Finite Group Databasebeta

← #483293

Subgroups — #483293

Order 512.

685 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62475C211yesmaximal, characteristic
2#33504C211yesmaximal, characteristic, complemented
3#17826C211yesmaximal, characteristic, complemented
4#11980C211yesmaximal, characteristic, complemented
5#11977C212yesmaximal, complemented
6#11969C212yesmaximal, complemented
7#11968C211yesmaximal, characteristic, complemented
8#11945C212yesmaximal, complemented
9#11944C211yesmaximal, characteristic, complemented
10#11936C211yesmaximal, 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—
28C8×#41C2211yescharacteristic
29#2906—21no—
30#2183—21no—
31#2168—22no—
32D4×C16—21no—
33#2163C2212yes—
34#2162C2211yescharacteristic
35#2150C2211yescharacteristic
36C2×C4⋊C16C2212yes—
37#1570—22no—
38#1570—21no—
39#1566C2211yescharacteristic
40#1566C2212yes—
41C2×#334C2212yes—
42C2×#334C2211yescharacteristic
43#1363C2211yescharacteristic
44#1363C2212yes—
45#1363C2212yes—
46#1363C2212yes—
47#1363C2212yes—
48#1363C2211yescharacteristic
49#1363C2212yes—
50#1360C2211yescharacteristic
51#1360C2211yescharacteristic
52#1360C2212yes—
53C2×#131—41no—
54#557—21no—
55#554—21no—
56#554—21no—
57#553—22no—
58#552—22no—
59#552—42no—
60#552—41no—
61#547—21no—
62#547—41no—
63#541—41no—
64#539—42no—
65#533—21no—
66#529—22no—
67#529—21no—
68C2×#125—22no—
69#520—41no—
70C4×#41D411yescharacteristic
71C2×M32—21no—
72C22×C16—22no—
73#500C2×C411yescharacteristic
74#500C2×C411yescharacteristic
75#499—22no—
76#499—22no—
77#499—21no—
78#499C2×C412yes—
79#499C2×C412yes—
80#499C2311yescharacteristic
81#499C2312yes—
82#498—21no—
83#495C2311yescharacteristic
84#488—21no—
85#488—22no—
86#485—22no—
87#484—22no—
88#483—22no—
89#482—22no—
90#481—22no—
91#480—22no—
92#479—22no—
93#479—22no—
94#478—22no—
95#478—22no—
96#477—21no—
97#476—21no—
98#475—21no—
99#474—21no—
100#467C2×C411yescharacteristic
101#467C2×C412yes—
102#466C2×C412yes—
103#466C2×C411yescharacteristic
104#465—22no—
105#465—21no—
106C8×Q8C2×C411yescharacteristic
107C8×Q8D411yescharacteristic
108C4×Q16C2×C412yes—
109C4×Q16C2×C411yescharacteristic
110C4×SD16—21no—
111C4×SD16—22no—
112C4×D8C2×C411yescharacteristic
113C4×D8C2×C412yes—
114C8×D4C2×C411yescharacteristic
115C8×D4C2×C412yes—
116C8×D4D411yescharacteristic
117C8×D4D412yes—
118#431D412yes—
119#431D411yescharacteristic
120#427—21no—
121#427C2×C412yes—
122C2×#107C2×C412yes—
123C2×#107C2×C412yes—
124C2×#106—22no—
125#424—41no—
126#424—42no—
127#423—22no—
128#423—21no—
129C2×C4⋊C8—22no—
130C2×C4⋊C8—21no—
131C2×C4×C8C2312yes—
132C2×C4×C8C2311yescharacteristic
133C4⋊C16—22no—
134C4⋊C16—21no—
135C4⋊C16C2311yescharacteristic
136C4⋊C16C2312yes—
137C4⋊C16C2311yescharacteristic
138C22⋊C16—21no—
139C22⋊C16—22no—
140#346—22no—
