Finite Group Databasebeta

← #483272

Subgroups — #483272

Order 512.

672 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#62475C211yesmaximal, characteristic
2#33502C211yesmaximal, characteristic, complemented
3#17834C211yesmaximal, characteristic, complemented
4#11978C212yesmaximal, complemented
5#11977C211yesmaximal, characteristic, complemented
6#11968C211yesmaximal, characteristic, complemented
7#11967C212yesmaximal, complemented
8#11944C212yesmaximal, complemented
9#11943C211yesmaximal, characteristic, complemented
10#11934C211yesmaximal, characteristic, complemented
11C2×#1363C212yesmaximal, complemented
12#3515—21no—
13#3383C411yescharacteristic
14#3381—21no—
15#3380C411yescharacteristic
16#3339C412yes—
17#3338C411yescharacteristic
18#3337—22no—
19#3336C412yes—
20#3335—21no—
21#3334C411yescharacteristic
22#3233—21no—
23#3232—22no—
24#3168C2211yescharacteristic
25#3168C2212yes—
26#3160C2211yescharacteristic
27C2×#499C2212yes—
28#2990—21no—
29C8×#41C2211yescharacteristic
30Q8×C16—21no—
31D4×C16—21no—
32D4×C16—22no—
33#2162C2211yescharacteristic
34#2162C2212yes—
35C2×C4⋊C16C2212yes—
36C2×C4⋊C16C2211yescharacteristic
37#1574—21no—
38#1573—22no—
39#1568C2212yes—
40#1566C2211yescharacteristic
41#1564C2211yescharacteristic
42C2×#334C2212yes—
43#1363C2211yescharacteristic
44#1363C2212yes—
45#1363C2212yes—
46#1363C2212yes—
47#1363C2212yes—
48#1363C2211yescharacteristic
49#1363C2212yes—
50#1360C2211yescharacteristic
51#1360C2211yescharacteristic
52#1360C2212yes—
53#557—21no—
54#554—21no—
55#554—21no—
56#553—22no—
57#552—22no—
58#552—42no—
59#552—41no—
60#547—21no—
61#547—41no—
62#541—41no—
63#539—42no—
64#533—21no—
65#529—22no—
66#529—21no—
67C2×#125—22no—
68#520—41no—
69C4×#41D411yescharacteristic
70C22×C16—22no—
71C22×C16—21no—
72#500C2×C411yescharacteristic
73#500C2×C411yescharacteristic
74#499—22no—
75#499—22no—
76#499—21no—
77#499C2×C412yes—
78#499C2×C412yes—
79#499C2311yescharacteristic
80#499C2312yes—
81#498—21no—
82#495C2311yescharacteristic
83#488—21no—
84#488—22no—
85#485—22no—
86#484—22no—
87#483—22no—
88#482—22no—
89#481—22no—
90#480—22no—
91#479—22no—
92#479—22no—
93#478—22no—
94#478—22no—
95#477—21no—
96#476—21no—
97#475—21no—
98#474—21no—
99#467C2×C411yescharacteristic
100#467C2×C412yes—
101#466C2×C412yes—
102#466C2×C411yescharacteristic
103#465—22no—
104#465—21no—
105#446—41no—
106C8×Q8C2×C411yescharacteristic
107C8×Q8D411yescharacteristic
108C4×Q16C2×C412yes—
109C4×Q16C2×C411yescharacteristic
110C4×SD16—21no—
111C4×SD16—22no—
112C4×D8C2×C411yescharacteristic
113C4×D8C2×C412yes—
114#435—42no—
115C8×D4—41no—
116C8×D4C2×C411yescharacteristic
117C8×D4C2×C412yes—
118C8×D4D411yescharacteristic
119C8×D4D412yes—
120#432—22no—
121#431—21no—
122#431D412yes—
123#431D411yescharacteristic
124#427—21no—
125#427C2×C412yes—
126C2×#107C2×C412yes—
127C2×#107C2×C412yes—
128C2×#106—22no—
129#423—21no—
130C2×C4⋊C8—22no—
