Finite Group Databasebeta

← #224677

Subgroups — #224677

Order 512.

877 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36593C211yesmaximal, characteristic
2#22821C211yesmaximal, characteristic
3#22722C211yesmaximal, characteristic
4#9702C212yesmaximal
5#9691C212yesmaximal
6#9691C211yesmaximal, characteristic
7#9691C211yesmaximal, characteristic
8#8590C212yesmaximal
9#8587C211yesmaximal, characteristic
10#8235C212yesmaximal
11C4×#333C211yesmaximal, characteristic
12C2×#557C412yes—
13C2×#446—21no—
14#2868C2211yescharacteristic
15#2868C2212yes—
16#2552—22no—
17#2552—21no—
18#2552—22no—
19#2552—21no—
20#2546C414yes—
21#2546C412yes—
22Q8×C4⋊C4—21no—
23Q8×C4⋊C4—21no—
24C4×C4⋊Q8C2211yescharacteristic
25C4×C4⋊Q8C2212yes—
26C42×Q8C2211yescharacteristic
27#1984—21no—
28#1940—22no—
29#1939—21no—
30#1916—22no—
31#1842C2211yescharacteristic
32#1842C2211yescharacteristic
33#1842C2212yes—
34#1831C2212yes—
35C8×C4⋊C4—21no—
36#1767C2212yes—
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2212yes—
40C2×#333C2211yescharacteristic
41C2×#333C2211yescharacteristic
42C2×#333C2212yes—
43C2×#333C2212yes—
44C2×#333C2212yes—
45C2×#333C2212yes—
46C2×#333C2212yes—
47#1296C2211yescharacteristic
48#1296C2211yescharacteristic
49#1296C2212yes—
50#557—48no—
51C2×C4⋊Q8C2×C412yes—
52C2×C4⋊Q8C2×C414yes—
53C2×C4⋊Q8C2×C412yes—
54C2×C4⋊Q8C2×C414yes—
55C2×C4⋊Q8C2311yescharacteristic
56C2×C4⋊Q8C2312yes—
57C2×#125—24no—
58C2×#125—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8C2×C414yes—
61C2×C4×Q8C2×C412yes—
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65#446—44no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8C2311yescharacteristic
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2311yescharacteristic
77C2×C4⋊C8C2311yescharacteristic
78C2×#97—22no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8—22no—
82#398—24no—
83#398—24no—
84#398—24no—
85#391C2×C412yes—
86#391C2×C414yes—
87#389—22no—
88#389—24no—
89#389—24no—
90#389—22no—
91#389—22no—
92#389—21no—
93#389—21no—
94#389—22no—
95#384—22no—
96#384—21no—
97#384—22no—
98#384—21no—
99#384—22no—
100#384—21no—
101#384—21no—
102#384—22no—
103#384C2311yescharacteristic
104C4×C4⋊C4—22no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4C2×C411yescharacteristic
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4D412yes—
111C4×C4⋊C4D411yescharacteristic
112C4×C4⋊C4D412yes—
113C4×C4⋊C4D411yescharacteristic
114C43C2312yes—
115C43C2311yescharacteristic
116#336—22no—
117#336—22no—
118#336—21no—
119#336—22no—
120#336—22no—
121#336—21no—
122#333—22no—
123#333—22no—
124#333—24no—
125#333—24no—
126#333—42no—
127#333—42no—
128#333—42no—
129C22×Q8C22⋊C411yescharacteristic
130C22×Q8C22×C414yes—
131C22×Q8C22×C412yes—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—22no—
138C4⋊Q8—21no—
139C4⋊Q8—22no—
140C4⋊Q8—21no—
141C4⋊Q8—22no—
142C4⋊Q8—21no—
143C4⋊Q8—22no—
144C4⋊Q8—21no—
145C4⋊Q8—22no—
146C4⋊Q8—48no—
147C4⋊Q8—44no—
148C4⋊Q8—82no—
149C4⋊Q8—84no—
150#125—416no—
151#125—48no—
152C4×Q8—24no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—48no—
156C4×Q8—44no—
157C4×Q8—48no—
158C4×Q8SD1614yes—
159C4×Q8Q1614yes—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22×C412yes—
183C2×C42—24no—
184C2×C42—22no—
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C412yes—
187C2×C42C22×C412yes—
188C2×C42C22×C411yescharacteristic
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D412yes—
196C2×C42C2×D411yescharacteristic
197C2×C42C2411yescharacteristic
198C4⋊C8—21no—
199C4⋊C8—22no—
200C4⋊C8—21no—
