Finite Group Databasebeta

← #326532

Subgroups — #326532

Order 512.

475 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33317C211yesmaximal, characteristic, complemented
2#21660C211yesmaximal, characteristic
3#10365C211yesmaximal, characteristic, complemented
4#10364C211yesmaximal, characteristic
5#10363C211yesmaximal, characteristic, complemented
6#10362C211yesmaximal, characteristic
7#9758C211yesmaximal, characteristic
8#9758C211yesmaximal, characteristic
9#8795C211yesmaximal, characteristic, complemented
10#8795C211yesmaximal, characteristic, complemented
11#8677C211yesmaximal, characteristic, complemented
12#8676C211yesmaximal, characteristic
13#8675C211yesmaximal, characteristic, complemented
14#8674C211yesmaximal, characteristic
15#8269C211yesmaximal, characteristic, complemented
16C2×#501—21no—
17C2×#501—21no—
18#3131C2211yescharacteristic
19#2960C2211yescharacteristic
20#2931C2211yescharacteristic
21#2862C2211yescharacteristic
22#2859C2211yescharacteristic
23#1994C2211yescharacteristic
24#1962C2211yescharacteristic
25#1938—21no—
26#1938—21no—
27#1847—21no—
28#1847—21no—
29#1846C2211yescharacteristic
30#1846C2211yescharacteristic
31#1846C2211yescharacteristic
32#1846C2211yescharacteristic
33#1837C2211yescharacteristic
34#1765—21no—
35#1765—21no—
36C4×#97C2211yescharacteristic
37#1564C2211yescharacteristic
38#1564C2211yescharacteristic
39#1518C2211yescharacteristic
40#1518C2211yescharacteristic
41#1518C2211yescharacteristic
42#1518C2211yescharacteristic
43#1517C2211yescharacteristic
44#1517C2211yescharacteristic
45#1517C2211yescharacteristic
46#1517C2211yescharacteristic
47#1452C2211yescharacteristic
48#1452C2211yescharacteristic
49#1286C2211yescharacteristic
50#1286C2211yescharacteristic
51#1286C2211yescharacteristic
52#1286C2211yescharacteristic
53#1286C2211yescharacteristic
54#1286C2211yescharacteristic
55#1286C2211yescharacteristic
56#1286C2211yescharacteristic
57#1276C2211yescharacteristic
58#1276C2211yescharacteristic
59C2×#131—21no—
60#533C2311yescharacteristic
61#532C2311yescharacteristic
62C2×C4⋊Q8—21no—
63C2×#125—21no—
64#501—44no—
65#501—44no—
66#488—21no—
67#488—21no—
68#487Q811yescharacteristic, complemented
69#487Q811yescharacteristic, complemented
70#486Q811yescharacteristic, complemented
71#486Q811yescharacteristic, complemented
72#446—21no—
73#446—21no—
74C8×Q8—21no—
75C8×Q8—21no—
76#436—21no—
77#436—21no—
78C8×D4—21no—
79C8×D4—21no—
80#431C2311yescharacteristic
81#431C2311yescharacteristic
82#431C2311yescharacteristic
83#431C2311yescharacteristic
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92#424—21no—
93#423C2311yescharacteristic
94#423C2311yescharacteristic
95C2×#104—21no—
96C2×#104—21no—
97#416—21no—
98#416—21no—
99#408—21no—
100#405—21no—
101C2×#97D411yescharacteristic
102C2×#97D411yescharacteristic
103C2×#97D411yescharacteristic
104C2×#97D411yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#391—41no—
108#391—41no—
109#389—41no—
110#389—41no—
111#384—21no—
112#382—21no—
113#382C2311yescharacteristic
114#382C2311yescharacteristic
115#382C2311yescharacteristic
116#382C2311yescharacteristic
117C43C2311yescharacteristic
118#341—21no—
119#341—21no—
120#339—21no—
121#339—21no—
122#335—21no—
123#335—21no—
124#334—21no—
125#334—21no—
126#328—41no—
127#328—41no—
128#328—41no—
129#328—41no—
130#327—41no—
131#327—41no—
132#327—41no—
133#327—41no—
134#322—21no—
135#322—21no—
136#322—21no—
137#322—21no—
138C2×#41—21no—
139C22×Q8—41no—
140#131—42no—
141#131—42no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—41no—
145C22×C8—41no—
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C4⋊Q8—42no—
151C4⋊Q8—42no—
152C4⋊Q8C2×Q811yescharacteristic
153C4⋊Q8C2×Q811yescharacteristic
154C4⋊Q8C2×Q811yescharacteristic
155#127C2×Q811yescharacteristic
156#125—21no—
157#125—42no—
158#125—42no—
159#124—21no—
160C22⋊Q8—21no—
161C22⋊Q8—21no—
162#121—21no—
163#121—21no—
164C4×Q8—21no—
165C4×Q8—21no—
166C4×D4—21no—
167C4×D4—21no—
168#117—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2411yescharacteristic
