Finite Group Databasebeta

← #117415

Subgroups — #117415

Order 512.

759 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21993C211yesmaximal, characteristic, complemented
2C2×#1962C211yesmaximal, characteristic
3#17717C211yesmaximal, characteristic
4#17478C211yesmaximal, characteristic
5#7934C211yesmaximal, characteristic, complemented
6#7867C211yesmaximal, characteristic, complemented
7C2×#1276C211yesmaximal, characteristic, complemented
8C2×#532C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C22×#104—22no—
15C2×C43C2211yescharacteristic
16#1962—22no—
17#1962—22no—
18#1950—22no—
19#1855—22no—
20#1848—21no—
21#1837—21no—
22#1553—22no—
23#1549C411yescharacteristic, complemented
24#1549C411yescharacteristic, complemented
25#1545C411yescharacteristic, complemented
26#1545C411yescharacteristic, complemented
27#1540—22no—
28#1496—22no—
29#1492—21no—
30#1492—21no—
31#1478—22no—
32#1283—22no—
33#1276—22no—
34#1276—22no—
35C22×#41—21no—
36C22×M16D412yes—
37#532—21no—
38#532—21no—
39#532—21no—
40#532—21no—
41#532—21no—
42#532—21no—
43#532D412yes—
44#532Q812yes—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#127C2×C411yescharacteristic
47C2×#124—21no—
48C2×#121—21no—
49C2×#121—21no—
50C2×C4×D4—21no—
51C2×C4×D4—21no—
52C22×C42—21no—
53C22×C42—21no—
54C22×C42—21no—
55C22×C42C2311yescharacteristic
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431C2×C411yescharacteristic
70#431C2×C411yescharacteristic
71#431D412yes—
72#431D412yes—
73C2×#108—24no—
74#423—21no—
75#423—21no—
76#423—21no—
77#423—21no—
78#423C2×C411yescharacteristic
79#423C2×C411yescharacteristic
80#423D411yescharacteristic
81#423D411yescharacteristic
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8D412yes—
86C2×#104—28no—
87C2×#104—48no—
88#407—21no—
89#407—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C4×M16—21no—
93C4×M16—21no—
94C2×C4×C8—21no—
95C2×C4×C8—21no—
96C43—21no—
97C43—21no—
98C43—21no—
99C43D412yes—
100#341—24no—
101#332—22no—
102#332—22no—
103#331—22no—
104#331—22no—
105D4⋊C8—24no—
106D4⋊C8—24no—
107C2×#41—22no—
108C2×#41—22no—
109C2×#41—44no—
110C2×#41—44no—
111C2×#41—41no—
112C2×#41—41no—
113C22×Q8—21no—
114C22×D4—21no—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4—41no—
118C23×C4—41no—
119C23×C4—41no—
120C23×C4C2×D411yescharacteristic
121C23×C4#4111yescharacteristic
122C23×C4#4111yescharacteristic
123C2×M16—21no—
124C2×M16—21no—
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—21no—
128C2×M16—21no—
129C2×M16—44no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—41no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8C4⋊C412yes—
137#127—21no—
138#127—21no—
139#127—21no—
140#127C4⋊C412yes—
141#124—22no—
142#124—22no—
143#124—41no—
144#124—41no—
145#121—22no—
146#121—22no—
147#121—22no—
148#121—22no—
149#121—41no—
150#121—41no—
151#121—41no—
152#121—41no—
153C4×D4—22no—
154C4×D4—22no—
155C4×D4—22no—
156C4×D4—22no—
157C4×D4—41no—
158C4×D4—41no—
159C4×D4—41no—
160C4×D4—41no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C42—24no—
166C2×C42—24no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—44no—
171C2×C42—42no—
172C2×C42—42no—
173C2×C42—41no—
174C2×C42—41no—
175C2×C42—41no—
176C2×C42C22×C411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×Q811yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188#108—84no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—22no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—22no—
199C4⋊C8—41no—
200C4⋊C8—41no—
201C4⋊C8—41no—
202C4⋊C8—41no—
203C4⋊C8—41no—
204C4⋊C8C22⋊C412yes—
205C4⋊C8C22⋊C412yes—
206#104—416no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8—41no—
212C22⋊C8—41no—
213C22⋊C8—41no—
214C22⋊C8—41no—
215C22⋊C8—41no—
216#97—41no—
217#97—41no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—22no—
