Finite Group Databasebeta

← #198363

Subgroups — #198363

Order 512.

1006 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2#60249C211yesmaximal, characteristic
3#21829C211yesmaximal, characteristic
4#17742C211yesmaximal, characteristic
5#17491C211yesmaximal, characteristic
6#9250C218yesmaximal, complemented
7#8000C211yesmaximal, characteristic
8#7936C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3230—22no—
12#3230—22no—
13#3230—22no—
14#3230—22no—
15C2×#500C412yes—
16C2×#487C412yes—
17C2×#484C414yes—
18C2×#477C2214yes—
19#2987—22no—
20#2986—22no—
21#2985—22no—
22#2985—22no—
23Q8×M16—22no—
24Q8×M16C2214yes—
25#2921—22no—
26#2920C2211yescharacteristic
27#2920C2211yescharacteristic
28#2920C2211yescharacteristic
29#2920C2211yescharacteristic
30C2×#431C2211yescharacteristic
31#1997—22no—
32#1997C2214yes—
33#1821—22no—
34#1556—22no—
35#1549C2214yes—
36#1546—22no—
37#1497—22no—
38#1487C2214yes—
39#1485—22no—
40#1292C2214yes—
41#1283C2214yes—
42C23×Q8—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533D412yes—
48#533D412yes—
49#533D412yes—
50#533D412yes—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×C4⋊Q8C2×C412yes—
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8C2312yes—
65C2×#125—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8C2×C412yes—
68C2×C4×Q8—21no—
69C2×C4×Q8C2312yes—
70C2×#117—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4C2×C412yes—
73C22×C42C2311yescharacteristic
74#500—24no—
75#500—24no—
76#487—24no—
77#487—24no—
78#484—28no—
79#484—28no—
80#477—28no—
81#477—28no—
82#446—24no—
83#446—28no—
84#446—24no—
85#446—28no—
86C8×Q8—24no—
87C8×Q8—24no—
88#433—28no—
89#433—28no—
90#432—21no—
91#432—21no—
92#432—21no—
93#432—21no—
94#432—21no—
95#432—21no—
96#432—21no—
97#432D412yes—
98#431—21no—
99#431—21no—
100#431—21no—
101#431—21no—
102#431—21no—
103#431—21no—
104#431—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431D412yes—
109#431C2312yes—
110C2×#107C2×C418yes—
111#423—24no—
112#423—21no—
113#423D412yes—
114#423C2312yes—
115#423C2312yes—
116C2×C4⋊C8C2312yes—
117C2×#103C2×C418yes—
118#407—21no—
119#407—21no—
120#407—21no—
121#407D412yes—
122C2×C22⋊C8C2×C412yes—
123C4×M16—24no—
124C4×M16—21no—
125C4×M16—21no—
126C4×M16—21no—
127C4×M16C2312yes—
128C2×C4×C8C2×C412yes—
129#343—28no—
130#332—28no—
131Q8⋊C8—28no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—41no—
136C22×Q8—41no—
137C22×Q8—41no—
138C22×Q8—41no—
139C22×Q8—84no—
140C22×Q8C22×C412yes—
141C23×C4—22no—
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4#4111yescharacteristic
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—41no—
148C2×M16C2×D414yes—
149C22×C8C22×C414yes—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—41no—
163C4⋊Q8—41no—
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C4⋊Q8—41no—
169C4⋊Q8—41no—
170C4⋊Q8C22⋊C412yes—
171C4⋊Q8C22⋊C412yes—
172C4⋊Q8C22⋊C412yes—
173C4⋊Q8C22⋊C412yes—
174#125—22no—
175#125—22no—
176#125—22no—
177#125—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8C22⋊C414yes—
185C22⋊Q8C22⋊C414yes—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8C2×D414yes—
193C4×Q8C2×D414yes—
194#117—22no—
195#117—22no—
196#117—22no—
197#117—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4C22×C414yes—
215C2×C4⋊C4C22×C412yes—
216C2×C22⋊C4—22no—
217C2×C42—22no—
218C2×C42C22×C411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42#4111yescharacteristic
227C2×C42#4111yescharacteristic
228C2×C42#4111yescharacteristic
229C2×C42C2411yescharacteristic
230#107—216no—
