
49C2×#117C2×C411yescharacteristic
50C2×#117C2×C411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C22⋊C4—22no—
54C22×C42C2311yescharacteristic
55C2×C4⋊C8—22no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8—22no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×#97—22no—
61C2×#97—22no—
62C2×C4×C8—22no—
63#397—22no—
64#397—22no—
65#397—22no—
66#397—22no—
67#397—22no—
68#397—22no—
69#396—22no—
70#396—22no—
71#390—21no—
72#390—22no—
73#390—21no—
74#390—22no—
75#390C2×C412yes—
76#390C2×C412yes—
77#390C2×C412yes—
78#390C2×C412yes—
79#390C2×C411yescharacteristic
80#390C2×C411yescharacteristic
81#390C2311yescharacteristic
82#390C2311yescharacteristic
83#389—22no—
84#389C2×C412yes—
85#389C2×C412yes—
86#389C2×C412yes—
87#389C2×C412yes—
88#389C2×C412yes—
89#389C2312yes—
90#388—22no—
91#388—22no—
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#385—22no—
97#385—22no—
98#384—21no—
99#382—21no—
100#380—21no—
101#380—21no—
102#379—22no—
103#379—21no—
104#379—21no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4D412yes—
107C4×C4⋊C4D412yes—
108C4×C22⋊C4D412yes—
109C4×C22⋊C4D412yes—
110#376C2×C412yes—
111C43C2312yes—
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336C2312yes—
117#336C2312yes—
118#336C2312yes—
119#336C2312yes—
120C25—21no—
121C22×D4—21no—
122C22×D4—42no—
123C22×D4—42no—
124C22×D4—42no—
125C22×D4C22×C411yescharacteristic
126C22×D4C22×C411yescharacteristic
127C23×C4—21no—
128C23×C4C22×C411yescharacteristic
129C23×C4C2×D412yes—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134#126—82no—
135#126—82no—
136#126—82no—
137#126—82no—
138#117—42no—
139#117—42no—
140#117—42no—
141#117—42no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4C22⋊C411yescharacteristic
151C2×C4⋊C4C22⋊C411yescharacteristic
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22×C412yes—
154C2×C4⋊C4C22×C411yescharacteristic
155C2×C4⋊C4C22×C411yescharacteristic
156C2×C4⋊C4C22×C412yes—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—82no—
172C2×C22⋊C4—82no—
173C2×C22⋊C4C22⋊C412yes—
174C2×C22⋊C4C22⋊C411yescharacteristic
175C2×C22⋊C4C22⋊C411yescharacteristic
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—42no—
179C2×C42C22×C412yes—
180C2×C42C22×C412yes—
181C2×C42C22×C411yescharacteristic
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2411yescharacteristic
188C4⋊C8—44no—
189C4⋊C8—44no—
190C4⋊C8—44no—
191C22⋊C8—44no—
192C22⋊C8—82no—
193#97—44no—
194#97—82no—
195C4×C8—82no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95C22⋊C412yes—
203#95C22⋊C412yes—
204#95C22⋊C412yes—
205#95C22⋊C412yes—
206C24—21no—
207C24—21no—
208C24—42no—
209C24—42no—
210C24—42no—
211C24—82no—
212C24—82no—
213C24C2×C22⋊C411yescharacteristic
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—88no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—82no—
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C23×C411yescharacteristic
244C22×C4C22×D412yes—
245C2×C8—24no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C4⋊C4—24no—
258C4⋊C4—42no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—82no—
268C4⋊C4—82no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C4⋊C4—82no—
272C22⋊C4—24no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—82no—
282C22⋊C4—82no—
283C22⋊C4—82no—
284C22⋊C4—82no—
285C22⋊C4—82no—
286C22⋊C4—82no—
287C22⋊C4—82no—
288C22⋊C4—82no—
289C22⋊C4—82no—
290C22⋊C4—82no—
291C42—24no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—24no—
297C42—42no—
298C42—42no—
299C42—41no—
300C42—41no—
301C42—42no—
302C42—42no—
303C42—42no—
304C23—21no—
305C23—21no—
306C23—22no—
307C23—22no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—84no—
319C23—88no—
320C23—162no—
321C23—162no—
322C23—164no—
323C23#41312yes—
324C23C22×C22⋊C411yescharacteristic
325D4—82no—
326D4—82no—
327D4—84no—
328C2×C4—28no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—21no—
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—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—162no—
363C2×C4#41012yes—
364C2×C4#56812yes—
365C2×C4#57612yes—
366C2×C4#57712yes—
367C8—44no—
368C8—82no—
369C8—82no—
370C8—82no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—42no—
375C22—42no—
376C22—81no—
377C22—81no—
378C22—82no—
379C22—82no—
380C22—81no—
381C22—81no—
382C22—82no—
383C22—82no—
384C22—81no—
385C22—81no—
386C22—168no—
387C22#287911yescharacteristic
388C22C2×#41011yescharacteristic
389C22C2×#41311yescharacteristic
390C22#289712yes—
391C22#290112yes—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—22no—
396C4—42no—
397C4—42no—
398C4—42no—
399C4—42no—
400C4—42no—
401C4—82no—
402C4—81no—
403C4—81no—
404C4—82no—
405C2—42no—
406C2—161no—
407C2—161no—
408C2#1739811yescharacteristic
409C2#1750512yes—
410C2#1760712yes—
411C2#1768611yescharacteristic
412C2#1777111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22641C2semidirect
3. #7726C2semidirect
4. #7712C2semidirect
5. #7686C2semidirect
6. #7672C2semidirect
9. #7613C2semidirect
28. #1839C22semidirect
29. #1839C22semidirect
32. #1767C22semidirect
33. C4×C4⋊C8C22semidirect
35. #1271C22semidirect
36. #1271C22semidirect
37. #1271C22semidirect
38. #1271C22semidirect

Maximal subgroups

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

subgroupindex
#352662
#226412
#77262
#77122
#76862
#76722
#76402
#76262
#76132