Finite Group Databasebeta

← #224682

Subgroups — #224682

Order 512.

737 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2#22825C211yesmaximal, characteristic
3#22821C211yesmaximal, characteristic
4#9709C212yesmaximal
5#9694C211yesmaximal, characteristic
6#9693C212yesmaximal
7#9692C211yesmaximal, characteristic
8#8593C212yesmaximal
9#8587C211yesmaximal, characteristic
10#8236C212yesmaximal
11C4×#333C211yesmaximal, characteristic
12#2868C2211yescharacteristic
13#2862C2212yes—
14#2582C412yes—
15#2571—21no—
16#2570—22no—
17#2568—21no—
18#2511C414yes—
19#2446—22no—
20#2365—21no—
21#2355—21no—
22#2355C412yes—
23C4×C4⋊Q8C2212yes—
24C4×C4⋊Q8C2211yescharacteristic
25C42×Q8C2211yescharacteristic
26#1984—21no—
27#1984—21no—
28#1917—22no—
29#1916—22no—
30#1842C2211yescharacteristic
31#1842C2211yescharacteristic
32#1831C2212yes—
33#1831C2212yes—
34#1770—21no—
35C8×C4⋊C4—21no—
36#1767C2212yes—
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2212yes—
40C2×#333C2211yescharacteristic
41C2×#333C2211yescharacteristic
42C2×#333C2212yes—
43C2×#332C2212yes—
44C2×#332C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47#1296C2211yescharacteristic
48#1296C2211yescharacteristic
49#1296C2212yes—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4⋊Q8C2312yes—
54C2×C4×Q8—22no—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8D411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2311yescharacteristic
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2311yescharacteristic
68C2×C4⋊C8C2311yescharacteristic
69C2×#97—22no—
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72C2×C4×C8—22no—
73#400—24no—
74#400—24no—
75#398—24no—
76#391—22no—
77#391—22no—
78#391—22no—
79#391—22no—
80#391C2×C412yes—
81#391C2×C414yes—
82#391C2×C414yes—
83#389—22no—
84#389—21no—
85#389—22no—
86#389—21no—
87#389C2×C414yes—
88#387—24no—
89#387—24no—
90#387—24no—
91#384—21no—
92#384—21no—
93#384C2×C412yes—
94#384C2×C412yes—
95#384C2311yescharacteristic
96#382—22no—
97#382—22no—
98#382—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4C2×C414yes—
104C4×C4⋊C4C2×C411yescharacteristic
105C4×C4⋊C4C2×C412yes—
106C4×C4⋊C4D412yes—
107C4×C4⋊C4D411yescharacteristic
108C4×C4⋊C4D411yescharacteristic
109C4×C4⋊C4D412yes—
110C43C2312yes—
111C43C2311yescharacteristic
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336—21no—
117#336—22no—
118#336—21no—
119#336—22no—
120#333—22no—
121#333—22no—
122#333—42no—
123#332—24no—
124#332—24no—
125#332—42no—
126#332—42no—
127C22×Q8—22no—
128C22×Q8C22⋊C411yescharacteristic
129C22×Q8C22×C412yes—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8—22no—
141C4⋊Q8—21no—
142C4⋊Q8—22no—
143C4⋊Q8—21no—
144C4⋊Q8—42no—
145C4⋊Q8—42no—
146C4⋊Q8—81no—
147C4⋊Q8—81no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—42no—
153C4×Q8—44no—
154C4×Q8—82no—
155C4×Q8—82no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4C22⋊C411yescharacteristic
168C2×C4⋊C4C22⋊C411yescharacteristic
169C2×C4⋊C4C22⋊C411yescharacteristic
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22×C414yes—
174C2×C4⋊C4C22×C412yes—
175C2×C42—22no—
176C2×C42—24no—
177C2×C42C22×C411yescharacteristic
178C2×C42C22×C412yes—
179C2×C42C22×C412yes—
180C2×C42C22×C411yescharacteristic
