Finite Group Databasebeta

← #321337

Subgroups — #321337

Order 512.

764 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33310C211yesmaximal, characteristic
2#28748C211yesmaximal, characteristic
3#28748C211yesmaximal, characteristic
4#28717C211yesmaximal, characteristic
5#28689C211yesmaximal, characteristic
6C4×#487C211yesmaximal, characteristic
7C2×#1718C212yesmaximal
8#10317C212yesmaximal
9#10314C212yesmaximal
10#9828C212yesmaximal
11#8261C211yesmaximal, characteristic
12C2×#500C2212yes—
13C2×#498C2212yes—
14C2×#498C2212yes—
15C2×#487C2212yes—
16C2×#487C2212yes—
17C2×#487C2212yes—
18C2×#487C2211yescharacteristic
19#2868C2211yescharacteristic
20#2868C2211yescharacteristic
21C4×C4⋊Q8C2211yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23#2083—22no—
24#2072—22no—
25#2043—22no—
26#2043—22no—
27#1990C2212yes—
28#1990C2212yes—
29#1990C2211yescharacteristic
30#1981—22no—
31#1981—22no—
32#1956C2212yes—
33#1956C2212yes—
34#1956C2211yescharacteristic
35#1947—22no—
36#1933—22no—
37#1844—22no—
38#1844—22no—
39#1834C2211yescharacteristic
40#1803—22no—
41C4×C4⋊C8C2211yescharacteristic
42C4×C4⋊C8C2211yescharacteristic
43C4×#103—22no—
44C42×C8C2211yescharacteristic
45#1718—28no—
46C2×#322C2212yes—
47#1273C2212yes—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2312yes—
53C2×C4⋊Q8C2312yes—
54C2×C4⋊Q8C2311yescharacteristic
55C2×C4⋊Q8C2311yescharacteristic
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58#500—24no—
59#500—42no—
60#498—24no—
61#498—24no—
62#498—42no—
63#498—42no—
64#487—24no—
65#487—24no—
66#487—24no—
67#487—22no—
68#487—42no—
69#487—42no—
70#487—42no—
71#487D414yes—
72C2×#107—24no—
73C2×#106—24no—
74C2×#106—24no—
75C2×C4⋊C8D412yes—
76C2×C4⋊C8D412yes—
77C2×#103—22no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—22no—
82C2×#103—24no—
83C2×#103—24no—
84C2×C4×C8D412yes—
85C2×C4×C8D412yes—
86C2×C4×C8C2312yes—
87C2×C4×C8C2312yes—
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90#391—21no—
91#391—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#384—21no—
97#384—21no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384—21no—
102#384—21no—
103#384—21no—
104#384C2311yescharacteristic
105#384C2311yescharacteristic
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C43C2311yescharacteristic
111#340—24no—
112#340—24no—
113#340—22no—
114#340—22no—
115#336D412yes—
116#336D412yes—
117#322—24no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×C8C2×D414yes—
123C22×C8C2×D414yes—
124C22×C8C2×D412yes—
125C22×C8C2×D412yes—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—44no—
141C4⋊Q8—44no—
142C4⋊Q8—82no—
143C4⋊Q8—82no—
144C4⋊Q8C2×D412yes—
145C4⋊Q8C2×D412yes—
146C4×Q8—44no—
147C4×Q8—44no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C42—41no—
173C2×C42—41no—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2411yescharacteristic
181#107—48no—
182#106—48no—
183#106—48no—
184C4⋊C8—24no—
185C4⋊C8—24no—
186#103—28no—
187#103—48no—
188#103—48no—
189#103—48no—
190#103—44no—
191#103—84no—
192#103—84no—
193C4×C8—24no—
194C4×C8—24no—
195C4×C8—22no—
196C4×C8—22no—
197C4×C8—22no—
198C4×C8—22no—
199C4×C8—21no—
200C4×C8—21no—
201C4×C8C2×D412yes—
202C4×C8#4112yes—
203#95—41no—
204#95—41no—
205#95—41no—
206#95—41no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—44no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—44no—
220C2×Q8—42no—
221C2×Q8—44no—
222C2×Q8—44no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4#12711yescharacteristic
230C22×C4#12711yescharacteristic
231C22×C4#12711yescharacteristic
232C22×C4#12711yescharacteristic
233C22×C4C22×D411yescharacteristic
234C22×C4C22×D411yescharacteristic
235C22×C4C22×D411yescharacteristic
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—22no—
240C2×C8—24no—
241C2×C8—24no—
242C2×C8—24no—
243C2×C8—24no—
244C2×C8—24no—
245C2×C8—24no—
246C2×C8#12114yes—
247C2×C8#13714yes—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
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—82no—
271C4⋊C4—82no—
272C4⋊C4—82no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4—82no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—82no—
284C42—82no—
285C42#13514yes—
286C42#13714yes—
287C42#13714yes—
288C42C22×D411yescharacteristic
289C42C2×#4111yescharacteristic
290C42H(5,𝔽2)11yescharacteristic
291C42H(5,𝔽2)11yescharacteristic
292C23C2×#12711yescharacteristic
293Q8—44no—
294Q8—44no—
295Q8—82no—
296Q8—82no—
297Q8—84no—
298Q8—84no—
299Q8—84no—
300Q8—84no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—82no—
315C2×C4—81no—
316C2×C4—82no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4#49512yes—
326C2×C4#49512yes—
327C2×C4#49712yes—
328C2×C4#49712yes—
329C2×C4#49712yes—
330C2×C4#49712yes—
331C2×C4C2×#12111yescharacteristic
332C2×C4#53511yescharacteristic
333C2×C4D4211yescharacteristic
334C2×C4D4211yescharacteristic
335C2×C4#55011yescharacteristic
336C2×C4#55011yescharacteristic
337C2×C4C2×#13512yes—
338C2×C4C2×#13712yes—
339C2×C4C2×#13712yes—
340C2×C4C2×#13712yes—
341C2×C4#57712yes—
342C2×C4#57712yes—
343C8—24no—
344C8—24no—
345C8—44no—
346C8—44no—
347C22#254111yescharacteristic
348C22C2×#49511yescharacteristic
349C22C2×#49711yescharacteristic
350C22C2×#49711yescharacteristic
351C22#315211yescharacteristic
352C22#315411yescharacteristic
353C22#315411yescharacteristic
354C4—22no—
355C4—22no—
356C4—22no—
357C4—82no—
358C4—82no—
359C4—162no—
360C4—162no—
361C4—161no—
362C4—161no—
363C4#171814yescomplemented
364C4#303912yes—
365C4#321812yes—
366C4#327812yes—
367C4#328412yes—
368C4#339412yes—
369C4#340012yes—
370C2C2×#171811yescharacteristic
371C2#1962611yescharacteristic
372C2#1968011yescharacteristic
373C2#1970211yescharacteristic
374C2#2844911yescharacteristic
375C2#2846011yescharacteristic
376C2#2847311yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#127
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
363. C4#1718semidirect

Maximal subgroups

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

subgroupindex
#333102
#287482
#287172
#286892
C4×#4872
C2×#17182
#103172
#103142
#98282
#82612