Finite Group Databasebeta

← #224656

Subgroups — #224656

Order 512.

749 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2#22821C211yesmaximal, characteristic
3#22821C211yesmaximal, characteristic
4#9700C212yesmaximal
5#9700C212yesmaximal
6#9694C211yesmaximal, characteristic
7#9692C211yesmaximal, characteristic
8#8593C212yesmaximal
9#8586C211yesmaximal, characteristic
10C4×#332C212yesmaximal
11C4×#332C211yesmaximal, characteristic
12#2868C2211yescharacteristic
13#2862C2212yes—
14#2582—21no—
15#2571C412yes—
16#2570C414yes—
17#2568—21no—
18#2511—22no—
19#2446—22no—
20#2365—21no—
21#2355—21no—
22#2355C412yes—
23C4×C4⋊Q8C2211yescharacteristic
24C4×C4⋊Q8C2212yes—
25C42×Q8C2211yescharacteristic
26#1984—21no—
27#1984—21no—
28#1916—22no—
29#1916—22no—
30#1842C2211yescharacteristic
31#1842C2211yescharacteristic
32#1831C2212yes—
33#1831C2212yes—
34C8×C4⋊C4—21no—
35C8×C4⋊C4—21no—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2212yes—
39C4×C4⋊C8C2212yes—
40C2×#332C2212yes—
41C2×#332C2212yes—
42C2×#332C2211yescharacteristic
43C2×#332C2211yescharacteristic
44C2×#332C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47#1296C2211yescharacteristic
48#1296C2212yes—
49#1296C2211yescharacteristic
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4⋊Q8C2312yes—
54C2×C4×Q8—21no—
55C2×C4×Q8—22no—
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×C4×C8—21no—
70C2×C4×C8—21no—
71C2×C4×C8—22no—
72C2×C4×C8—22no—
73#400—24no—
74#400—24no—
75#398—24no—
76#391—22no—
77#391—22no—
78#391—21no—
79#391—22no—
80#391—22no—
81#391—22no—
82#391—22no—
83#389—22no—
84#389—22no—
85#389—21no—
86#389C2×C412yes—
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#382C2×C414yes—
98#382C2×C414yes—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4C2×C414yes—
103C4×C4⋊C4C2×C412yes—
104C4×C4⋊C4C2×C412yes—
105C4×C4⋊C4C2×C411yescharacteristic
106C4×C4⋊C4D411yescharacteristic
107C4×C4⋊C4D411yescharacteristic
108C4×C4⋊C4D412yes—
109C4×C4⋊C4D412yes—
110C43C2312yes—
111C43C2311yescharacteristic
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336—21no—
117#336—21no—
118#336—22no—
119#336—22no—
120#332—24no—
121#332—24no—
122#332—22no—
123#332—22no—
124#332—42no—
125#332—42no—
126#332—42no—
127C22×Q8—22no—
128C22×Q8—21no—
129C22×Q8C22⋊C411yescharacteristic
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8—22no—
136C4⋊Q8—21no—
137C4⋊Q8—22no—
138C4⋊Q8—21no—
139C4⋊Q8—22no—
140C4⋊Q8—21no—
141C4⋊Q8—22no—
142C4⋊Q8—21no—
143C4⋊Q8—22no—
144C4⋊Q8—42no—
145C4⋊Q8—42no—
146C4⋊Q8—81no—
147C4⋊Q8—81no—
148C4×Q8—24no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—42no—
154C4×Q8—42no—
155C4×Q8—82no—
156C4×Q8—82no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C411yescharacteristic
169C2×C4⋊C4C22⋊C411yescharacteristic
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22⋊C411yescharacteristic
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C414yes—
175C2×C4⋊C4C22×C412yes—
176C2×C42—22no—
177C2×C42—24no—
178C2×C42C22×C412yes—
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C412yes—
181C2×C42C22×C411yescharacteristic
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D412yes—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2411yescharacteristic
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—42no—
202C4⋊C8—42no—
203C4⋊C8—81no—
204C4⋊C8—81no—
205C4⋊C8#4112yes—
206C4⋊C8#4112yes—
207C4×C8—42no—
208C4×C8—42no—
209C4×C8—82no—
210C4×C8—82no—
211#95—24no—
212#95—24no—
213#95—22no—
214#95—22no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95C22⋊C412yes—
220#95C22⋊C412yes—
221#95C22⋊C412yes—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—41no—
226C2×Q8—42no—
227C2×Q8—41no—
228C2×Q8—42no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—41no—
236C2×Q8—41no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C23×C411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—42no—
259C2×C8—41no—
260C2×C8—42no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—41no—
268C2×C8—41no—
269C4⋊C4—24no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—22no—
275C4⋊C4—24no—
276C4⋊C4—22no—
277C4⋊C4—24no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—84no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4#10114yes—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—21no—
300C42—22no—
301C42—21no—
302C42—22no—
303C42—21no—
304C42—22no—
305C42—21no—
306C42—22no—
307C42—24no—
308C42—41no—
309C42—42no—
310C42—44no—
311C42—84no—
312C42#13512yes—
313C42#13512yes—
314C42#13512yes—
315C42#13512yes—
316C42C2×#4111yescharacteristic
317C42C2×#4111yescharacteristic
318C42H(5,𝔽2)11yescharacteristic
319C422−1+411yescharacteristic
320C23C22×C22⋊C411yescharacteristic
321Q8—44no—
322Q8—81no—
323Q8—82no—
324Q8—81no—
325Q8—82no—
326Q8—81no—
327Q8—81no—
328Q8—82no—
329Q8—82no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4C2×#10114yes—
360C2×C4C2×#10112yes—
361C2×C4#41612yes—
362C2×C4#41612yes—
363C2×C4#41612yes—
364C2×C4#41612yes—
365C2×C4C4×#4111yescharacteristic
366C2×C4#52011yescharacteristic
367C2×C4#54811yescharacteristic
368C2×C4#54811yescharacteristic
369C2×C4#55011yescharacteristic
370C2×C4#55011yescharacteristic
371C2×C4C2×#13511yescharacteristic
372C2×C4C2×#13511yescharacteristic
373C2×C4C2×#13511yescharacteristic
374C2×C4C2×#13511yescharacteristic
375C2×C4#57611yescharacteristic
376C2×C4#57711yescharacteristic
377C2×C4#57711yescharacteristic
378C2×C4#57811yescharacteristic
379C8—42no—
380C8—42no—
381C8—81no—
382C8—82no—
383C8—81no—
384C8—82no—
385C22#234111yescharacteristic
386C22C22×#10111yescharacteristic
387C22#288911yescharacteristic
388C22C2×#41611yescharacteristic
389C22C2×#41611yescharacteristic
390C22#289711yescharacteristic
391C22#289811yescharacteristic
392C4—21no—
393C4—21no—
394C4—22no—
395C4—22no—
396C4—24no—
397C4—42no—
398C4—42no—
399C4—42no—
400C4—41no—
401C4—82no—
402C4—84no—
403C4—82no—
404C4#154414yes—
405C4#336011yescharacteristic
406C4#336111yescharacteristic
407C4#336411yescharacteristic
408C4#336511yescharacteristic
409C4#339311yescharacteristic
410C4#339411yescharacteristic
411C4#339411yescharacteristic
412C4#339611yescharacteristic
413C2C2×#154411yescharacteristic
414C2#1772511yescharacteristic
415C2#1772711yescharacteristic
416C2#1777411yescharacteristic
417C2#2253811yescharacteristic
418C2#2254111yescharacteristic
419C2#2254311yescharacteristic
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
#228212
#97002
#96942
#96922
#85932
#85862
C4×#3322