Finite Group Databasebeta

← #224672

Subgroups — #224672

Order 512.

737 conjugacy classes of subgroups.

The subgroup lattice has 414 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#9693C212yesmaximal
6#9691C211yesmaximal, characteristic
7#9691C211yesmaximal, characteristic
8#8594C212yesmaximal
9#8586C211yesmaximal, characteristic
10#8235C212yesmaximal
11C4×#332C211yesmaximal, characteristic
12#2868C2211yescharacteristic
13#2862C2212yes—
14#2582—21no—
15#2571—21no—
16#2570—22no—
17#2568C412yes—
18#2511—22no—
19#2446C414yes—
20#2365C412yes—
21#2355—21no—
22#2355—21no—
23C4×C4⋊Q8C2211yescharacteristic
24C4×C4⋊Q8C2212yes—
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⋊C8C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C2×#333C2212yes—
41C2×#333C2212yes—
42C2×#333C2212yes—
43C2×#333C2212yes—
44C2×#332C2211yescharacteristic
45C2×#332C2211yescharacteristic
46C2×#332C2212yes—
47#1296C2211yescharacteristic
48#1296C2211yescharacteristic
49#1296C2212yes—
50C2×C4⋊Q8C2×C412yes—
51C2×C4⋊Q8C2×C412yes—
52C2×C4⋊Q8C2312yes—
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4×Q8—21no—
55C2×C4×Q8C2×C414yes—
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—21no—
78#391—22no—
79#391—22no—
80#391C2×C414yes—
81#391C2×C414yes—
82#391C2×C414yes—
83#389—22no—
84#389—21no—
85#389—22no—
86#389—22no—
87#389C2×C412yes—
88#387—24no—
89#387—24no—
90#387—24no—
91#384—21no—
92#384—21no—
93#384—21no—
94#384—21no—
95#384C2311yescharacteristic
96#382—22no—
97#382—22no—
98#382—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4C2×C412yes—
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—24no—
121#333—24no—
122#333—42no—
123#333—42no—
124#332—22no—
125#332—22no—
126#332—42no—
127C22×Q8—21no—
128C22×Q8C22⋊C411yescharacteristic
129C22×Q8C22×C414yes—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8—22no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—21no—
141C4⋊Q8—22no—
142C4⋊Q8—21no—
143C4⋊Q8—22no—
144C4⋊Q8—44no—
145C4⋊Q8—82no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—42no—
151C4×Q8—42no—
152C4×Q8—42no—
153C4×Q8—84no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C4⋊C4C22⋊C411yescharacteristic
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C411yescharacteristic
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C412yes—
173C2×C42—22no—
174C2×C42—24no—
175C2×C42C22×C411yescharacteristic
176C2×C42C22×C412yes—
177C2×C42C22×C412yes—
178C2×C42C22×C411yescharacteristic
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C2×C42C2×D412yes—
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2411yescharacteristic
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—42no—
199C4⋊C8—42no—
200C4⋊C8—81no—
201C4⋊C8—81no—
202C4⋊C8#4112yes—
203C4⋊C8#4112yes—
204#97—82no—
205C4×C8—42no—
206C4×C8—42no—
207C4×C8—82no—
208#95—24no—
209#95—24no—
210#95—22no—
211#95—24no—
212#95—22no—
213#95—24no—
214#95—24no—
215#95—24no—
216#95C22⋊C412yes—
217#95C22⋊C412yes—
218#95C22⋊C412yes—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—24no—
223C2×Q8—24no—
224C2×Q8—44no—
225C2×Q8—44no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4C2×C22⋊C412yes—
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C23×C411yescharacteristic
244C22×C4C22×D411yescharacteristic
245C22×C4C22×D411yescharacteristic
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—42no—
254C2×C8—41no—
255C2×C8—42no—
256C2×C8—41no—
257C2×C8—42no—
258C2×C8—41no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—41no—
263C2×C8—41no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—24no—
267C4⋊C4—22no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—44no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—84no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—42no—
303C42—42no—
304C42—41no—
305C42—44no—
306C42—84no—
307C42#13512yes—
308C42#13512yes—
309C42#13712yes—
310C42#13712yes—
311C42C2×#4111yescharacteristic
312C42C2×#4111yescharacteristic
313C42H(5,𝔽2)11yescharacteristic
314C422−1+411yescharacteristic
315C23C22×C22⋊C411yescharacteristic
316Q8—44no—
317Q8—84no—
318Q8—84no—
319Q8—81no—
320Q8—81no—
321Q8—81no—
322Q8—81no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
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—21no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4#41312yes—
354C2×C4#41612yes—
355C2×C4#41612yes—
356C2×C4#41912yes—
357C2×C4#41912yes—
358C2×C4C4×#4111yescharacteristic
359C2×C4#52011yescharacteristic
360C2×C4#54811yescharacteristic
361C2×C4#54811yescharacteristic
362C2×C4#55011yescharacteristic
363C2×C4#55011yescharacteristic
364C2×C4C2×#13511yescharacteristic
365C2×C4C2×#13511yescharacteristic
366C2×C4C2×#13711yescharacteristic
367C2×C4C2×#13711yescharacteristic
368C2×C4#57711yescharacteristic
369C2×C4#57711yescharacteristic
370C2×C4#57811yescharacteristic
371C2×C4#57811yescharacteristic
372C8—42no—
373C8—42no—
374C8—81no—
375C8—82no—
376C8—81no—
377C8—82no—
378C22#234111yescharacteristic
379C22C2×#41311yescharacteristic
380C22#288811yescharacteristic
381C22C2×#41611yescharacteristic
382C22C2×#41911yescharacteristic
383C22#289811yescharacteristic
384C22#289811yescharacteristic
385C4—21no—
386C4—21no—
387C4—22no—
388C4—22no—
389C4—42no—
390C4—42no—
391C4—42no—
392C4—41no—
393C4—42no—
394C4—84no—
395C4—82no—
396C4—82no—
397C4#155414yes—
398C4#336011yescharacteristic
399C4#336411yescharacteristic
400C4#337211yescharacteristic
401C4#337411yescharacteristic
402C4#339311yescharacteristic
403C4#339411yescharacteristic
404C4#339811yescharacteristic
405C4#340011yescharacteristic
406C2C2×#155411yescharacteristic
407C2#1772611yescharacteristic
408C2#1776011yescharacteristic
409C2#1777811yescharacteristic
410C2#2253911yescharacteristic
411C2#2254111yescharacteristic
412C2#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
#96932
#96912
#85942
#85862
#82352
C4×#3322