Finite Group Databasebeta

← #317697

Subgroups — #317697

Order 512.

558 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29290C211yesmaximal, characteristic
2#29287C211yesmaximal, characteristic
3#21712C211yesmaximal, characteristic
4C4×#487C211yesmaximal, characteristic
5#10655C212yesmaximal
6#10541C212yesmaximal
7#9575C212yesmaximal
8#9569C212yesmaximal
9#8784C211yesmaximal, characteristic
10#8587C211yesmaximal, characteristic
11#8586C211yesmaximal, characteristic
12C2×#487C2211yescharacteristic
13C2×#446—21no—
14C4×C4⋊Q8C2211yescharacteristic
15C4×C4⋊Q8C2211yescharacteristic
16C4×C4⋊Q8C2211yescharacteristic
17#2088—22no—
18#2087—22no—
19#1990C2211yescharacteristic
20#1988C2212yes—
21#1987—21no—
22#1987—21no—
23#1987C2212yes—
24#1984—21no—
25#1956C2211yescharacteristic
26#1948—21no—
27#1948C2212yes—
28#1947—21no—
29#1947C2212yes—
30#1939—21no—
31#1939C2212yes—
32#1933—21no—
33#1933—21no—
34#1842C2211yescharacteristic
35#1842C2211yescharacteristic
36#1842C2211yescharacteristic
37#1834C2211yescharacteristic
38#1770—21no—
39#1770C2212yes—
40C4×#103—21no—
41C4×#103—21no—
42C42×C8C2211yescharacteristic
43C2×#333C2212yes—
44C2×#332C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1277C2212yes—
48#1277C2212yes—
49C2×C4⋊Q8D411yescharacteristic
50C2×C4⋊Q8C2311yescharacteristic
51C2×C4⋊Q8C2311yescharacteristic
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55#487—22no—
56#487—22no—
57#446—44no—
58C2×#107—22no—
59C2×#106—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2312yes—
70C2×#103—21no—
71C2×#103—21no—
72C2×#103—21no—
73C2×#103—21no—
74C2×#97—22no—
75C2×C4×C8D411yescharacteristic
76C2×C4×C8D411yescharacteristic
77C2×C4×C8D411yescharacteristic
78C2×C4×C8D411yescharacteristic
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2311yescharacteristic
81#391—21no—
82#391—21no—
83#391—21no—
84#389—21no—
85#389—21no—
86#389—21no—
87#389—21no—
88#389—21no—
89#389D412yes—
90#384D411yescharacteristic
91#384C2311yescharacteristic
92#384C2311yescharacteristic
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4C2312yes—
99C43C2311yescharacteristic
100#340—21no—
101#340—21no—
102#340—21no—
103#340—21no—
104#340—22no—
105#340—22no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—22no—
110#336—22no—
111#336—22no—
112#333—24no—
113#332—24no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8C2×D411yescharacteristic
121C22×C8C2×D411yescharacteristic
122C22×C8C2×D411yescharacteristic
123C22×C8C2×D411yescharacteristic
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8#4112yes—
133C4⋊Q8#4112yes—
134C4×Q8—42no—
135C4×Q8—42no—
136C4×Q8—42no—
137C4×Q8—42no—
138C4×Q8—82no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
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—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4C2×D412yes—
156C2×C4⋊C4C2×D412yes—
157C2×C42—22no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42C2411yescharacteristic
167#107—82no—
168#106—82no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—42no—
176C4⋊C8—44no—
177C4⋊C8—42no—
178C4⋊C8—42no—
179C4⋊C8—42no—
180C4⋊C8—81no—
181C4⋊C8—81no—
182#103—24no—
183#103—24no—
184#103—42no—
185#103—42no—
186#97—44no—
187C4×C8—22no—
188C4×C8—22no—
189C4×C8—22no—
190C4×C8—21no—
191C4×C8—21no—
192C4×C8—21no—
193C4×C8—21no—
194C4×C8—41no—
195#95—22no—
196#95—22no—
197#95—41no—
198#95—41no—
199#95—41no—
200#95—41no—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—41no—
206C2×Q8—41no—
207C2×Q8—42no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8—81no—
213C2×Q8—81no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4#12111yescharacteristic
223C22×C4#12111yescharacteristic
224C22×C4C22×D411yescharacteristic
225C22×C4C22×D411yescharacteristic
226C22×C4C2×#4111yescharacteristic
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8AGL(1,ℤ/8)12yes—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—81no—
273C4⋊C4—81no—
274C4⋊C4—82no—
275C4⋊C4—82no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—22no—
281C42—41no—
282C42—42no—
283C42—42no—
284C42—82no—
285C42—82no—
286C42#13512yes—
287C42#13712yes—
288C42#13712yes—
289C42#13712yes—
290C42C2×#4111yescharacteristic
291C42C2×#4111yescharacteristic
292C422−1+411yescharacteristic
293C422−1+411yescharacteristic
294C23C2×#12111yescharacteristic
295Q8—42no—
296Q8—42no—
297Q8—82no—
298Q8—84no—
299Q8—82no—
300Q8—81no—
301Q8—81no—
302C2×C4—21no—
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—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4#46412yes—
330C2×C4#46412yes—
331C2×C4#47012yes—
332C2×C4#47012yes—
333C2×C4#47112yes—
334C2×C4#52511yescharacteristic
335C2×C4#53611yescharacteristic
336C2×C4#54811yescharacteristic
337C2×C4#54811yescharacteristic
338C2×C4#55211yescharacteristic
339C2×C4#55211yescharacteristic
340C2×C4C2×#13511yescharacteristic
341C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
342C2×C4C2×#13711yescharacteristic
343C2×C4C2×#13711yescharacteristic
344C2×C4C2×#13711yescharacteristic
345C2×C4#57511yescharacteristic
346C2×C4#57611yescharacteristic
347C2×C4#57811yescharacteristic
348C8—22no—
349C8—41no—
350C8—41no—
351C8—41no—
352C8—82no—
353C8—82no—
354C8—82no—
355C22#253611yescharacteristic
356C22C2×#46411yescharacteristic
357C22#303611yescharacteristic
358C22C2×#47011yescharacteristic
359C22#306111yescharacteristic
360C22C2×#47111yescharacteristic
361C22#306711yescharacteristic
362C4—22no—
363C4—21no—
364C4—21no—
365C4—22no—
366C4—41no—
367C4—82no—
368C4—82no—
369C4—82no—
370C4—82no—
371C4—161no—
372C4—161no—
373C4#170414yes—
374C4#310611yescharacteristic
375C4#311611yescharacteristic
376C4#323311yescharacteristic
377C4#323811yescharacteristic
378C4#337411yescharacteristic
379C4#337411yescharacteristic
380C4#337511yescharacteristic
381C4#337511yescharacteristic
382C2C2×#170411yescharacteristic
383C2#1921711yescharacteristic
384C2#1928111yescharacteristic
385C2#1933911yescharacteristic
386C2#2819611yescharacteristic
387C2#2820411yescharacteristic
388C2#2821811yescharacteristic
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

Maximal subgroups

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

subgroupindex
#292902
#292872
#217122
C4×#4872
#106552
#105412
#95752
#95692
#87842
#85872
#85862