221C4×C8—22no—
222C4×C8—41no—
223C4×C8—41no—
224C4×C8—41no—
225C24—41no—
226C24#12411yescharacteristic
227#41—44no—
228#41—48no—
229#41—44no—
230#41—84no—
231#41—82no—
232#41—82no—
233C2×Q8—22no—
234C2×Q8—44no—
235C2×Q8—41no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—44no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—82no—
248C2×D4—84no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4#11711yescharacteristic
264C22×C4C4×D411yescharacteristic
265C22×C4C4×D411yescharacteristic
266C22×C4#12011yescharacteristic
267C22×C4#12011yescharacteristic
268C22×C4#12111yescharacteristic
269C22×C4#12111yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4C22⋊Q811yescharacteristic
273C22×C4C22⋊Q811yescharacteristic
274C22×C4C22⋊Q811yescharacteristic
275C22×C4#12311yescharacteristic
276C22×C4#12311yescharacteristic
277C22×C4#12311yescharacteristic
278C22×C4#12411yescharacteristic
279C22×C4#12411yescharacteristic
280C22×C4#12611yescharacteristic
281M16—44no—
282M16—42no—
283M16—42no—
284M16—42no—
285M16—44no—
286M16—42no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—82no—
298C2×C8—82no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C42—24no—
308C42—24no—
309C42—24no—
310C42—44no—
311C42—44no—
312C42—42no—
313C42—42no—
314C42—42no—
315C42—44no—
316C42C2×C22⋊C411yescharacteristic
317C42C2×C4⋊C411yescharacteristic
318C42C4×D411yescharacteristic
319C42C4×D411yescharacteristic
320C42#12011yescharacteristic
321C42#12111yescharacteristic
322C42#12111yescharacteristic
323C42#12111yescharacteristic
324C42#12111yescharacteristic
325C42C22⋊Q811yescharacteristic
326C42C22⋊Q811yescharacteristic
327C42C22⋊Q811yescharacteristic
328C42#12311yescharacteristic
329C42#12311yescharacteristic
330C42#12611yescharacteristic
331C42#12711yescharacteristic
332C23—22no—
333C23—22no—
334C23—41no—
335C23—42no—
336C23—82no—
337C23—82no—
338C23—82no—
339C23#38811yescharacteristic
340C23#39311yescharacteristic
341C23#39411yescharacteristic
342Q8—44no—
343Q8—82no—
344D4—44no—
345D4—44no—
346D4—82no—
347D4—84no—
348D4—84no—
349D4—82no—
350D4—82no—
351D4—81no—
352D4—81no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—416no—
377C2×C4—42no—
378C2×C4—84no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4#38011yescharacteristic
383C2×C4#38511yescharacteristic
384C2×C4#38511yescharacteristic
385C2×C4#38611yescharacteristic
386C2×C4#38811yescharacteristic
387C2×C4#39011yescharacteristic
388C2×C4#39211yescharacteristic
389C2×C4#39411yescharacteristic
390C2×C4#39611yescharacteristic
391C2×C4#39611yescharacteristic
392C2×C4#39711yescharacteristic
393C2×C4#39911yescharacteristic
394C2×C4#44312yes—
395C2×C4#44412yes—
396C2×C4#45312yes—
397C2×C4#45412yes—
398C8—42no—
399C8—42no—
400C8—82no—
401C8—81no—
402C8—81no—
403C8—82no—
404C8—81no—
405C8—81no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—44no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22—164no—
418C22#143911yescharacteristic
419C22#192912yes—
420C22#202712yes—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—48no—
429C4—82no—
430C4#186612yes—
431C4#188312yes—
432C4#193212yes—
433C4#197411yescharacteristic
434C4#197511yescharacteristic
435C4#200211yescharacteristic
436C4#200311yescharacteristic
437C2—22no—
438C2—22no—
439C2—41no—
440C2—162no—
441C2#1106311yescharacteristic
442C2#1112511yescharacteristic
443C2#1114611yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC2×#532C22
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. #21993C2semidirect
5. #7934C2semidirect
6. #7867C2semidirect
7. C2×#1276C2semidirect
23. #1549C4semidirect
24. #1549C4semidirect
25. #1545C4semidirect
26. #1545C4semidirect

Maximal subgroups

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

subgroupindex
#219932
C2×#19622
#177172
#174782
#79342
#78672
C2×#12762