131C2×C4×C8C2312yes—
132C2×C4×C8C2311yescharacteristic
133C4⋊C16—21no—
134C4⋊C16—22no—
135C4⋊C16C2311yescharacteristic
136C4⋊C16C2312yes—
137C4⋊C16C2311yescharacteristic
138C22⋊C16—22no—
139C22⋊C16—21no—
140C4×C16—21no—
141C4×C16—22no—
142#335—21no—
143#335—22no—
144#334—22no—
145#334—21no—
146#334C2311yescharacteristic
147#334C2312yes—
148#334C2311yescharacteristic
149C2×#41—41no—
150C2×#41SD1611yescharacteristic
151C2×Q16C22×C411yescharacteristic
152C2×SD16—21no—
153C2×D8C22×C411yescharacteristic
154C2×M16—41no—
155C22×C8—42no—
156C22×C8C22×C412yes—
157C22×C8C22×C411yescharacteristic
158C22×C8C2×D412yes—
159C22×C8C2×D411yescharacteristic
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—41no—
163#126—42no—
164#126—42no—
165#126—42no—
166#126—41no—
167#125—22no—
168#125—22no—
169#125—21no—
170#125—22no—
171#125—42no—
172#125—42no—
173#125—41no—
174#125—42no—
175#125—42no—
176#124—21no—
177#123—42no—
178#123—41no—
179#123—42no—
180#123—42no—
181#123—42no—
182C22⋊Q8—21no—
183C22⋊Q8—22no—
184C22⋊Q8—41no—
185C22⋊Q8—41no—
186C22⋊Q8—41no—
187C22⋊Q8—42no—
188C22⋊Q8—41no—
189C22⋊Q8—42no—
190#121—22no—
191#121—21no—
192#121—42no—
193C4×Q8—22no—
194C4×Q8—21no—
195C4×Q8—41no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×Q8C22⋊C411yescharacteristic
199C4×Q8D811yescharacteristic
200C4×D4—21no—
201C4×D4—22no—
202C4×D4—41no—
203C4×D4—42no—
204C4×D4—41no—
205C4×D4—42no—
206C4×D4C22⋊C411yescharacteristic
207C4×D4C22⋊C412yes—
208C4×D4D811yescharacteristic
209C4×D4D812yes—
210#117—22no—
211#117—42no—
212#117—41no—
213#117—42no—
214#117SD1611yescharacteristic
215#117SD1612yes—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—41no—
221C2×C42C2×D412yes—
222C2×C42C2×D411yescharacteristic
223C2×C16—21no—
224C2×C16—21no—
225C2×C16—22no—
226C2×C16—82no—
227C2×C16—81no—
228#107—22no—
229#107—22no—
230#107—21no—
231#107—21no—
232#107C22×C411yescharacteristic
233#107C22×C411yescharacteristic
234#107C22×C412yes—
235#107C22×C412yes—
236#106—22no—
237#106—22no—
238#106—22no—
239#106—21no—
240#106—22no—
241C4⋊C8—21no—
242C4⋊C8—22no—
243C4⋊C8—21no—
244C4⋊C8—42no—
245C4⋊C8—41no—
246C4⋊C8C22⋊C411yescharacteristic
247C4⋊C8C22⋊C412yes—
248C4⋊C8C22⋊C411yescharacteristic
249C4⋊C8C22⋊C412yes—
250#103—22no—
251#103—21no—
252#103—22no—
253#103—21no—
254#102—22no—
255#102—21no—
256#102—22no—
257#102—21no—
258C22⋊C8—41no—
259C22⋊C8—42no—
260C22⋊C8C22⋊C412yes—
261C22⋊C8C22⋊C412yes—
262C22⋊C8C22⋊C412yes—
263#97—42no—
264C4×C8—41no—
265C4×C8C22×C412yes—
266C4×C8C22×C411yescharacteristic
267C4×C8C2×D411yescharacteristic
268C4×C8C2×D412yes—
269C4×C8C2×D412yes—
270C4×C8C2×D411yescharacteristic
271C4×C8C2411yescharacteristic
272#41—24no—
273#41—84no—
274C2×Q8—21no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8#10211yescharacteristic
278C2×D4—21no—
279C2×D4—42no—
280C2×D4—41no—
281C2×D4#10212yes—
282C2×D4#10211yescharacteristic
