Finite Group Databasebeta

← #196935

Subgroups — #196935

Order 512.

1194 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35248C211yesmaximal, characteristic
2C2×#2320C211yesmaximal, characteristic
3#21820C211yesmaximal, characteristic
4C2×#1506C211yesmaximal, characteristic
5C2×#1503C211yesmaximal, characteristic
6#9153C214yesmaximal
7#9152C214yesmaximal
8#8021C211yesmaximal, characteristic
9#8013C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2855C2214yes—
15#2634—22no—
16#2581C414yescomplemented
17#2568C412yescomplemented
18#2546C412yescomplemented
19#2503—21no—
20#2484—22no—
21#2434—21no—
22#2327—22no—
23#2320—24no—
24#2320—24no—
25C2×#391—21no—
26C2×#389—21no—
27C2×#387—22no—
28#2283C2211yescharacteristic
29C2×#383C2212yes—
30C2×#382C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#1985—22no—
34#1984—22no—
35#1817C2214yes—
36#1506—24no—
37#1503—24no—
38C2×#328C2214yes—
39C2×#328C2212yes—
40C2×#328C2212yes—
41#1294C2212yes—
42#1294C2212yes—
43#1294C2214yes—
44C23×Q8D411yescharacteristic
45#533—22no—
46#533—22no—
47#533—22no—
48#533—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×#125—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8C2312yes—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—21no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8C2×C412yes—
61C2×C4×Q8C2×C412yes—
62C2×C4×Q8C2×C412yes—
63C2×C4×Q8D412yes—
64C2×C4×Q8D412yes—
65C2×#117—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4D412yes—
69C22×C4⋊C4D411yescharacteristic
70C22×C42—21no—
71C22×C42—21no—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74#432—22no—
75#432—22no—
76#431—22no—
77#431—22no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C22⋊C8C2312yes—
81C2×C22⋊C8C2312yes—
82C2×#97—21no—
83C2×C4×C8—21no—
84#400—24no—
85#400—24no—
86#398—24no—
87#395—24no—
88#391—28no—
89#391—22no—
90#391—22no—
91#391C2×C418yes—
92#389—28no—
93#389C2×C418yes—
94#387—216no—
95#387—28no—
96#387—28no—
97#385—24no—
98#383—21no—
99#383—24no—
100#383—24no—
101#382—22no—
102#382—24no—
103#382—24no—
104#382C2×C412yes—
105#381—24no—
106#381—21no—
107#380C2312yes—
108C4×C4⋊C4—22no—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4D414yes—
111C4×C22⋊C4—21no—
112#376—22no—
113#376—21no—
114#376D414yes—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95C2311yescharacteristic
120C2×#95C2311yescharacteristic
121C2×#95C2312yes—
122#336—28no—
123#328—28no—
124#328—24no—
125#328—24no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—44no—
130C22×Q8C22⋊C412yes—
131C22×Q8C22⋊C412yes—
132C22×Q8C22×C412yes—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2411yescharacteristic
141C22×C8—22no—
142C22×C8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—48no—
148#125—42no—
149#125—42no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C22⋊Q8—42no—
155C22⋊Q8—42no—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—24no—
164C4×Q8—24no—
165C4×Q8—24no—
166C4×Q8—42no—
167C4×Q8—42no—
168#117—42no—
169#117—42no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—84no—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C414yes—
186C2×C4⋊C4C22⋊C414yes—
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C414yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—44no—
201C2×C42—82no—
202C2×C42C22×C414yes—
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D414yes—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42#4112yes—
211C2×C42#4112yes—
212C4⋊C8—42no—
213C4⋊C8—42no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—22no—
218#97—42no—
219C4×C8—42no—
220#95—28no—
221#95—28no—
222#95—28no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—28no—
227#95—28no—
228#95—28no—
229#95—24no—
230#95—21no—
231#95—21no—
232#95—22no—
233#95—22no—
234#95C2×D412yes—
235#95C2×D412yes—
236#95C2×D414yes—
237C24C22×D411yescharacteristic
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—44no—
245C2×Q8—48no—
246C2×Q8—48no—
247C2×Q8—44no—
248C2×Q8—42no—
249C2×Q8—44no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—44no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—44no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4C2×C22⋊C414yes—
277C22×C4C2×C22⋊C414yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4#12012yes—
283C22×C4#12012yes—
284C22×C4#12312yes—
285C22×C4#12312yes—
286C22×C4C23×C411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C2×#4111yescharacteristic
291C22×C4C2×#4111yescharacteristic
292C2×C8—44no—
293C2×C8—44no—
294C2×C8—42no—
295C2×C8—42no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—28no—
299C4⋊C4—28no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—44no—
303C4⋊C4—48no—
304C4⋊C4—416no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4—88no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C42—24no—
317C42—24no—
318C42—24no—
319C42—24no—
320C42—48no—
321C42—44no—
322C42—44no—
323C42—84no—
324C42C4×D412yes—
325C42C4×D412yes—
326C42C4×D412yes—
327C42C4×D412yes—
328C42#12112yes—
329C42#12112yes—
330C42#12112yes—
331C42#12112yes—
332C23—22no—
333C23—22no—
334C23C22×C22⋊C411yescharacteristic
335C23C2×#12011yescharacteristic
336C23C2×#12311yescharacteristic
337C23#57712yes—
338C23#57812yes—
339Q8—48no—
340Q8—48no—
341Q8—82no—
342Q8—82no—
343Q8—84no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—28no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—84no—
385C2×C4—88no—
386C2×C4—84no—
387C2×C4#38612yes—
388C2×C4#38612yes—
389C2×C4#38612yes—
390C2×C4#38612yes—
391C2×C4#42114yes—
392C2×C4C2×C4×D411yescharacteristic
393C2×C4C2×C4×D411yescharacteristic
394C2×C4C2×#12111yescharacteristic
395C2×C4C2×#12111yescharacteristic
396C8—82no—
397C8—82no—
398C22—24no—
399C22—24no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22C2×#38611yescharacteristic
405C22#288412yes—
406C22#289812yes—
407C22C2×#42112yes—
408C22#300911yescharacteristic
409C22#301111yescharacteristic
410C22#309712yes—
411C4—22no—
412C4—22no—
413C4—22no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—44no—
419C4—44no—
420C4—82no—
421C4—84no—
422C4—82no—
423C4—84no—
424C4—82no—
425C4—82no—
426C4#189714yes—
427C2—21no—
428C2—21no—
429C2—22no—
430C2#1760114yes—
431C2#2091311yescharacteristic
432C2C2×#189711yescharacteristic
433C2#2098611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #2581C4semidirect
17. #2568C4semidirect
18. #2546C4semidirect

Maximal subgroups

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

subgroupindex
#352482
C2×#23202
#218202
C2×#15062
C2×#15032
#91532
#91522
#80212
#80132