Finite Group Databasebeta

← #164982

Subgroups — #164982

Order 512.

675 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36286C211yesmaximal, characteristic
2#22832C211yesmaximal, characteristic
3#22828C211yesmaximal, characteristic
4#7729C212yesmaximal
5#7693C212yesmaximal
6#7662C211yesmaximal, characteristic
7#7662C211yesmaximal, characteristic
8#7652C211yesmaximal, characteristic
9#7611C212yesmaximal
10#7581C212yesmaximal
11#7570C211yesmaximal, characteristic
12#2863C2212yes—
13#2862C2211yescharacteristic
14#2605—22no—
15#2588—22no—
16#2581—21no—
17#2570—21no—
18#2568C411yescharacteristic
19#2567C411yescharacteristic
20#2511C412yes—
21#2451C412yes—
22#2369C411yescharacteristic
23#2365C411yescharacteristic
24#2356—21no—
25#2355—21no—
26C4×C4⋊Q8C2211yescharacteristic
27#2295C2212yes—
28C42×Q8C2211yescharacteristic
29#1984—21no—
30#1984—21no—
31#1917—22no—
32#1916—22no—
33#1839C2212yes—
34#1839C2212yes—
35#1831C2211yescharacteristic
36#1831C2211yescharacteristic
37#1770—21no—
38C8×C4⋊C4—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1745C2212yes—
42#1744C2212yes—
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C4×Q8—21no—
57C2×C4×Q8C2×C412yes—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×#97—22no—
66C2×C4×C8—22no—
67C2×C4×C8—21no—
68C2×C4×C8—21no—
69#400—22no—
70#400—22no—
71#400—22no—
72#398—22no—
73#395—22no—
74#395—22no—
75#391—21no—
76#391—21no—
77#391—21no—
78#391—21no—
79#391—21no—
80#391—22no—
81#391—22no—
82#391C2×C412yes—
83#391C2×C412yes—
84#391C2×C412yes—
85#391C2×C412yes—
86#391C2312yes—
87#389—21no—
88#389—21no—
89#389—21no—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2311yescharacteristic
94#389C2311yescharacteristic
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#384—21no—
105#384C2×C411yescharacteristic
106#383—22no—
107#382—21no—
108#382—21no—
109#382—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4D411yescharacteristic
120C4×C4⋊C4D412yes—
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D411yescharacteristic
123#376—22no—
124C43C2311yescharacteristic
125C43C2312yes—
126#336—21no—
127#336—21no—
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#336C2312yes—
135#336C2312yes—
136#336C2311yescharacteristic
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8C22⋊C411yescharacteristic
143C22×Q8C22×C412yes—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—41no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—24no—
159C4×Q8—42no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×Q8—42no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4C22⋊C411yescharacteristic
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C412yes—
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2411yescharacteristic
203C4⋊C8—42no—
204C4⋊C8—42no—
205C4⋊C8—42no—
206C4⋊C8—42no—
207#97—44no—
208C4×C8—44no—
209C4×C8—42no—
210C4×C8—42no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—21no—
217#95—22no—
218#95—21no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—21no—
226#95—22no—
227#95—22no—
228#95—21no—
229#95C22⋊C412yes—
230#95C22⋊C412yes—
231#95C22⋊C412yes—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—81no—
243C2×Q8—81no—
244C2×Q8—81no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C23×C411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—41no—
277C2×C8—42no—
278C2×C8—41no—
279C2×C8—42no—
280C4⋊C4—21no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—24no—
284C4⋊C4—22no—
285C4⋊C4—24no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—42no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—81no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4#13112yes—
307C42—22no—
308C42—21no—
309C42—22no—
310C42—21no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—42no—
317C42—42no—
318C42—41no—
319C42—41no—
320C42—42no—
321C42—41no—
322C42—44no—
323C42—42no—
324C42—42no—
325C42—82no—
326C42#13512yes—
327C42#13512yes—
328C42#13712yes—
329C42#13712yes—
330C23C22×C22⋊C411yescharacteristic
331Q8—44no—
332Q8—84no—
333Q8—82no—
334Q8—82no—
335Q8—162no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4#41312yes—
380C2×C4#41612yes—
381C2×C4#41612yes—
382C2×C4#41912yes—
383C2×C4#41912yes—
384C2×C4C2×#13111yescharacteristic
385C2×C4#56811yescharacteristic
386C2×C4C2×#13511yescharacteristic
387C2×C4C2×#13511yescharacteristic
388C2×C4C2×#13711yescharacteristic
389C2×C4C2×#13711yescharacteristic
390C8—42no—
391C8—42no—
392C8—81no—
393C8—82no—
394C8—81no—
395C8—82no—
396C22#288011yescharacteristic
397C22#288411yescharacteristic
398C22C2×#41311yescharacteristic
399C22C2×#41611yescharacteristic
400C22C2×#41911yescharacteristic
401C22#290111yescharacteristic
402C22#290111yescharacteristic
403C4—21no—
404C4—21no—
405C4—22no—
406C4—22no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—42no—
412C4—41no—
413C4—42no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4—82no—
418C4—82no—
419C4—81no—
420C4—81no—
421C4—81no—
422C4#155414yes—
423C2#1742811yescharacteristic
424C2#1750811yescharacteristic
425C2#1751411yescharacteristic
426C2#1761611yescharacteristic
427C2#1763211yescharacteristic
428C2#1769111yescharacteristic
429C2C2×#155411yescharacteristic
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 ▷ C23 ▷ C1

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#362862
#228322
#228282
#77292
#76932
#76622
#76522
#76112
#75812
#75702