Finite Group Databasebeta

← #326544

Subgroups — #326544

Order 512.

428 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30306C211yesmaximal, characteristic, complemented
2#22234C211yesmaximal, characteristic
3#10369C211yesmaximal, characteristic, complemented
4#10366C211yesmaximal, characteristic
5#10363C211yesmaximal, characteristic, complemented
6#10362C211yesmaximal, characteristic
7#9758C211yesmaximal, characteristic
8#9757C211yesmaximal, characteristic
9#8794C211yesmaximal, characteristic, complemented
10#8790C211yesmaximal, characteristic, complemented
11#8681C211yesmaximal, characteristic, complemented
12#8680C211yesmaximal, characteristic
13#8676C211yesmaximal, characteristic
14#8675C211yesmaximal, characteristic, complemented
15#8270C211yesmaximal, characteristic, complemented
16#3131C2211yescharacteristic
17#2960C2211yescharacteristic
18#2931C2211yescharacteristic
19#2864C2211yescharacteristic
20#2862C2211yescharacteristic
21#2085—21no—
22#2085—21no—
23#2058—21no—
24#2057—21no—
25#1994C2211yescharacteristic
26#1962C2211yescharacteristic
27#1929—21no—
28#1929—21no—
29#1866—21no—
30#1866—21no—
31#1848C2211yescharacteristic
32#1846C2211yescharacteristic
33#1846C2211yescharacteristic
34#1835C2211yescharacteristic
35#1835C2211yescharacteristic
36#1744C2211yescharacteristic
37#1564C2211yescharacteristic
38#1564C2211yescharacteristic
39#1518C2211yescharacteristic
40#1518C2211yescharacteristic
41#1518C2211yescharacteristic
42#1518C2211yescharacteristic
43#1517C2211yescharacteristic
44#1517C2211yescharacteristic
45#1517C2211yescharacteristic
46#1517C2211yescharacteristic
47#1452C2211yescharacteristic
48#1450C2211yescharacteristic
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—
63#488—21no—
64#488—21no—
65#487Q811yescharacteristic, complemented
66#487Q811yescharacteristic, complemented
67#486Q811yescharacteristic, complemented
68#486Q811yescharacteristic, complemented
69#446—21no—
70#446—21no—
71C8×Q8—21no—
72C8×Q8—21no—
73#436—21no—
74#436—21no—
75C8×D4—21no—
76C8×D4—21no—
77#431C2311yescharacteristic
78#431C2311yescharacteristic
79#431C2311yescharacteristic
80#431C2311yescharacteristic
81C2×#108—21no—
82C2×#108—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#106—21no—
86C2×#106—21no—
87#424—21no—
88#423C2311yescharacteristic
89#423C2311yescharacteristic
90C2×#104—21no—
91C2×#104—21no—
92#416—21no—
93#416—21no—
94#408—21no—
95#405—21no—
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#391—41no—
99#391—41no—
100#389—21no—
101#389—21no—
102#387—41no—
103#387—41no—
104#382—21no—
105#382C2311yescharacteristic
106#382C2311yescharacteristic
107#382C2311yescharacteristic
108#382C2311yescharacteristic
109C43C2311yescharacteristic
110#340—21no—
111#340—21no—
112#340—21no—
113#340—21no—
114#339—21no—
115#339—21no—
116#336D411yescharacteristic
117#336D411yescharacteristic
118#336D411yescharacteristic
119#336D411yescharacteristic
120#335—21no—
121#335—21no—
122#334—21no—
123#334—21no—
124#328—41no—
125#328—41no—
126#328—41no—
127#328—41no—
128#327—41no—
129#327—41no—
130#327—41no—
131#327—41no—
132#322—21no—
133#322—21no—
134#322—21no—
135C82—21no—
136C2×#41—21no—
137C22×Q8—41no—
138#131—81no—
139#131—81no—
140C2×M16—21no—
141C2×M16—21no—
142C2×M16—41no—
143C22×C8—41no—
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C4⋊Q8—81no—
149C4⋊Q8—81no—
150C4⋊Q8C2×Q811yescharacteristic
151C4⋊Q8C2×Q811yescharacteristic
152C4⋊Q8C2×Q811yescharacteristic
153#127C2×Q811yescharacteristic
154#125—21no—
155#124—21no—
156C22⋊Q8—21no—
157C22⋊Q8—21no—
158#121—21no—
159#121—21no—
160C4×Q8—21no—
161C4×Q8—21no—
162C4×D4—21no—
163C4×D4—21no—
