Finite Group Databasebeta

← #224657

Subgroups — #224657

Order 512.

777 conjugacy classes of subgroups.

The subgroup lattice has 417 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#9693C212yesmaximal
6#9691C211yesmaximal, characteristic
7#9691C211yesmaximal, characteristic
8#8590C212yesmaximal
9#8586C211yesmaximal, characteristic
10C4×#332C211yesmaximal, characteristic
11C4×#333C212yesmaximal
12C2×C8×Q8—21no—
13#2868C2211yescharacteristic
14#2862C2212yes—
15#2582—21no—
16#2571—21no—
17#2570—22no—
18#2568C412yes—
19#2511—22no—
20#2446C414yes—
21#2365C412yes—
22#2355—21no—
23#2355—21no—
24C4×C4⋊Q8C2211yescharacteristic
25C4×C4⋊Q8C2212yes—
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⋊C8C2212yes—
37C4×C4⋊C8C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C2×#333C2212yes—
41C2×#333C2212yes—
42C2×#333C2212yes—
43C2×#333C2212yes—
44C2×#332C2211yescharacteristic
45C2×#332C2211yescharacteristic
46C2×#332C2212yes—
47#1296C2211yescharacteristic
48#1296C2212yes—
49#1296C2211yescharacteristic
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
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⋊C8C2311yescharacteristic
69C2×C4⋊C8C2312yes—
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—21no—
81#391—22no—
82#391—22no—
83#391C2×C414yes—
84#391C2×C414yes—
85#391C2×C414yes—
86#389—22no—
87#389—21no—
88#389—22no—
89#389—22no—
90#389C2×C412yes—
91#387—24no—
92#387—24no—
93#387—24no—
94#384—21no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384C2311yescharacteristic
99#382—22no—
100#382—22no—
101#382—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4C2×C412yes—
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—24no—
122#333—24no—
123#333—42no—
124#333—42no—
125#332—22no—
126#332—22no—
127#332—42no—
128C22×Q8—21no—
129C22×Q8C22⋊C411yescharacteristic
130C22×Q8C22×C414yes—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—22no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—21no—
142C4⋊Q8—22no—
143C4⋊Q8—21no—
144C4⋊Q8—22no—
145C4⋊Q8—44no—
146C4⋊Q8—82no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—42no—
150C4×Q8—42no—
151C4×Q8—42no—
152C4×Q8—84no—
153C4×Q8SD1614yes—
154C4×Q8Q1614yes—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4C22⋊C411yescharacteristic
167C2×C4⋊C4C22⋊C411yescharacteristic
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C411yescharacteristic
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C42—22no—
175C2×C42—24no—
176C2×C42C22×C411yescharacteristic
177C2×C42C22×C412yes—
178C2×C42C22×C412yes—
179C2×C42C22×C411yescharacteristic
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C2×C42C2×D412yes—
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D412yes—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2411yescharacteristic
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—44no—
200C4⋊C8—42no—
201C4⋊C8—42no—
202C4⋊C8—81no—
203C4⋊C8—81no—
204C4⋊C8—82no—
205C4⋊C8#4112yes—
206C4⋊C8#4112yes—
207C4×C8—44no—
208C4×C8—42no—
209C4×C8—42no—
210C4×C8—82no—
211#95—24no—
212#95—24no—
213#95—22no—
214#95—24no—
215#95—22no—
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—24no—
225C2×Q8—24no—
226C2×Q8—44no—
227C2×Q8—44no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8#10314yes—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C412yes—
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C23×C411yescharacteristic
247C22×C4C22×D411yescharacteristic
248C22×C4C22×D411yescharacteristic
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—42no—
257C2×C8—41no—
258C2×C8—42no—
259C2×C8—41no—
260C2×C8—42no—
261C2×C8—41no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—41no—
265C2×C8—42no—
266C2×C8—41no—
267C4⋊C4—22no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—44no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—84no—
288C4⋊C4—82no—
289C4⋊C4—82no—
290C4⋊C4#10318yes—
291C4⋊C4#10314yes—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—21no—
296C42—21no—
297C42—22no—
298C42—21no—
299C42—22no—
300C42—21no—
301C42—22no—
302C42—22no—
303C42—42no—
304C42—42no—
305C42—41no—
306C42—44no—
307C42—84no—
308C42C2×SD1614yes—
309C42C2×SD1612yes—
310C42C2×Q1614yes—
311C42C2×Q1612yes—
312C42#13512yes—
313C42#13512yes—
314C42C2×#4111yescharacteristic
315C42C2×#4111yescharacteristic
316C42H(5,𝔽2)11yescharacteristic
317C422−1+411yescharacteristic
318C23C22×C22⋊C411yescharacteristic
319Q8—28no—
320Q8—84no—
321Q8—84no—
322Q8—81no—
323Q8—81no—
324Q8—81no—
325Q8—81no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4#41312yes—
355C2×C4C2×#10314yes—
356C2×C4C2×#10312yes—
357C2×C4C2×#10314yes—
358C2×C4C2×#10312yes—
359C2×C4#41612yes—
360C2×C4#41612yes—
361C2×C4C4×#4111yescharacteristic
362C2×C4#52011yescharacteristic
363C2×C4#54811yescharacteristic
364C2×C4#54811yescharacteristic
365C2×C4#55011yescharacteristic
366C2×C4#55011yescharacteristic
367C2×C4C22×SD1611yescharacteristic
368C2×C4C22×Q1611yescharacteristic
369C2×C4C2×#13511yescharacteristic
370C2×C4C2×#13511yescharacteristic
371C2×C4#57711yescharacteristic
372C2×C4#57711yescharacteristic
373C2×C4#57811yescharacteristic
374C2×C4#57811yescharacteristic
375C8—42no—
376C8—42no—
377C8—81no—
378C8—82no—
379C8—81no—
380C8—82no—
381C22#234111yescharacteristic
382C22C2×#41311yescharacteristic
383C22#288811yescharacteristic
384C22C22×#10311yescharacteristic
385C22C2×#41611yescharacteristic
386C22#289811yescharacteristic
387C22#289811yescharacteristic
388C4—21no—
389C4—21no—
390C4—24no—
391C4—24no—
392C4—22no—
393C4—22no—
394C4—42no—
395C4—41no—
396C4—42no—
397C4—84no—
398C4—82no—
399C4—82no—
400C4#154214yes—
401C4#335911yescharacteristic
402C4#336011yescharacteristic
403C4#336311yescharacteristic
404C4#336411yescharacteristic
405C4#339211yescharacteristic
406C4#339311yescharacteristic
407C4#339411yescharacteristic
408C4#339511yescharacteristic
409C2C2×#154211yescharacteristic
410C2#1772211yescharacteristic
411C2#1772611yescharacteristic
412C2#1777811yescharacteristic
413C2#2253911yescharacteristic
414C2#2254111yescharacteristic
415C2#2254211yescharacteristic
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
#96932
#96912
#85902
#85862
C4×#3322
C4×#3332