Finite Group Databasebeta

← #117909

Subgroups — #117909

Order 512.

592 conjugacy classes of subgroups.

The subgroup lattice has 463 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17793C211yesmaximal, characteristic, complemented
2#17537C211yesmaximal, characteristic
3C2×#1494C211yesmaximal, characteristic
4#17380C211yesmaximal, characteristic
5#7949C211yesmaximal, characteristic, complemented
6#7890C211yesmaximal, characteristic, complemented
7#7864C211yesmaximal, characteristic, complemented
8C2×#529C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1568—22no—
16#1567—21no—
17#1566—21no—
18#1564—22no—
19#1509—22no—
20#1508—21no—
21#1507—21no—
22#1500—22no—
23#1494—21no—
24#1494—21no—
25#1494—21no—
26#1494—21no—
27C2×Q8⋊C8C411yescharacteristic, complemented
28C2×Q8⋊C8C411yescharacteristic, complemented
29C2×D4⋊C8C411yescharacteristic, complemented
30C2×D4⋊C8C411yescharacteristic, complemented
31#1473—22no—
32#1472—21no—
33#1471—21no—
34#1458—22no—
35#1285—22no—
36#1279—21no—
37#1279—21no—
38#1275—22no—
39C23×C8—21no—
40#529—21no—
41#529—21no—
42#529—21no—
43#529—21no—
44#529—21no—
45#529—21no—
46#529—21no—
47#529—21no—
48C2×#126—21no—
49C2×#125Q811yescharacteristic
50C2×#124Q811yescharacteristic
51C2×#123—21no—
52C2×C22⋊Q8D411yescharacteristic
53C2×#121D411yescharacteristic
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×D4—21no—
56C2×C4×D4C2×C411yescharacteristic
57C2×#117—21no—
58C2×#117—21no—
59C22×C42C2311yescharacteristic
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432—21no—
66#432—21no—
67#432D412yes—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431D412yes—
75#431D412yes—
76#423—21no—
77#423—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8C2×C411yescharacteristic
80C2×C4⋊C8C2×C411yescharacteristic
81#407—21no—
82#407—21no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C4×M16—21no—
86C4×M16—21no—
87C2×#97—21no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91C2×C4×C8—21no—
92C2×C4×C8C2×C411yescharacteristic
93C2×C4×C8C2×C411yescharacteristic
94C2×C4×C8D411yescharacteristic
95C2×C4×C8D411yescharacteristic
96C2×C4×C8D411yescharacteristic
97C2×C4×C8D411yescharacteristic
98C4×C4⋊C4—21no—
99C4×C4⋊C4D412yes—
100C4×C22⋊C4—21no—
101C4×C22⋊C4—21no—
102#376—21no—
103C43—21no—
104#335—24no—
105#334—24no—
106#328—24no—
107#327—24no—
108Q8⋊C8—21no—
109Q8⋊C8—21no—
110Q8⋊C8—21no—
111Q8⋊C8—21no—
112D4⋊C8—21no—
113D4⋊C8—21no—
114D4⋊C8—21no—
115D4⋊C8—21no—
116#324—24no—
117#323—24no—
118C22×Q8C4⋊C411yescharacteristic
119C22×D4—21no—
120C22×D4C4⋊C411yescharacteristic
121C23×C4—21no—
122C23×C4C2×D411yescharacteristic
123C23×C4#4111yescharacteristic
124C23×C4#4111yescharacteristic
125C2×M16—41no—
126C2×M16—41no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—44no—
136C22×C8—41no—
137#126—41no—
138#126—41no—
139#126—41no—
140#126—41no—
141#125—21no—
142#125—21no—
143#125—21no—
144#125—21no—
145#124—21no—
146#124—21no—
147#124—21no—
148#124—21no—
149#123—41no—
150#123—41no—
151#123—41no—
152#123—41no—
153C22⋊Q8—21no—
154C22⋊Q8—21no—
155C22⋊Q8—21no—
156C22⋊Q8—21no—
157#121—21no—
158#121—21no—
159#121—21no—
160#121—21no—
161C4×Q8—21no—
162C4×Q8—21no—
163C4×Q8—21no—
164C4×Q8—21no—
165C4×D4—21no—
166C4×D4—21no—
167C4×D4—21no—
168C4×D4—21no—
169C4×D4—41no—
170C4×D4—41no—
171C4×D4—41no—
172C4×D4—41no—
173#117—24no—
174#117—42no—
175#117—41no—
176#117—41no—
177#117—41no—
178#117—41no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4C4⋊C411yescharacteristic
185C2×C4⋊C4C4⋊C411yescharacteristic
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—41no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—41no—
195C2×C42—41no—
196C2×C42C22×C411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×Q811yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42#4111yescharacteristic
206C2×C42#4111yescharacteristic
207C2×C42#4111yescharacteristic
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—41no—
216C4⋊C8—41no—
