Finite Group Databasebeta

← #224661

Subgroups — #224661

Order 512.

777 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2#22821C211yesmaximal, characteristic
3Q8×C4⋊C8C211yesmaximal, characteristic
4#9697C212yesmaximal
5#9694C211yesmaximal, characteristic
6#9693C212yesmaximal
7#9692C211yesmaximal, characteristic
8#8591C212yesmaximal
9#8587C211yesmaximal, characteristic
10C4×#332C212yesmaximal
11C4×#333C211yesmaximal, characteristic
12C2×C8×Q8—21no—
13#2868C2211yescharacteristic
14#2862C2212yes—
15#2582C412yes—
16#2571—21no—
17#2570—22no—
18#2568—21no—
19#2511C414yes—
20#2446—22no—
21#2365—21no—
22#2355—21no—
23#2355C412yes—
24C4×C4⋊Q8C2212yes—
25C4×C4⋊Q8C2211yescharacteristic
26C42×Q8C2211yescharacteristic
27#1984—21no—
28#1939—22no—
29#1939—21no—
30#1916—22no—
31#1842C2211yescharacteristic
32#1842C2211yescharacteristic
33#1842C2212yes—
34#1831C2212yes—
35C8×C4⋊C4—21no—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2212yes—
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
59C8×Q8—44no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8C2311yescharacteristic
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2311yescharacteristic
70C2×C4⋊C8C2311yescharacteristic
71C2×C4⋊C8C2311yescharacteristic
72C2×C4×C8—22no—
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75C2×C4×C8—22no—
76#400—24no—
77#400—24no—
78#398—24no—
79#391—22no—
80#391—22no—
81#391—22no—
82#391—22no—
83#391C2×C412yes—
84#391C2×C414yes—
85#391C2×C414yes—
86#389—22no—
87#389—21no—
88#389—22no—
89#389—21no—
90#389C2×C414yes—
91#387—24no—
92#387—24no—
93#387—24no—
94#384—21no—
95#384—21no—
96#384C2×C412yes—
97#384C2×C412yes—
98#384C2311yescharacteristic
99#382—22no—
100#382—22no—
101#382—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4C2×C414yes—
107C4×C4⋊C4C2×C411yescharacteristic
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4D412yes—
110C4×C4⋊C4D411yescharacteristic
111C4×C4⋊C4D411yescharacteristic
112C4×C4⋊C4D412yes—
113C43C2312yes—
114C43C2311yescharacteristic
115#336—22no—
116#336—22no—
117#336—21no—
118#336—22no—
119#336—22no—
120#336—21no—
121#333—22no—
122#333—22no—
123#333—42no—
124#332—24no—
125#332—24no—
126#332—42no—
127#332—42no—
128C22×Q8—22no—
129C22×Q8C22⋊C411yescharacteristic
130C22×Q8C22×C412yes—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—21no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—22no—
142C4⋊Q8—21no—
143C4⋊Q8—22no—
144C4⋊Q8—21no—
145C4⋊Q8—42no—
146C4⋊Q8—42no—
147C4⋊Q8—81no—
148C4⋊Q8—81no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—42no—
152C4×Q8—44no—
153C4×Q8—82no—
154C4×Q8—82no—
155C4×Q8SD1614yes—
156C4×Q8Q1614yes—
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—21no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
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×C411yescharacteristic
179C2×C42C22×C412yes—
180C2×C42C22×C412yes—
181C2×C42C22×C411yescharacteristic
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D412yes—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D412yes—
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2411yescharacteristic
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—44no—
202C4⋊C8—42no—
203C4⋊C8—42no—
204C4⋊C8—81no—
205C4⋊C8—81no—
206C4⋊C8—82no—
207C4⋊C8#4112yes—
208C4⋊C8#4112yes—
209C4×C8—44no—
210C4×C8—42no—
211C4×C8—42no—
212C4×C8—82no—
213#95—24no—
214#95—22no—
215#95—22no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95C22⋊C412yes—
222#95C22⋊C412yes—
223#95C22⋊C412yes—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8#10314yes—
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—42no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—41no—
268C2×C8—41no—
269C4⋊C4—22no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—84no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4#10318yes—
293C4⋊C4#10314yes—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—21no—
298C42—21no—
299C42—22no—
300C42—22no—
301C42—21no—
302C42—21no—
303C42—22no—
304C42—22no—
305C42—41no—
306C42—42no—
307C42—42no—
308C42—44no—
309C42—84no—
310C42C2×SD1614yes—
311C42C2×SD1612yes—
312C42C2×Q1614yes—
313C42C2×Q1612yes—
314C42#13512yes—
315C42#13512yes—
316C42C2×#4111yescharacteristic
317C42C2×#4111yescharacteristic
318C42H(5,𝔽2)11yescharacteristic
319C422−1+411yescharacteristic
320C23C22×C22⋊C411yescharacteristic
321Q8—28no—
322Q8—82no—
323Q8—82no—
324Q8—82no—
325Q8—82no—
326Q8—82no—
327Q8—82no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4#41312yes—
357C2×C4C2×#10314yes—
358C2×C4C2×#10312yes—
359C2×C4C2×#10314yes—
360C2×C4C2×#10312yes—
361C2×C4#41612yes—
362C2×C4#41612yes—
363C2×C4C4×#4111yescharacteristic
364C2×C4#52011yescharacteristic
365C2×C4#54811yescharacteristic
366C2×C4#54811yescharacteristic
367C2×C4#55011yescharacteristic
368C2×C4#55011yescharacteristic
369C2×C4C22×SD1611yescharacteristic
370C2×C4C22×Q1611yescharacteristic
371C2×C4C2×#13511yescharacteristic
372C2×C4C2×#13511yescharacteristic
373C2×C4#57611yescharacteristic
374C2×C4#57711yescharacteristic
375C2×C4#57711yescharacteristic
376C2×C4#57811yescharacteristic
377C8—42no—
378C8—42no—
379C8—82no—
380C8—81no—
381C8—81no—
382C8—82no—
383C22#234111yescharacteristic
384C22C2×#41311yescharacteristic
385C22#288911yescharacteristic
386C22C22×#10311yescharacteristic
387C22C2×#41611yescharacteristic
388C22#289711yescharacteristic
389C22#289811yescharacteristic
390C4—21no—
391C4—21no—
392C4—24no—
393C4—24no—
394C4—22no—
395C4—22no—
396C4—42no—
397C4—41no—
398C4—42no—
399C4—82no—
400C4—84no—
401C4—82no—
402C4#154214yes—
403C4#335911yescharacteristic
404C4#336111yescharacteristic
405C4#336311yescharacteristic
406C4#336511yescharacteristic
407C4#339211yescharacteristic
408C4#339411yescharacteristic
409C4#339511yescharacteristic
410C4#339611yescharacteristic
411C2C2×#154211yescharacteristic
412C2#1772311yescharacteristic
413C2#1772511yescharacteristic
414C2#1777211yescharacteristic
415C2#2253711yescharacteristic
416C2#2254211yescharacteristic
417C2#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
Q8×C4⋊C82
#96972
#96942
#96932
#96922
#85912
#85872
C4×#3322
C4×#3332