Finite Group Databasebeta

← #316795

Subgroups — #316795

Order 512.

616 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28206C211yesmaximal, characteristic
2#28161C211yesmaximal, characteristic
3C4×#498C211yesmaximal, characteristic
4C8×C4⋊Q8C211yesmaximal, characteristic
5#10620C212yesmaximal
6#10273C212yesmaximal
7#9218C212yesmaximal
8#9216C212yesmaximal
9#8785C211yesmaximal, characteristic
10#8591C211yesmaximal, characteristic
11#8590C211yesmaximal, characteristic
12C2×#498C2211yescharacteristic
13C2×#477C2212yes—
14C2×#462C2212yes—
15C2×C8×Q8C2212yes—
16C4×C4⋊Q8C2211yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19#2083—21no—
20#2072—21no—
21#2043—21no—
22#2016—21no—
23#1990C2211yescharacteristic
24#1990C2211yescharacteristic
25#1984C2212yes—
26#1941—21no—
27#1941—21no—
28#1939—21no—
29#1939—21no—
30#1928—22no—
31#1878—22no—
32#1844C2211yescharacteristic
33#1842C2211yescharacteristic
34#1831C2211yescharacteristic
35#1831C2211yescharacteristic
36#1803C2212yes—
37#1803C2212yes—
38C4×#106—21no—
39C4×#106—21no—
40C8×C4⋊C4—21no—
41C8×C4⋊C4—21no—
42C42×C8C2211yescharacteristic
43C2×#333C2212yes—
44C2×#332C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1277C2212yes—
48#1277C2212yes—
49C22×Q16—22no—
50C2×C4⋊Q8D411yescharacteristic
51C2×C4⋊Q8C2311yescharacteristic
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8C2312yes—
56#498—22no—
57#498—22no—
58#477—24no—
59#477—42no—
60#462—24no—
61#462—42no—
62C8×Q8—42no—
63C8×Q8—42no—
64C2×#107—22no—
65C2×#106—21no—
66C2×#106—21no—
67C2×#106—21no—
68C2×#106—21no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×#103—22no—
76C2×#103—22no—
77C2×#103—22no—
78C2×#103—22no—
79C2×C4×C8—22no—
80C2×C4×C8D411yescharacteristic
81C2×C4×C8D411yescharacteristic
82C2×C4×C8D411yescharacteristic
83C2×C4×C8D411yescharacteristic
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2311yescharacteristic
86#391—21no—
87#391—21no—
88#391D412yes—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#384D411yescharacteristic
96#384C2311yescharacteristic
97#384C2311yescharacteristic
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C43C2311yescharacteristic
105#340—21no—
106#340—21no—
107#340—21no—
108#340—21no—
109#340—22no—
110#340—22no—
111#336—22no—
112#336—22no—
113#336—21no—
114#336—21no—
115#336—21no—
116#336—21no—
117#333—42no—
118#332—42no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8C2×D412yes—
122C2×Q16—44no—
123C2×Q16—44no—
124C2×Q16—82no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8C2×D411yescharacteristic
129C22×C8C2×D411yescharacteristic
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D411yescharacteristic
132C4⋊Q8—24no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8#4112yes—
140C4⋊Q8#4112yes—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—82no—
146C4×Q8—82no—
147C2×C4⋊C4—21no—
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—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
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⋊C4C2×D412yes—
165C2×C42—22no—
166C2×C42—41no—
167C2×C42—41no—
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42C2411yescharacteristic
175#107—44no—
176#106—24no—
177#106—24no—
178#106—42no—
179#106—42no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—42no—
187C4⋊C8—44no—
188C4⋊C8—42no—
189#103—44no—
190#103—44no—
191#103—44no—
192#103—82no—
193C4×C8—22no—
194C4×C8—22no—
195C4×C8—21no—
196C4×C8—21no—
197C4×C8—21no—
198C4×C8—21no—
199C4×C8—41no—
200C4×C8—44no—
201C4×C8Q1614yescomplemented
202#95—22no—
203#95—22no—
204#95—41no—
205#95—41no—
206#95—41no—
207#95—41no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—81no—
220C2×Q8—81no—
221C2×Q8#13514yes—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4C22×D411yescharacteristic
233C22×C4C22×D411yescharacteristic
234C22×C4C2×#4111yescharacteristic
235Q16—88no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8C2×Q1612yes—
255C2×C8C2×Q1612yes—
256C2×C8#13712yes—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—81no—
280C4⋊C4—81no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C42—22no—
284C42—21no—
285C42—21no—
286C42—22no—
287C42—22no—
288C42—41no—
289C42—42no—
290C42—42no—
291C42—82no—
292C42—82no—
293C42C2×Q1612yes—
294C42#13712yes—
295C42C2×#4111yescharacteristic
296C42C2×#4111yescharacteristic
297C42H(5,𝔽2)11yescharacteristic
298C42H(5,𝔽2)11yescharacteristic
299C23C2×#12111yescharacteristic
300Q8—24no—
301Q8—24no—
302Q8—42no—
303Q8—42no—
304Q8—44no—
305Q8—82no—
306Q8—82no—
307Q8—82no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—24no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4#46212yes—
338C2×C4#46212yes—
339C2×C4#47112yes—
340C2×C4#52511yescharacteristic
341C2×C4#53511yescharacteristic
342C2×C4#54611yescharacteristic
343C2×C4#54611yescharacteristic
344C2×C4#55311yescharacteristic
345C2×C4#55311yescharacteristic
346C2×C4C22×Q1611yescharacteristic
347C2×C4C2×#13512yes—
348C2×C4C2×#13711yescharacteristic
349C2×C4C2×#13711yescharacteristic
350C2×C4#57511yescharacteristic
351C2×C4#57712yes—
352C8—22no—
353C8—22no—
354C8—22no—
355C8—41no—
356C8—82no—
357C8—82no—
358C8—82no—
359C22#253511yescharacteristic
360C22C2×#46211yescharacteristic
361C22#303711yescharacteristic
362C22#305011yescharacteristic
363C22#305711yescharacteristic
364C22C2×#47111yescharacteristic
365C22#306711yescharacteristic
366C4—22no—
367C4—21no—
368C4—21no—
369C4—22no—
370C4—22no—
371C4—41no—
372C4—44no—
373C4—82no—
374C4—82no—
375C4—82no—
376C4—161no—
377C4—161no—
378C4#310411yescharacteristic
379C4#311311yescharacteristic
380C4#320911yescharacteristic
381C4#321611yescharacteristic
382C4#329612yes—
383C4#330112yes—
384C2#1911811yescharacteristic
385C2#1920711yescharacteristic
386C2#1926311yescharacteristic
387C2#1934611yescharacteristic
388C2#2814411yescharacteristic
389C2#2816111yescharacteristic
390C2#2817511yescharacteristic
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
201. C4×C8Q16semidirect

Maximal subgroups

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

subgroupindex
#282062
#281612
C4×#4982
C8×C4⋊Q82
#106202
#102732
#92182
#92162
#87852
#85912
#85902