141C4×C16—21no—
142#334—22no—
143#334—21no—
144#334—21no—
145#334—22no—
146#334C2311yescharacteristic
147#334C2312yes—
148#334C2311yescharacteristic
149C2×#41—41no—
150C2×#41D811yescharacteristic
151C2×Q16C22×C411yescharacteristic
152C2×SD16—21no—
153C2×D8C22×C411yescharacteristic
154#131—84no—
155C2×M16—42no—
156C2×M16—41no—
157C22×C8—42no—
158C22×C8—41no—
159C22×C8C22×C412yes—
160C22×C8C22×C411yescharacteristic
161C22×C8C2×D412yes—
162C22×C8C2×D411yescharacteristic
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—41no—
166#126—42no—
167#126—42no—
168#126—41no—
169#126—42no—
170#125—22no—
171#125—22no—
172#125—21no—
173#125—22no—
174#125—42no—
175#125—42no—
176#125—41no—
177#125—42no—
178#125—42no—
179#124—21no—
180#123—42no—
181#123—42no—
182#123—41no—
183#123—42no—
184#123—42no—
185C22⋊Q8—21no—
186C22⋊Q8—22no—
187C22⋊Q8—41no—
188C22⋊Q8—41no—
189C22⋊Q8—41no—
190C22⋊Q8—42no—
191C22⋊Q8—41no—
192C22⋊Q8—42no—
193#121—22no—
194#121—21no—
195#121—42no—
196C4×Q8—22no—
197C4×Q8—21no—
198C4×Q8—41no—
199C4×Q8—41no—
200C4×Q8—41no—
201C4×Q8C22⋊C411yescharacteristic
202C4×Q8SD1611yescharacteristic
203C4×D4—21no—
204C4×D4—22no—
205C4×D4—41no—
206C4×D4—41no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4C22⋊C411yescharacteristic
210C4×D4C22⋊C412yes—
211C4×D4SD1611yescharacteristic
212C4×D4SD1612yes—
213#117—22no—
214#117—42no—
215#117—42no—
216#117—41no—
217#117D811yescharacteristic
218#117D812yes—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—41no—
224C2×C42C2×D412yes—
225C2×C42C2×D411yescharacteristic
226M32—81no—
227C2×C16—21no—
228C2×C16—21no—
229C2×C16—22no—
230C2×C16—82no—
231#107—22no—
232#107—22no—
233#107—21no—
234#107—21no—
235#107C22×C411yescharacteristic
236#107C22×C411yescharacteristic
237#107C22×C412yes—
238#107C22×C412yes—
239#106—22no—
240#106—22no—
241#106—22no—
242#106—21no—
243#106—22no—
244C4⋊C8—21no—
245C4⋊C8—22no—
246C4⋊C8—21no—
247C4⋊C8—42no—
248C4⋊C8—41no—
249C4⋊C8—42no—
250C4⋊C8—41no—
251C4⋊C8C22⋊C411yescharacteristic
252C4⋊C8C22⋊C412yes—
253C4⋊C8C22⋊C411yescharacteristic
254C4⋊C8C22⋊C412yes—
255#103—22no—
256#103—21no—
257#103—22no—
258#103—21no—
259#102—22no—
260#102—21no—
261#102—22no—
262#102—21no—
263C22⋊C8C22⋊C412yes—
264C22⋊C8C22⋊C412yes—
265C22⋊C8C22⋊C412yes—
266C4×C8C22×C412yes—
267C4×C8C22×C411yescharacteristic
268C4×C8C2×D411yescharacteristic
269C4×C8C2×D412yes—
270C4×C8C2×D412yes—
271C4×C8C2×D411yescharacteristic
272C4×C8C2411yescharacteristic
273#41—24no—
274#41—84no—
275C2×Q8—21no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8#10211yescharacteristic
279C2×D4—21no—
280C2×D4—42no—
281C2×D4—41no—
282C2×D4#10212yes—
283C2×D4#10211yescharacteristic
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×D812yes—
291C22×C4C2×D811yescharacteristic
292C22×C4C2×SD1612yes—