182#107—42no—
183#107—42no—
184#107—42no—
185#107—42no—
186#106—42no—
187#106—42no—
188#106—42no—
189#106—42no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—41no—
199C4⋊C8—41no—
200#104—42no—
201#104—42no—
202#103—21no—
203#103—21no—
204#103—21no—
205#103—21no—
206#102—21no—
207#102—21no—
208#102—21no—
209#102—21no—
210C22⋊C8—21no—
211C22⋊C8—21no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214C22⋊C8—41no—
215C22⋊C8—41no—
216#97—22no—
217#97—22no—
218#97—22no—
219#97—22no—
220#97—41no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—41no—
226C4×C8C2×Q811yescharacteristic
227C4×C8C2×Q811yescharacteristic
228C4×C8#4111yescharacteristic
229C4×C8#4111yescharacteristic
230C4×C8#4111yescharacteristic
231C4×C8#4111yescharacteristic
232C4×C8#4111yescharacteristic
233C4×C8#4111yescharacteristic
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—42no—
243#41—41no—
244#41—41no—
245#41—41no—
246#41—41no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8—82no—
251C2×Q8—82no—
252C2×Q8—82no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—41no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4C22×D411yescharacteristic
265C22×C4H(5,𝔽2)11yescharacteristic
266C22×C4H(5,𝔽2)11yescharacteristic
267M16—41no—
268M16—41no—
269M16—41no—
270M16—41no—
271M16—82no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—82no—
285C2×C8C22⋊Q811yescharacteristic
286C2×C8C22⋊Q811yescharacteristic
287C2×C8C22⋊Q811yescharacteristic
288C2×C8C22⋊Q811yescharacteristic
289C2×C8#12411yescharacteristic
290C2×C8#12411yescharacteristic
291C2×C8#12411yescharacteristic
292C2×C8#12411yescharacteristic
293C2×C8AGL(1,ℤ/8)12yes—
294C2×C8#13712yes—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—84no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C42—21no—
319C42—21no—
320C42—41no—
321C42—81no—
322C42—81no—
323C42AGL(1,ℤ/8)12yes—
324C42#13712yes—
325C42C22×Q811yescharacteristic
326C42C2×#4111yescharacteristic
327C42C2×#4111yescharacteristic
328C42C2×#4111yescharacteristic
329C23—41no—
330C23#53511yescharacteristic
331Q8—41no—
332Q8—41no—
333Q8—81no—
334Q8—81no—
335Q8—164no—
336D4—41no—
337D4—41no—
338D4—81no—
339D4—81no—
340D4—81no—
341D4—81no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
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×C4C2×C22⋊Q811yescharacteristic
368C2×C4C2×#12411yescharacteristic
369C2×C4#55411yescharacteristic
370C2×C4#55411yescharacteristic
371C2×C4#55511yescharacteristic
372C2×C4#55511yescharacteristic
373C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
374C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
375C2×C4C2×#13711yescharacteristic
376C2×C4C2×#13711yescharacteristic
377C8—21no—
378C8—21no—
379C8—21no—
380C8—21no—
381C8—21no—
382C8—21no—
383C8—81no—
384C8—81no—
385C8—81no—
386C8—81no—
387C8—81no—
388C8—81no—
389C8#48912yes—
390C8#48912yes—
391C22—21no—
392C22—21no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22#256311yescharacteristic
397C22#321711yescharacteristic
398C22#321811yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—42no—
405C4—81no—
406C4—81no—
407C4—81no—
408C4—161no—
409C4—161no—
410C4—161no—
411C4—161no—
412C4#307911yescharacteristic
413C4#307911yescharacteristic
414C4C2×#48911yescharacteristic
415C4#314111yescharacteristic
416C2—21no—
417C2—21no—
418C2—161no—
419C2#2914811yescharacteristic
420C2#2915511yescharacteristic
421C2#2915511yescharacteristic
subgroupG/H
center, socleC22#2563
commutator, Frattini, cosocleC2×C42C24
prime radical#532C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33317C2semidirect
3. #10365C2semidirect
5. #10363C2semidirect
9. #8795C2semidirect
10. #8795C2semidirect
11. #8677C2semidirect
13. #8675C2semidirect
15. #8269C2semidirect
68. #487Q8semidirect
69. #487Q8semidirect
70. #486Q8semidirect
71. #486Q8semidirect

Maximal subgroups

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

subgroupindex
#333172
#216602
#103652
#103642
#103632
#103622
#97582
#87952
#86772
#86762
#86752
#86742
#82692