231C4⋊C8—28no—
232C4⋊C8—28no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—41no—
242C4⋊C8C2×D414yes—
243#103—216no—
244C22⋊C8—22no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—41no—
252C22⋊C8—41no—
253C22⋊C8—41no—
254#97—28no—
255#97—22no—
256#97—22no—
257#97—41no—
258#97—41no—
259C4×C8—28no—
260C4×C8—22no—
261C4×C8—22no—
262C4×C8—22no—
263C4×C8—22no—
264C4×C8—22no—
265C4×C8—41no—
266C24C4×D411yescharacteristic
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—44no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—82no—
276C2×Q8—84no—
277C2×Q8—84no—
278C2×Q8—84no—
279C2×Q8—84no—
280C2×Q8—84no—
281C2×Q8—84no—
282C2×Q8C2×C22⋊C414yes—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—84no—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C4×D411yescharacteristic
295C22×C4C4×D411yescharacteristic
296C22×C4C4×D411yescharacteristic
297C22×C4C4×D411yescharacteristic
298C22×C4C4×D411yescharacteristic
299C22×C4#12011yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4#12111yescharacteristic
302C22×C4#12111yescharacteristic
303C22×C4#12111yescharacteristic
304C22×C4#12111yescharacteristic
305C22×C4#12311yescharacteristic
306C22×C4#12311yescharacteristic
307C22×C4C23×C411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C2×#4111yescharacteristic
310M16—28no—
311M16—44no—
312M16—44no—
313M16—82no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—44no—
318C2×C8—42no—
319C2×C8—42no—
320C4⋊C4—22no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4C2×C22⋊C414yes—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C42—24no—
343C42—24no—
344C42C2×C22⋊C411yescharacteristic
345C42C2×C22⋊C411yescharacteristic
346C42C4×D411yescharacteristic
347C42C4×D411yescharacteristic
348C42#12011yescharacteristic
349C42#12011yescharacteristic
350C42#12111yescharacteristic
351C42#12111yescharacteristic
352C42#12111yescharacteristic
353C42#12111yescharacteristic
354C42#12311yescharacteristic
355C42#12311yescharacteristic
356C42C22×D411yescharacteristic
357C42C22×D411yescharacteristic
358C42C22×D411yescharacteristic
359C42C2×#4111yescharacteristic
360C23—22no—
361C23—41no—
362C23—41no—
363C23#38611yescharacteristic
364C23#38611yescharacteristic
365C23C2×C4×D411yescharacteristic
366Q8—28no—
367Q8—44no—
368Q8—82no—
369Q8—82no—
370Q8—84no—
371Q8—84no—
372Q8—84no—
373Q8—84no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—44no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—84no—
397C2×C4—84no—
398C2×C4—84no—
399C2×C4#38611yescharacteristic
400C2×C4#38611yescharacteristic
401C2×C4#38611yescharacteristic
402C2×C4#38611yescharacteristic
403C2×C4#38611yescharacteristic
404C2×C4#38611yescharacteristic
405C2×C4C22×C22⋊C411yescharacteristic
406C2×C4C2×C4×D411yescharacteristic
407C2×C4C2×#12011yescharacteristic
408C2×C4C2×#12111yescharacteristic
409C2×C4C2×#12111yescharacteristic
410C2×C4C2×#12311yescharacteristic
411C2×C4#57814yes—
412C8—44no—
413C8—44no—
414C8—44no—
415C8—82no—
416C8—82no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—44no—
426C22C2×#38611yescharacteristic
427C22#295012yes—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—44no—
437C4—44no—
438C4—84no—
439C4#190312yes—
440C4#288912yes—
441C4#301112yes—
442C4#304412yes—
443C2—22no—
444C2—41no—
445C2—41no—
446C2#2093511yescharacteristic
447C2#2099111yescharacteristic
448C2C2×#190311yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9250C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
#602492
#218292
#177422
#174912
#92502
#80002
#79362