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×D812yes—
290C22×C4C2×D811yescharacteristic
291C22×C4C2×SD1612yes—
292C22×C4C2×SD1611yescharacteristic
293Q16—41no—
294SD16—41no—
295SD16—41no—
296D8—41no—
297M16—161no—
298C2×C8—22no—
299C2×C8—21no—
300C2×C8—22no—
301C2×C8—21no—
302C2×C8—42no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8—162no—
308C2×C8C2×C22⋊C412yes—
309C2×C8C2×C22⋊C411yescharacteristic
310C2×C8C2×C22⋊C411yescharacteristic
311C2×C8C2×C22⋊C412yes—
312C2×C8C23×C411yescharacteristic
313C2×C8C22×D411yescharacteristic
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—41no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—41no—
334C4⋊C4#10211yescharacteristic
335C4⋊C4#10211yescharacteristic
336C4⋊C4#10212yes—
337C4⋊C4#10212yes—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—41no—
347C22⋊C4#10212yes—
348C22⋊C4#10212yes—
349C22⋊C4#10212yes—
350C42—42no—
351C42—42no—
352C42—41no—
353C42C2×C22⋊C412yes—
354C42C2×C22⋊C411yescharacteristic
355C42C2×D811yescharacteristic
356C42C2×D812yes—
357C42C2×SD1612yes—
358C42C2×SD1611yescharacteristic
359C42C22×D411yescharacteristic
360C16—41no—
361C16—42no—
362C16—41no—
363C23—41no—
364C23C2×#10211yescharacteristic
365C23C2×#10212yes—
366Q8—22no—
367Q8—41no—
368Q8—41no—
369Q8—82no—
370Q8—161no—
371D4—24no—
372D4—41no—
373D4—41no—
374D4—84no—
375D4—161no—
376D4#35914yes—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—84no—
391C2×C4—82no—
392C2×C4—162no—
393C2×C4—162no—
394C2×C4C2×#10211yescharacteristic
395C2×C4C2×#10212yes—
396C2×C4C2×#10212yes—
397C2×C4C2×#10211yescharacteristic
398C2×C4C2×#10212yes—
399C2×C4C2×#10211yescharacteristic
400C2×C4C22×C22⋊C411yescharacteristic
401C2×C4C22×D811yescharacteristic
402C2×C4C22×SD1611yescharacteristic
403C8—21no—
404C8—21no—
405C8—22no—
406C8—22no—
407C8—21no—
408C8—81no—
409C8—81no—
410C8—82no—
411C8#57711yescharacteristic
412C8#57811yescharacteristic
413C22—22no—
414C22—22no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22C2×#35914yes—
422C22C22×#10211yescharacteristic
423C4—21no—
424C4—22no—
425C4—21no—
426C4—22no—
427C4—21no—
428C4—22no—
429C4—21no—
430C4—81no—
431C4—81no—
432C4—82no—
433C4—81no—
434C4—82no—
435C4C2×#35912yes—
436C4#288911yescharacteristic
437C4#289811yescharacteristic
438C2—24no—
439C2—41no—
440C2—161no—
441C2#1772111yescharacteristic
442C2C22×#35911yescharacteristic
443C2#3338811yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
prime radical#62475C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33502C2semidirect
3. #17834C2semidirect
4. #11978C2semidirect
5. #11977C2semidirect
6. #11968C2semidirect
7. #11967C2semidirect
8. #11944C2semidirect
9. #11943C2semidirect
10. #11934C2semidirect
11. C2×#1363C2semidirect

Maximal subgroups

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

subgroupindex
#624752
#335022
#178342
#119782
#119772
#119682
#119672
#119442
#119432
#119342
C2×#13632