Finite Group Databasebeta

← #189296

Subgroups — #189296

Order 512.

858 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60192C211yesmaximal, characteristic
2#20765C211yesmaximal, characteristic
3#17787C211yesmaximal, characteristic
4#17575C211yesmaximal, characteristic
5C2×#1518C211yesmaximal, characteristic
6#8681C214yesmaximal, complemented
7#8679C214yesmaximal, complemented
8#7957C211yesmaximal, characteristic
9C2×#1286C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11Q8×M16—22no—
12Q8×M16C414yescomplemented
13#2962—22no—
14#2962C414yescomplemented
15#2961—22no—
16#2960C2214yes—
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19#2905—21no—
20C2×#423C2211yescharacteristic
21#2873—21no—
22C2×C4×M16C2211yescharacteristic
23#2285C2211yescharacteristic
24C2×#382C2211yescharacteristic
25#1836—22no—
26#1835C2212yes—
27#1835C2212yes—
28#1736—21no—
29#1736—21no—
30#1565—22no—
31#1565—22no—
32#1564C2214yes—
33#1520—22no—
34#1518—22no—
35#1518C2214yes—
36#1518C2212yes—
37#1518C2212yes—
38#1503—22no—
39C2×#328—22no—
40#1286C2214yes—
41#1286C2212yes—
42#1286C2212yes—
43C22×M16—21no—
44C22×M16D411yescharacteristic
45C22×M16D411yescharacteristic
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533C2×C412yes—
51#533D412yes—
52#533Q812yes—
53#533C2312yes—
54C2×C4⋊Q8Q811yescharacteristic
55C2×C4⋊Q8Q811yescharacteristic
56C2×#125—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8D412yes—
59C2×C4×Q8—21no—
60C2×C4×Q8C2×C412yes—
61C2×#117—21no—
62C2×#117—21no—
63C22×C4⋊C4—21no—
64C22×C42C2311yescharacteristic
65#446—28no—
66#446—28no—
67#446—24no—
68#446C2×C418yescomplemented
69C8×Q8—24no—
70C8×Q8C2×C418yescomplemented
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431—21no—
76#431—21no—
77#431C2312yes—
78#431C2312yes—
79#424—24no—
80#424—24no—
81#423—24no—
82#423—21no—
83#423—21no—
84#423C2×C412yes—
85#423C2312yes—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2×C412yes—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92#405—24no—
93#405—24no—
94C4×M16—24no—
95C4×M16C2×C412yes—
96C4×M16C2×C412yes—
97C4×M16D412yes—
98C4×M16D412yes—
99C2×#97D412yes—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8C2312yes—
103#384—21no—
104#382—21no—
105#382—21no—
106#382—21no—
107#382C2312yes—
108#382C2311yescharacteristic
109#382C2311yescharacteristic
110#380—21no—
111#380—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C2×#95—21no—
115C2×#95—21no—
116#340—22no—
117#340—22no—
118#340—22no—
119#340—22no—
120#335—24no—
121#334—24no—
122#328—28no—
123#328—24no—
124#328—24no—
125C22×Q8—21no—
126C22×Q8C4⋊C412yes—
127C23×C4—41no—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4#4111yescharacteristic
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—42no—
138C2×M16—42no—
139C2×M16C22⋊C412yes—
140C2×M16C22⋊C412yes—
141C2×M16C22⋊C412yes—
142C2×M16C22⋊C412yes—
143C22×C8—22no—
144C22×C8—41no—
145C22×C8—41no—
146C22×C8C2×D412yes—
147C22×C8C2×D412yes—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8C4⋊C412yes—
153C4⋊Q8C4⋊C412yes—
154C4⋊Q8C2×Q812yes—
155C4⋊Q8C2×Q812yes—
156#125—22no—
157#125—22no—
158#125—22no—
159#125—22no—
160C22⋊Q8—22no—
161C22⋊Q8—22no—
162C22⋊Q8—22no—
163C22⋊Q8—22no—
164C22⋊Q8—22no—
165C22⋊Q8—22no—
166C22⋊Q8C4⋊C414yes—
167C22⋊Q8C2×D414yes—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8C4⋊C414yes—
175C4×Q8C22×C414yes—
176#117—22no—
177#117—22no—
178#117—22no—
179#117—22no—
180#117—42no—
181#117—42no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4C4⋊C412yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—41no—
198C2×C42—22no—
199C2×C42—41no—
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×Q811yescharacteristic
207C2×C42C2×Q811yescharacteristic
208C2×C42#4111yescharacteristic
209C2×C42#4111yescharacteristic
210C2×C42#4111yescharacteristic
211C2×C42C2411yescharacteristic
212C4⋊C8—28no—
213C4⋊C8—28no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—41no—
220C4⋊C8—41no—
221C4⋊C8C22×C414yes—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8—41no—
