Finite Group Databasebeta

← #330447

Subgroups — #330447

Order 512.

554 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33289C211yesmaximal, characteristic
2#29984C211yesmaximal, characteristic
3#29799C211yesmaximal, characteristic
4#21341C211yesmaximal, characteristic
5#10594C212yesmaximal
6#10590C212yesmaximal
7#10479C212yesmaximal
8#10478C212yesmaximal
9#9903C211yesmaximal, characteristic
10#9703C211yesmaximal, characteristic
11#8807C211yesmaximal, characteristic
12C2×#462C2212yes—
13#2868C2211yescharacteristic
14#2868C2211yescharacteristic
15#2295C2211yescharacteristic
16#2100—21no—
17#2095—21no—
18#2057—21no—
19#2043—21no—
20#2043—21no—
21#2024—21no—
22#2016—21no—
23#2016—21no—
24#1990—21no—
25#1990—21no—
26#1990C2212yes—
27#1990C2212yes—
28#1984C2211yescharacteristic
29#1984C2211yescharacteristic
30#1981C2212yes—
31#1947C2212yes—
32#1939C2211yescharacteristic
33#1939C2211yescharacteristic
34#1916—21no—
35#1916—21no—
36#1834—21no—
37#1834—21no—
38#1831C2211yescharacteristic
39#1831C2211yescharacteristic
40#1803C2212yes—
41#1768—21no—
42#1767C2211yescharacteristic
43#1746—21no—
44C42×C8C2211yescharacteristic
45C2×#335C2212yes—
46C2×#334C2212yes—
47C2×#332C2212yes—
48C2×#332C2212yes—
49#1296C2212yes—
50#1296C2212yes—
51C22×Q16—22no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8C2312yes—
58C2×C4×Q8C2311yescharacteristic
59#462—42no—
60#462—42no—
61C2×#107—22no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2311yescharacteristic
66C2×C4⋊C8C2311yescharacteristic
67C2×C4⋊C8C2312yes—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—22no—
71C2×#103—22no—
72C2×#97—21no—
73C2×C4×C8D411yescharacteristic
74C2×C4×C8D411yescharacteristic
75C2×C4×C8D411yescharacteristic
76C2×C4×C8D411yescharacteristic
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79#391D411yescharacteristic
80#391D411yescharacteristic
81#389D411yescharacteristic
82#389D411yescharacteristic
83#387—21no—
84#387—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384C2312yes—
93C4×C4⋊C4—21no—
94C4×C4⋊C4C2311yescharacteristic
95#376—21no—
96C43C2311yescharacteristic
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#336—21no—
104#336—21no—
105#336—21no—
106#336—21no—
107#336—21no—
108#336—21no—
109#336—21no—
110#336—21no—
111#336—21no—
112#336—21no—
113#336—21no—
114#335—42no—
115#334—42no—
116#332—42no—
117#332—42no—
118C22×Q8—22no—
119C22×Q8—41no—
120C22×Q8C2×D411yescharacteristic
121C2×Q16—44no—
122C2×Q16—82no—
123C2×Q16—82no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—22no—
128C22×C8C2×D411yescharacteristic
129C22×C8C2×D411yescharacteristic
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D411yescharacteristic
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—42no—
137C4⋊Q8—42no—
138C4⋊Q8—82no—
139C4⋊Q8—82no—
140C4⋊Q8—82no—
141C4⋊Q8—82no—
142C4×Q8—21no—
143C4×Q8—21no—
144C4×Q8—21no—
145C4×Q8—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4C2×D411yescharacteristic
162C2×C4⋊C4C2×D411yescharacteristic
163C2×C4⋊C4C2×D411yescharacteristic
164C2×C42—21no—
165C2×C42—21no—
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42C2411yescharacteristic
173#107—82no—
174C4⋊C8—21no—
175C4⋊C8—21no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—42no—
183C4⋊C8—42no—
184#103—44no—
185#103—82no—
186#103—82no—
187#103—82no—
188#97—42no—
189C4×C8—22no—
190C4×C8—21no—
191C4×C8—21no—
192C4×C8—21no—
193C4×C8—21no—
194C4×C8—41no—
195C4×C8—41no—
196C4×C8Q1614yescomplemented
197#95—21no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95—21no—
203#95—21no—
204C2×Q8—21no—
205C2×Q8—21no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—82no—
215C2×Q8—82no—
216C2×Q8—82no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4#12111yescharacteristic
224C22×C4#12111yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4C22×D411yescharacteristic
228C22×C4C22×D411yescharacteristic
229C22×C4C2×#4111yescharacteristic
230Q16—88no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8C2×Q1612yes—
254C2×C8C2×Q1612yes—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—82no—
272C4⋊C4—82no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4—82no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C42—21no—
280C42—21no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—41no—
286C42—41no—
287C42—41no—
288C42—41no—
289C42C2×Q1612yes—
290C42#13712yes—
291C42H(5,𝔽2)11yescharacteristic
292C42H(5,𝔽2)11yescharacteristic
293C422−1+411yescharacteristic
294C422−1+411yescharacteristic
295C23C2×#12111yescharacteristic
296Q8—42no—
297Q8—42no—
298Q8—82no—
299Q8—82no—
300Q8—82no—
301Q8—82no—
302Q8—164no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4#46212yes—
332C2×C4#46212yes—
333C2×C4#47112yes—
334C2×C4#47312yes—
335C2×C4#53511yescharacteristic
336C2×C4#53611yescharacteristic
337C2×C4#53711yescharacteristic
338C2×C4#53711yescharacteristic
339C2×C4#54211yescharacteristic
340C2×C4#54211yescharacteristic
341C2×C4C22×Q1611yescharacteristic
342C2×C4C2×#13711yescharacteristic
343C2×C4#57511yescharacteristic
344C2×C4#57511yescharacteristic
345C2×C4#57612yes—
346C2×C4#57712yes—
347C8—22no—
348C8—22no—
349C8—41no—
350C8—41no—
351C8—82no—
352C8—82no—
353C8—81no—
354C8—81no—
355C22#262011yescharacteristic
356C22C2×#46211yescharacteristic
357C22#304011yescharacteristic
358C22#305711yescharacteristic
359C22#305811yescharacteristic
360C22C2×#47111yescharacteristic
361C22C2×#47311yescharacteristic
362C4—22no—
363C4—22no—
364C4—22no—
365C4—21no—
366C4—21no—
367C4—22no—
368C4—81no—
369C4—81no—
370C4—81no—
371C4—81no—
372C4—162no—
373C4—162no—
374C4#322412yes—
375C4#324112yes—
376C4#324911yescharacteristic
377C4#324911yescharacteristic
378C4#325211yescharacteristic
379C4#325211yescharacteristic
380C2#1920711yescharacteristic
381C2#1920811yescharacteristic
382C2#1936311yescharacteristic
383C2#1936411yescharacteristic
384C2#2983011yescharacteristic
385C2#2984711yescharacteristic
386C2#2985911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
196. C4×C8Q16semidirect

Maximal subgroups

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

subgroupindex
#332892
#299842
#297992
#213412
#105942
#105902
#104792
#104782
#99032
#97032
#88072