201C4⋊C8—22no—
202C4⋊C8—21no—
203C4⋊C8—22no—
204C4⋊C8—21no—
205C4⋊C8—22no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—42no—
209C4⋊C8—44no—
210C4⋊C8—42no—
211C4⋊C8—81no—
212C4⋊C8—82no—
213C4⋊C8—81no—
214C4⋊C8#4112yes—
215C4⋊C8#4112yes—
216#97—44no—
217C4×C8—42no—
218C4×C8—42no—
219C4×C8—82no—
220#95—24no—
221#95—24no—
222#95—22no—
223#95—22no—
224#95C22⋊C412yes—
225#95C22⋊C412yes—
226#95C22⋊C412yes—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—24no—
230C2×Q8—22no—
231C2×Q8—24no—
232C2×Q8—22no—
233C2×Q8—44no—
234C2×Q8—42no—
235C2×Q8—44no—
236C2×Q8—42no—
237C2×Q8#10314yes—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C23×C411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C22×C4C22×D411yescharacteristic
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—42no—
260C2×C8—41no—
261C2×C8—42no—
262C2×C8—41no—
263C2×C8—42no—
264C2×C8—41no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—41no—
268C2×C8—42no—
269C2×C8—41no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—42no—
274C4⋊C4—44no—
275C4⋊C4—48no—
276C4⋊C4—44no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—44no—
290C4⋊C4—48no—
291C4⋊C4—48no—
292C4⋊C4—82no—
293C4⋊C4—84no—
294C4⋊C4—82no—
295C4⋊C4—84no—
296C4⋊C4#10014yes—
297C4⋊C4#10318yes—
298C4⋊C4#10314yes—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—21no—
303C42—22no—
304C42—21no—
305C42—22no—
306C42—21no—
307C42—22no—
308C42—21no—
309C42—24no—
310C42—22no—
311C42—41no—
312C42—42no—
313C42—48no—
314C42—44no—
315C42C2×SD1612yes—
316C42C2×SD1614yes—
317C42C2×Q1614yes—
318C42C2×Q1612yes—
319C42#13712yes—
320C42#13712yes—
321C42C2×#4111yescharacteristic
322C42C2×#4111yescharacteristic
323C42H(5,𝔽2)11yescharacteristic
324C422−1+411yescharacteristic
325C23C22×C22⋊C411yescharacteristic
326Q8—28no—
327Q8—84no—
328Q8—82no—
329Q8—84no—
330Q8—82no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4C2×#10014yes—
359C2×C4C2×#10012yes—
360C2×C4C2×#10312yes—
361C2×C4C2×#10314yes—
362C2×C4C2×#10314yes—
363C2×C4C2×#10312yes—
364C2×C4#41912yes—
365C2×C4#41912yes—
366C2×C4C4×#4111yescharacteristic
367C2×C4#52011yescharacteristic
368C2×C4#54811yescharacteristic
369C2×C4#54811yescharacteristic
370C2×C4#55011yescharacteristic
371C2×C4#55011yescharacteristic
372C2×C4C22×SD1611yescharacteristic
373C2×C4C22×Q1611yescharacteristic
374C2×C4C2×#13711yescharacteristic
375C2×C4C2×#13711yescharacteristic
376C2×C4#57711yescharacteristic
377C2×C4#57711yescharacteristic
378C2×C4#57811yescharacteristic
379C2×C4#57811yescharacteristic
380C8—42no—
381C8—42no—
382C8—81no—
383C8—82no—
384C8—81no—
385C8—82no—
386C22#234111yescharacteristic
387C22C22×#10011yescharacteristic
388C22#288811yescharacteristic
389C22C22×#10311yescharacteristic
390C22C2×#41911yescharacteristic
391C22#289811yescharacteristic
392C22#289811yescharacteristic
393C4—21no—
394C4—21no—
395C4—24no—
396C4—24no—
397C4—22no—
398C4—22no—
399C4—24no—
400C4—42no—
401C4—41no—
402C4—84no—
403C4—82no—
404C4—82no—
405C4#154814yes—
406C4#335911yescharacteristic
407C4#336311yescharacteristic
408C4#337211yescharacteristic
409C4#337411yescharacteristic
410C4#339211yescharacteristic
411C4#339511yescharacteristic
412C4#339811yescharacteristic
413C4#340011yescharacteristic
414C2C2×#154811yescharacteristic
415C2#1772211yescharacteristic
416C2#1776011yescharacteristic
417C2#1777911yescharacteristic
418C2#2253611yescharacteristic
419C2#2254211yescharacteristic
420C2#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
#365932
#228212
#227222
#97022
#96912
#85902
#85872
#82352
C4×#3332