Finite Group Databasebeta

← #117659

Subgroups — #117659

Order 512.

649 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21807C211yesmaximal, characteristic
2#17819C211yesmaximal, characteristic, complemented
3#17544C211yesmaximal, characteristic
4#17463C211yesmaximal, characteristic
5#7941C211yesmaximal, characteristic, complemented
6#7924C211yesmaximal, characteristic, complemented
7#7858C211yesmaximal, characteristic, complemented
8C2×#532C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2283C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1982C411yescharacteristic, complemented
16#1982C411yescharacteristic, complemented
17#1981C411yescharacteristic, complemented
18#1981C411yescharacteristic, complemented
19#1958—22no—
20#1788—22no—
21#1574—22no—
22#1574—22no—
23#1568—22no—
24#1507—22no—
25#1505—22no—
26#1502—22no—
27#1484—22no—
28#1480—22no—
29#1478—22no—
30#1284—22no—
31#1282—22no—
32#1274—22no—
33C22×#41—21no—
34#532—21no—
35#532—21no—
36#532—21no—
37#532—21no—
38#532—21no—
39#532—21no—
40#532D412yes—
41#532Q812yes—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#127C2×C411yescharacteristic
44C2×#124—21no—
45C2×#121—21no—
46C2×#121—21no—
47C2×C4×D4—21no—
48C2×C4×D4—21no—
49C2×#117—21no—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431D412yes—
65#431D412yes—
66#431D412yes—
67#423—21no—
68#423—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8C2×C411yescharacteristic
71C2×C4⋊C8C2×C411yescharacteristic
72C2×#103—22no—
73C2×#103—22no—
74C2×#102—22no—
75C2×#102—22no—
76#407—21no—
77#407—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8D412yes—
80C4×M16—21no—
81C4×M16—21no—
82C2×C4×C8—21no—
83C2×C4×C8D412yes—
84#383—21no—
85#382—21no—
86#381—21no—
87#380—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4C2×C411yescharacteristic
91C4×C4⋊C4C2×C411yescharacteristic
92C4×C22⋊C4—21no—
93C4×C22⋊C4—21no—
94C4×C22⋊C4D412yes—
95#376—21no—
96#376—21no—
97C43—21no—
98#335—24no—
99#334—24no—
100#327—28no—
101D4⋊C8—28no—
102C2×#41—22no—
103C2×#41—22no—
104C2×#41—41no—
105C2×#41—41no—
106C2×#41—82no—
107C2×#41—82no—
108C22×Q8—21no—
109C22×D4—21no—
110C22×D4—21no—
111C22×D4—21no—
112C22×D4—41no—
113C23×C4—41no—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4#4111yescharacteristic
117C2×M16—41no—
118C2×M16—41no—
119C22×C8—41no—
120C22×C8C22⋊C412yes—
121C22×C8C22⋊C412yes—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8C4⋊C412yes—
126#127—21no—
127#127—21no—
128#127—21no—
129#127C4⋊C412yes—
130#124—22no—
131#124—22no—
132#124—41no—
133#124—41no—
134#121—22no—
135#121—22no—
136#121—22no—
137#121—22no—
138#121—41no—
139#121—41no—
140#121—41no—
141#121—41no—
142C4×D4—22no—
143C4×D4—22no—
144C4×D4—22no—
145C4×D4—22no—
146C4×D4—41no—
147C4×D4—41no—
148C4×D4—41no—
149C4×D4—41no—
150#117—24no—
151#117—42no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—41no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—22no—
164C2×C42—41no—
165C2×C42—41no—
166C2×C42C22×C411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×Q811yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187C4⋊C8—41no—
188#103—82no—
189#103—82no—
190#102—82no—
191#102—82no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200#97—41no—
201#97—41no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—22no—
207C4×C8—41no—
208#95—21no—
209#95—21no—
210#95—22no—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—41no—
215C24—41no—
216C24#12111yescharacteristic
217#41—82no—
218#41—82no—
219#41—82no—
