Finite Group Databasebeta

← #495307

Subgroups — #495307

Order 512.

598 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2175C211yesmaximal, characteristic
2#21276C211yesmaximal, characteristic, complemented
3#21275C211yesmaximal, characteristic
4#13238C211yesmaximal, characteristic
5#13237C211yesmaximal, characteristic
6#13168C211yesmaximal, characteristic, complemented
7#13165C211yesmaximal, characteristic, complemented
8#13150C211yesmaximal, characteristic, complemented
9#13149C211yesmaximal, characteristic
10#13148C211yesmaximal, characteristic, complemented
11#13147C211yesmaximal, characteristic
12#12674C211yesmaximal, characteristic, complemented
13#12638C211yesmaximal, characteristic, complemented
14#12637C211yesmaximal, characteristic
15C4×#346C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C2×C4×Q16C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19#2295—21no—
20#2293—21no—
21#2175C418yescomplemented
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#346C2211yescharacteristic
32C2×#346C2211yescharacteristic
33C2×#346C2211yescharacteristic
34C2×#346C2211yescharacteristic
35C2×C4×C16C2211yescharacteristic
36C2×C4×C16C2211yescharacteristic
37#1970C2211yescharacteristic
38#1970C2211yescharacteristic
39#1969C2211yescharacteristic
40#1969C2211yescharacteristic
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1934—21no—
46#1934—21no—
47#1933—21no—
48#1933—21no—
49C4×#107C2211yescharacteristic
50C4×#107C2211yescharacteristic
51C4×#103—21no—
52C4×#102—21no—
53C42×C8C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60C22×Q16C2311yescharacteristic
61C22×D8C2311yescharacteristic
62C2×C4×Q8—21no—
63C2×C4×D4—21no—
64C2×#117—41no—
65C2×SD32—22no—
66C2×SD32—22no—
67C2×SD32C2×C414yes—
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C4×Q16—22no—
73C4×Q16C2×C414yes—
74C4×D8—22no—
75C4×D8C2×C414yes—
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#103—21no—
83C2×#103—21no—
84C2×#103—21no—
85C2×#102—21no—
86C2×#102—21no—
87C2×#102—21no—
88C2×C4×C8D411yescharacteristic
89C2×C4×C8D411yescharacteristic
90C2×C4×C8D411yescharacteristic
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94#391—21no—
95#391—21no—
96#390—21no—
97#390—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#388—41no—
103#388—41no—
104#387—41no—
105#387—41no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—41no—
109C4×C22⋊C4—41no—
110#376—41no—
111C43D411yescharacteristic
112#367—22no—
113#366—22no—
114#366C2×C414yes—
115#358—22no—
116#358—22no—
117#358C2×C414yes—
118#357—22no—
119#357—22no—
120#357C2×C414yes—
121#346—22no—
122#346—22no—
123#346—22no—
124#346C2×C414yes—
125C4×C16—22no—
126C4×C16—22no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133C22×Q8—21no—
134C22×D4—21no—
135C23×C4—41no—
136C2×Q16—21no—
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16C22×C412yes—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8C22×C412yes—
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2411yescharacteristic
152C4×Q8—24no—
153C4×Q8—42no—
154C4×D4—24no—
155C4×D4—42no—
156#117—84no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C42—41no—
174C2×C42—41no—
175C2×C42—41no—
176C2×C42D811yescharacteristic
177C2×C42D811yescharacteristic
178C2×C42D811yescharacteristic
179C2×C42D811yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183SD32—28no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16C22×C412yes—
191C2×C16C22×C412yes—
192C2×C16#4112yes—
193C2×C16#4112yes—
194C2×C16#4112yes—
195C2×C16#4112yes—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107C22×C412yes—
207#107C22×C412yes—
208#103—24no—
209#103—42no—
210#103—42no—
211#102—24no—
212#102—42no—
213#102—42no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—22no—
221C4×C8C22×C412yes—
222C4×C8#4112yes—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233C24—41no—
234C2×Q8—22no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×D4—22no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—44no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C22×D411yescharacteristic
263Q16—22no—
264Q16—22no—
265Q16—41no—
266Q16—41no—
267D8—22no—
268D8—22no—
269D8—41no—
270D8—41no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8C4×D412yes—
278C2×C8#13512yes—
279C2×C8#13512yes—
280C2×C8C23×C411yescharacteristic
281C2×C8C2×#4111yescharacteristic
282C2×C8C2×#4111yescharacteristic
283C2×C8C2×#4111yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C2×C8C2×#4111yescharacteristic
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—44no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C22⋊C4—44no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—22no—
313C42—22no—
314C42—41no—
315C42—41no—
316C42—44no—
317C42—82no—
318C42—82no—
319C42C4×D412yes—
320C42#13512yes—
321C16—22no—
322C16—22no—
323C16—22no—
324C16#11714yescomplemented
325C23—42no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23C22×D811yescharacteristic
333Q8—42no—
334Q8—42no—
335Q8—81no—
336Q8—81no—
337D4—42no—
338D4—42no—
339D4—81no—
340D4—81no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—88no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4C4×D812yes—
378C2×C4C4×D812yes—
379C2×C4#50912yes—
380C2×C4#51012yes—
381C2×C4C2×C4×D411yescharacteristic
382C2×C4#53211yescharacteristic
383C2×C4#53211yescharacteristic
384C2×C4C2×#13511yescharacteristic
385C2×C4C2×#13511yescharacteristic
386C2×C4C2×#13511yescharacteristic
387C8—22no—
388C8—22no—
389C8—22no—
390C8C2×#11711yescharacteristic
391C8C4×#4111yescharacteristic
392C8#52911yescharacteristic
393C8#52911yescharacteristic
394C22—82no—
395C22—82no—
396C22—82no—
397C22—161no—
398C22—161no—
399C22—161no—
400C22—161no—
401C22C2×C4×D811yescharacteristic
402C22#316911yescharacteristic
403C22#316911yescharacteristic
404C22C2×#50911yescharacteristic
405C22C2×#51011yescharacteristic
406C22#341411yescharacteristic
407C22#341411yescharacteristic
408C4—22no—
409C4—22no—
410C4—22no—
411C4—22no—
412C4—41no—
413C4—41no—
414C4—82no—
415C4—82no—
416C4—82no—
417C4—161no—
418C4—161no—
419C4—161no—
420C4—161no—
421C4#229311yescharacteristic
422C4C4×#13511yescharacteristic
423C4#316811yescharacteristic
424C4#316811yescharacteristic
425C2—162no—
426C2#2127511yescharacteristic
427C2C4×#50911yescharacteristic
428C2C4×#51011yescharacteristic
429C2#3374611yescharacteristic
430C2#3374711yescharacteristic
431C2#3375011yescharacteristic
432C2#3375111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21276C2semidirect
6. #13168C2semidirect
7. #13165C2semidirect
8. #13150C2semidirect
10. #13148C2semidirect
12. #12674C2semidirect
13. #12638C2semidirect
15. C4×#346C2semidirect
21. #2175C4semidirect
324. C16#117semidirect

Maximal subgroups

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

subgroupindex
C2×#21752
#212762
#212752
#132382
#132372
#131682
#131652
#131502
#131492
#131482
#131472
#126742
#126382
#126372
C4×#3462