293C22×C4C2×SD1611yescharacteristic
294Q16—41no—
295SD16—41no—
296SD16—41no—
297D8—41no—
298M16—84no—
299M16—82no—
300C2×C8—22no—
301C2×C8—21no—
302C2×C8—22no—
303C2×C8—21no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—42no—
308C2×C8—41no—
309C2×C8—84no—
310C2×C8—82no—
311C2×C8C2×C22⋊C412yes—
312C2×C8C2×C22⋊C411yescharacteristic
313C2×C8C2×C22⋊C411yescharacteristic
314C2×C8C2×C22⋊C412yes—
315C2×C8C23×C411yescharacteristic
316C2×C8C22×D411yescharacteristic
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—41no—
333C4⋊C4—42no—
334C4⋊C4—41no—
335C4⋊C4—42no—
336C4⋊C4—41no—
337C4⋊C4#10211yescharacteristic
338C4⋊C4#10211yescharacteristic
339C4⋊C4#10212yes—
340C4⋊C4#10212yes—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—41no—
350C22⋊C4#10212yes—
351C22⋊C4#10212yes—
352C22⋊C4#10212yes—
353C42—42no—
354C42—42no—
355C42—41no—
356C42C2×C22⋊C412yes—
357C42C2×C22⋊C411yescharacteristic
358C42C2×D812yes—
359C42C2×D811yescharacteristic
360C42C2×SD1611yescharacteristic
361C42C2×SD1612yes—
362C42C22×D411yescharacteristic
363C16—41no—
364C16—42no—
365C16—41no—
366C23—41no—
367C23C2×#10211yescharacteristic
368C23C2×#10212yes—
369Q8—22no—
370Q8—41no—
371Q8—41no—
372Q8—82no—
373Q8—161no—
374D4—24no—
375D4—41no—
376D4—41no—
377D4—84no—
378D4—161no—
379D4#36014yes—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—84no—
394C2×C4—82no—
395C2×C4—162no—
396C2×C4—162no—
397C2×C4C2×#10211yescharacteristic
398C2×C4C2×#10212yes—
399C2×C4C2×#10211yescharacteristic
400C2×C4C2×#10212yes—
401C2×C4C2×#10212yes—
402C2×C4C2×#10211yescharacteristic
403C2×C4C22×C22⋊C411yescharacteristic
404C2×C4C22×D811yescharacteristic
405C2×C4C22×SD1611yescharacteristic
406C8—21no—
407C8—21no—
408C8—22no—
409C8—22no—
410C8—21no—
411C8—81no—
412C8—82no—
413C8—81no—
414C8#57711yescharacteristic
415C8#57811yescharacteristic
416C22—22no—
417C22—22no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22C2×#36014yes—
425C22C22×#10211yescharacteristic
426C4—21no—
427C4—22no—
428C4—21no—
429C4—22no—
430C4—21no—
431C4—21no—
432C4—22no—
433C4—81no—
434C4—81no—
435C4—82no—
436C4—81no—
437C4—82no—
438C4C2×#36012yes—
439C4#288911yescharacteristic
440C4#289811yescharacteristic
441C2—24no—
442C2—41no—
443C2—161no—
444C2#1772111yescharacteristic
445C2#3338511yescharacteristic
446C2C22×#36011yescharacteristic
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. #33504C2semidirect
3. #17826C2semidirect
4. #11980C2semidirect
5. #11977C2semidirect
6. #11969C2semidirect
7. #11968C2semidirect
8. #11945C2semidirect
9. #11944C2semidirect
10. #11936C2semidirect
11. C2×#1363C2semidirect

Maximal subgroups

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

subgroupindex
#624752
#335042
#178262
#119802
#119772
#119692
#119682
#119452
#119442
#119362
C2×#13632