217C4⋊C8—41no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—41no—
222C22⋊C8—41no—
223C22⋊C8—41no—
224C22⋊C8—41no—
225C22⋊C8—41no—
226#97—22no—
227#97—41no—
228#97—41no—
229#97—41no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—22no—
237C4×C8—22no—
238C4×C8—41no—
239C4×C8—41no—
240C4×C8C22⋊C411yescharacteristic
241C4×C8C22⋊C411yescharacteristic
242C4×C8C22⋊C411yescharacteristic
243C4×C8C22⋊C411yescharacteristic
244#95—22no—
245#95—41no—
246C24—21no—
247C24#12311yescharacteristic
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8—42no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—42no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—82no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4C2×C4⋊C411yescharacteristic
270C22×C4C4×D411yescharacteristic
271C22×C4C4×D411yescharacteristic
272C22×C4#12011yescharacteristic
273C22×C4#12011yescharacteristic
274C22×C4#12111yescharacteristic
275C22×C4#12111yescharacteristic
276C22×C4#12111yescharacteristic
277C22×C4#12111yescharacteristic
278C22×C4C22⋊Q811yescharacteristic
279C22×C4C22⋊Q811yescharacteristic
280C22×C4#12311yescharacteristic
281C22×C4#12311yescharacteristic
282C22×C4#12311yescharacteristic
283C22×C4#12411yescharacteristic
284C22×C4#12411yescharacteristic
285C22×C4#12611yescharacteristic
286C22×C4#12611yescharacteristic
287M16—82no—
288M16—82no—
289C2×C8—24no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—82no—
307C4⋊C4—24no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C22⋊C4—44no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C42—44no—
333C42—41no—
334C42—41no—
335C42—82no—
336C42—82no—
337C42C2×C22⋊C411yescharacteristic
338C42#11711yescharacteristic
339C42C4×D411yescharacteristic
340C42C4×D411yescharacteristic
341C42#12011yescharacteristic
342C42#12111yescharacteristic
343C42#12111yescharacteristic
344C42#12111yescharacteristic
345C42#12111yescharacteristic
346C42C22⋊Q811yescharacteristic
347C42C22⋊Q811yescharacteristic
348C42C22⋊Q811yescharacteristic
349C42C22⋊Q811yescharacteristic
350C42#12311yescharacteristic
351C42#12411yescharacteristic
352C42#12711yescharacteristic
353C23—22no—
354C23—22no—
355C23—41no—
356C23—41no—
357C23—42no—
358C23—81no—
359C23—81no—
360C23#38511yescharacteristic
361C23#39411yescharacteristic
362C23#39711yescharacteristic
363Q8—42no—
364Q8—81no—
365D4—42no—
366D4—81no—
367D4—82no—
368D4—82no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4#38011yescharacteristic
401C2×C4#38511yescharacteristic
402C2×C4#38611yescharacteristic
403C2×C4#38811yescharacteristic
404C2×C4#38811yescharacteristic
405C2×C4#39011yescharacteristic
406C2×C4#39211yescharacteristic
407C2×C4#39311yescharacteristic
408C2×C4#39411yescharacteristic
409C2×C4#39611yescharacteristic
410C2×C4#39611yescharacteristic
411C2×C4#39911yescharacteristic
412C2×C4#44312yes—
413C2×C4#44412yes—
414C8—22no—
415C8—22no—
416C8—41no—
417C8—41no—
418C8—82no—
419C8—82no—
420C8—82no—
421C8—81no—
422C8—81no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—44no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—162no—
435C22#143911yescharacteristic
436C22#186612yes—
437C22#205012yes—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—84no—
448C4—81no—
449C4—81no—
450C4—81no—
451C4#181212yes—
452C4#192912yes—
453C4#197411yescharacteristic
454C4#197511yescharacteristic
455C2—22no—
456C2—22no—
457C2—41no—
458C2—161no—
459C2#1108211yescharacteristic
460C2#1109811yescharacteristic
461C2#1112511yescharacteristic
subgroupG/H
centerC2×C4#388
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC2×#121D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17793C2semidirect
5. #7949C2semidirect
6. #7890C2semidirect
7. #7864C2semidirect
27. C2×Q8⋊C8C4semidirect
28. C2×Q8⋊C8C4semidirect
29. C2×D4⋊C8C4semidirect
30. C2×D4⋊C8C4semidirect

Maximal subgroups

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

subgroupindex
#177932
#175372
C2×#14942
#173802
#79492
#78902
#78642