225C22⋊C8—41no—
226#97—28no—
227#97C22⋊C414yes—
228#97C22⋊C414yes—
229C4×C8—28no—
230C4×C8—22no—
231C4×C8—22no—
232C4×C8—41no—
233C4×C8—41no—
234C4×C8C22×C414yes—
235C4×C8C2×D414yes—
236#95—22no—
237#95—21no—
238#95—21no—
239#95—22no—
240#95—21no—
241#95—21no—
242#95—41no—
243#95—41no—
244C24#12111yescharacteristic
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—82no—
249C2×Q8—82no—
250C2×Q8C2×C4⋊C414yes—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C4⋊C411yescharacteristic
264C22×C4C2×C4⋊C411yescharacteristic
265C22×C4#11711yescharacteristic
266C22×C4#11711yescharacteristic
267C22×C4#12111yescharacteristic
268C22×C4#12111yescharacteristic
269C22×C4#12111yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4C22⋊Q811yescharacteristic
273C22×C4C22⋊Q811yescharacteristic
274C22×C4C22⋊Q811yescharacteristic
275C22×C4C22⋊Q811yescharacteristic
276C22×C4#13512yes—
277C22×C4#13512yes—
278C22×C4#13712yes—
279C22×C4#13712yes—
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C2×#4111yescharacteristic
283M16—24no—
284M16—24no—
285M16—24no—
286M16—24no—
287M16—42no—
288M16—42no—
289M16—44no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—24no—
295C2×C8—44no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—82no—
299C2×C8—82no—
300C2×C8C2×C22⋊C414yes—
301C2×C8C2×C22⋊C414yes—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4C2×C4⋊C414yes—
323C22⋊C4—24no—
324C22⋊C4—24no—
325C22⋊C4—82no—
326C42—24no—
327C42—24no—
328C42—82no—
329C42C2×C22⋊C411yescharacteristic
330C42C2×C22⋊C411yescharacteristic
331C42C2×C4⋊C411yescharacteristic
332C42C2×C4⋊C411yescharacteristic
333C42#11711yescharacteristic
334C42#11711yescharacteristic
335C42#12111yescharacteristic
336C42#12111yescharacteristic
337C42C22⋊Q811yescharacteristic
338C42C22⋊Q811yescharacteristic
339C42C22⋊Q811yescharacteristic
340C42C22⋊Q811yescharacteristic
341C42C23×C411yescharacteristic
342C42C22×D411yescharacteristic
343C42C22×Q811yescharacteristic
344C42C2×#4111yescharacteristic
345C23—41no—
346C23—41no—
347C23#38011yescharacteristic
348C23#38011yescharacteristic
349C23#46412yes—
350C23#46412yes—
351C23C2×#12111yescharacteristic
352Q8—44no—
353Q8—44no—
354Q8—82no—
355Q8—82no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—82no—
381C2×C4—164no—
382C2×C4#38011yescharacteristic
383C2×C4#38011yescharacteristic
384C2×C4#38011yescharacteristic
385C2×C4#38011yescharacteristic
386C2×C4#38011yescharacteristic
387C2×C4#38011yescharacteristic
388C2×C4#41612yes—
389C2×C4#41612yes—
390C2×C4#41912yes—
391C2×C4#41912yes—
392C2×C4#46412yes—
393C2×C4#46412yes—
394C2×C4#47912yes—
395C2×C4#47912yes—
396C2×C4#47912yes—
397C2×C4#47912yes—
398C2×C4C22×C22⋊C411yescharacteristic
399C2×C4C22×C4⋊C411yescharacteristic
400C2×C4C2×#11711yescharacteristic
401C2×C4C2×#12111yescharacteristic
402C2×C4C2×C22⋊Q811yescharacteristic
403C2×C4C2×C22⋊Q811yescharacteristic
404C2×C4C2×#13511yescharacteristic
405C2×C4C2×#13511yescharacteristic
406C2×C4C2×#13711yescharacteristic
407C2×C4C2×#13711yescharacteristic
408C8—24no—
409C8—24no—
410C8—44no—
411C8—82no—
412C8—82no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—42no—
420C22—42no—
421C22#180512yes—
422C22#180512yes—
423C22C2×#38011yescharacteristic
424C22C2×#46411yescharacteristic
425C22C2×#46411yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—44no—
435C4—44no—
436C4—161no—
437C4—161no—
438C4—162no—
439C4#180512yes—
440C4#180512yes—
441C4C2×#41611yescharacteristic
442C4C2×#41911yescharacteristic
443C4C2×#47911yescharacteristic
444C4C2×#47911yescharacteristic
445C2—21no—
446C2—21no—
447C2—22no—
448C2—22no—
449C2C2×#180511yescharacteristic
450C2#2067111yescharacteristic
451C2#2067511yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C42 ▷ 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. #8681C2semidirect
7. #8679C2semidirect
12. Q8×M16C4semidirect
14. #2962C4semidirect
68. #446C2×C4semidirect
70. C8×Q8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#601922
#207652
#177872
#175752
C2×#15182
#86812
#86792
#79572
C2×#12862