164#117—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2411yescharacteristic
177#108—81no—
178#108—81no—
179#107—81no—
180#107—81no—
181#106—81no—
182#106—81no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—41no—
192C4⋊C8—41no—
193#104—81no—
194#104—81no—
195#103—21no—
196#103—21no—
197#103—21no—
198#103—21no—
199#102—21no—
200#102—21no—
201#102—21no—
202#102—21no—
203C22⋊C8—21no—
204C22⋊C8—21no—
205C22⋊C8—21no—
206C22⋊C8—21no—
207C22⋊C8—41no—
208C22⋊C8—41no—
209#97—41no—
210C4×C8—21no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—41no—
215C4×C8C2×Q811yescharacteristic
216C4×C8C2×Q811yescharacteristic
217C4×C8#4111yescharacteristic
218C4×C8#4111yescharacteristic
219C4×C8#4111yescharacteristic
220C4×C8#4111yescharacteristic
221C4×C8#4111yescharacteristic
222C4×C8#4111yescharacteristic
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—42no—
232#95—41no—
233#95—41no—
234#41—41no—
235#41—41no—
236#41—41no—
237#41—41no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—21no—
241C2×Q8—82no—
242C2×Q8—82no—
243C2×Q8—82no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—41no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4C22×D411yescharacteristic
256C22×C4H(5,𝔽2)11yescharacteristic
257C22×C4H(5,𝔽2)11yescharacteristic
258M16—41no—
259M16—41no—
260M16—41no—
261M16—41no—
262M16—82no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—82no—
278C2×C8C22⋊Q811yescharacteristic
279C2×C8C22⋊Q811yescharacteristic
280C2×C8C22⋊Q811yescharacteristic
281C2×C8C22⋊Q811yescharacteristic
282C2×C8#12411yescharacteristic
283C2×C8#12411yescharacteristic
284C2×C8#12411yescharacteristic
285C2×C8#12411yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—42no—
313C42—41no—
314C42—81no—
315C42C22×Q811yescharacteristic
316C42C2×#4111yescharacteristic
317C42C2×#4111yescharacteristic
318C42C2×#4111yescharacteristic
319C23—41no—
320C23#53511yescharacteristic
321Q8—41no—
322Q8—41no—
323Q8—81no—
324Q8—81no—
325Q8—164no—
326D4—41no—
327D4—41no—
328D4—81no—
329D4—81no—
330D4—81no—
331D4—81no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4C2×C22⋊Q811yescharacteristic
358C2×C4C2×#12411yescharacteristic
359C2×C4#55411yescharacteristic
360C2×C4#55411yescharacteristic
361C2×C4#55511yescharacteristic
362C2×C4#55511yescharacteristic
363C2×C4#57611yescharacteristic
364C2×C4#57711yescharacteristic
365C2×C4#57711yescharacteristic
366C2×C4#57811yescharacteristic
367C8—21no—
368C8—21no—
369C8—21no—
370C8—21no—
371C8—21no—
372C8—21no—
373C8—21no—
374C8—21no—
375C8—81no—
376C8—81no—
377C8—81no—
378C8—81no—
379C8—81no—
380C8—81no—
381C22—21no—
382C22—21no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#256311yescharacteristic
387C22#322411yescharacteristic
388C22#322511yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—42no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—161no—
399C4—161no—
400C4—161no—
401C4—161no—
402C4#308311yescharacteristic
403C4#308411yescharacteristic
404C4#313611yescharacteristic
405C4#313711yescharacteristic
406C2—21no—
407C2—21no—
408C2—161no—
409C2#2915111yescharacteristic
410C2#2915411yescharacteristic
411C2#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. #30306C2semidirect
3. #10369C2semidirect
5. #10363C2semidirect
9. #8794C2semidirect
10. #8790C2semidirect
11. #8681C2semidirect
14. #8675C2semidirect
15. #8270C2semidirect
65. #487Q8semidirect
66. #487Q8semidirect
67. #486Q8semidirect
68. #486Q8semidirect

Maximal subgroups

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

subgroupindex
#303062
#222342
#103692
#103662
#103632
#103622
#97582
#97572
#87942
#87902
#86812
#86802
#86762
#86752
#82702