220#41—82no—
221#41—84no—
222#41—84no—
223C2×Q8—22no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—41no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—82no—
240C2×D4—84no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—81no—
252C22×C4—84no—
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C4×D411yescharacteristic
255C22×C4C4×D411yescharacteristic
256C22×C4#12011yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4#12111yescharacteristic
262C22×C4C22⋊Q811yescharacteristic
263C22×C4C22⋊Q811yescharacteristic
264C22×C4C22⋊Q811yescharacteristic
265C22×C4#12311yescharacteristic
266C22×C4#12311yescharacteristic
267C22×C4#12411yescharacteristic
268C22×C4#12411yescharacteristic
269C22×C4#12611yescharacteristic
270C22×C4#12711yescharacteristic
271M16—82no—
272M16—82no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—81no—
281C2×C8—81no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—44no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C22⋊C4—24no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—82no—
298C42—44no—
299C42—42no—
300C42—42no—
301C42—81no—
302C42—81no—
303C42—82no—
304C42C2×C4⋊C411yescharacteristic
305C42#11711yescharacteristic
306C42C4×D411yescharacteristic
307C42C4×D411yescharacteristic
308C42#12011yescharacteristic
309C42#12011yescharacteristic
310C42#12111yescharacteristic
311C42#12111yescharacteristic
312C42C22⋊Q811yescharacteristic
313C42C22⋊Q811yescharacteristic
314C42C22⋊Q811yescharacteristic
315C42#12311yescharacteristic
316C42#12311yescharacteristic
317C42#12311yescharacteristic
318C42#12411yescharacteristic
319C42#12611yescharacteristic
320C23—21no—
321C23—21no—
322C23—41no—
323C23—41no—
324C23—42no—
325C23—82no—
326C23—82no—
327C23—81no—
328C23—81no—
329C23#39011yescharacteristic
330C23#39411yescharacteristic
331C23#39611yescharacteristic
332Q8—44no—
333Q8—81no—
334Q8—81no—
335D4—44no—
336D4—44no—
337D4—82no—
338D4—82no—
339D4—84no—
340D4—84no—
341D4—81no—
342D4—81no—
343D4—82no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4#38011yescharacteristic
379C2×C4#38511yescharacteristic
380C2×C4#38511yescharacteristic
381C2×C4#38611yescharacteristic
382C2×C4#38811yescharacteristic
383C2×C4#38811yescharacteristic
384C2×C4#39211yescharacteristic
385C2×C4#39311yescharacteristic
386C2×C4#39411yescharacteristic
387C2×C4#39611yescharacteristic
388C2×C4#39711yescharacteristic
389C2×C4#39911yescharacteristic
390C2×C4#44312yes—
391C2×C4#44412yes—
392C8—42no—
393C8—42no—
394C8—82no—
395C8—82no—
396C8—82no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—81no—
406C22—81no—
407C22—82no—
408C22—82no—
409C22—82no—
410C22—162no—
411C22—162no—
412C22#143911yescharacteristic
413C22#197411yescharacteristic
414C22#197511yescharacteristic
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—84no—
425C4—82no—
426C4—161no—
427C4—161no—
428C4#186612yes—
429C4#191011yescharacteristic
430C4#191011yescharacteristic
431C4#192912yes—
432C2—21no—
433C2—21no—
434C2—41no—
435C2—41no—
436C2—162no—
437C2#1107311yescharacteristic
438C2#1112511yescharacteristic
439C2#1113211yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC2×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17819C2semidirect
5. #7941C2semidirect
6. #7924C2semidirect
7. #7858C2semidirect
15. #1982C4semidirect
16. #1982C4semidirect
17. #1981C4semidirect
18. #1981C4semidirect

Maximal subgroups

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

subgroupindex
#218072
#178192
#175442
#174632
#79412
#79242
#78582