181C2×C42C2×D412yes—
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D412yes—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2411yescharacteristic
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—42no—
201C4⋊C8—42no—
202C4⋊C8—81no—
203C4⋊C8—81no—
204C4⋊C8#4112yes—
205C4⋊C8#4112yes—
206#97—82no—
207C4×C8—42no—
208C4×C8—42no—
209C4×C8—82no—
210#95—24no—
211#95—22no—
212#95—22no—
213#95—24no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95C22⋊C412yes—
219#95C22⋊C412yes—
220#95C22⋊C412yes—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C23×C411yescharacteristic
246C22×C4C22×D411yescharacteristic
247C22×C4C22×D411yescharacteristic
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—42no—
256C2×C8—41no—
257C2×C8—42no—
258C2×C8—41no—
259C2×C8—42no—
260C2×C8—41no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—41no—
265C2×C8—41no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—84no—
289C4⋊C4—82no—
290C4⋊C4—82no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—21no—
297C42—22no—
298C42—21no—
299C42—22no—
300C42—21no—
301C42—21no—
302C42—22no—
303C42—22no—
304C42—41no—
305C42—42no—
306C42—42no—
307C42—44no—
308C42—84no—
309C42#13512yes—
310C42#13512yes—
311C42#13712yes—
312C42#13712yes—
313C42C2×#4111yescharacteristic
314C42C2×#4111yescharacteristic
315C42H(5,𝔽2)11yescharacteristic
316C422−1+411yescharacteristic
317C23C22×C22⋊C411yescharacteristic
318Q8—44no—
319Q8—82no—
320Q8—82no—
321Q8—82no—
322Q8—82no—
323Q8—82no—
324Q8—82no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4#41312yes—
356C2×C4#41612yes—
357C2×C4#41612yes—
358C2×C4#41912yes—
359C2×C4#41912yes—
360C2×C4C4×#4111yescharacteristic
361C2×C4#52011yescharacteristic
362C2×C4#54811yescharacteristic
363C2×C4#54811yescharacteristic
364C2×C4#55011yescharacteristic
365C2×C4#55011yescharacteristic
366C2×C4C2×#13511yescharacteristic
367C2×C4C2×#13511yescharacteristic
368C2×C4C2×#13711yescharacteristic
369C2×C4C2×#13711yescharacteristic
370C2×C4#57611yescharacteristic
371C2×C4#57711yescharacteristic
372C2×C4#57711yescharacteristic
373C2×C4#57811yescharacteristic
374C8—42no—
375C8—42no—
376C8—81no—
377C8—82no—
378C8—81no—
379C8—82no—
380C22#234111yescharacteristic
381C22C2×#41311yescharacteristic
382C22#288911yescharacteristic
383C22C2×#41611yescharacteristic
384C22C2×#41911yescharacteristic
385C22#289711yescharacteristic
386C22#289811yescharacteristic
387C4—21no—
388C4—21no—
389C4—22no—
390C4—22no—
391C4—42no—
392C4—42no—
393C4—42no—
394C4—41no—
395C4—42no—
396C4—82no—
397C4—84no—
398C4—82no—
399C4#155414yes—
400C4#336111yescharacteristic
401C4#336511yescharacteristic
402C4#337211yescharacteristic
403C4#337411yescharacteristic
404C4#339411yescharacteristic
405C4#339611yescharacteristic
406C4#339811yescharacteristic
407C4#340011yescharacteristic
408C2C2×#155411yescharacteristic
409C2#1772511yescharacteristic
410C2#1776111yescharacteristic
411C2#1778111yescharacteristic
412C2#2253711yescharacteristic
413C2#2254311yescharacteristic
414C2#2254711yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ 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
#365982
#228252
#228212
#97092
#96942
#96932
#96922
#85932
#85872
#82362
C4×#3332