Finite Group Databasebeta

← #161110

Subgroups — #161110

Order 512.

774 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34328C211yesmaximal, characteristic
2#22832C211yesmaximal, characteristic
3#22018C211yesmaximal, characteristic
4#7650C212yesmaximal
5#7644C212yesmaximal
6#7619C212yesmaximal
7#7611C212yesmaximal
8#7588C211yesmaximal, characteristic
9#7577C211yesmaximal, characteristic
10#7570C211yesmaximal, characteristic
11#7562C211yesmaximal, characteristic
12C2×#557C411yescharacteristic
13#2371—22no—
14#2370C412yes—
15#2365—21no—
16#2365C411yescharacteristic
17#2356—21no—
18#2356C412yes—
19#2355C411yescharacteristic
20#2355C411yescharacteristic
21#2336—22no—
22#2327—21no—
23C4×C4⋊Q8C2211yescharacteristic
24C4×#125C2212yes—
25#2295C2211yescharacteristic
26#2295C2212yes—
27#2276—21no—
28C42×Q8C2211yescharacteristic
29#1984—21no—
30#1984—21no—
31#1939—21no—
32#1917—22no—
33#1916—22no—
34#1839C2212yes—
35#1839C2212yes—
36#1831C2212yes—
37#1831C2211yescharacteristic
38#1770—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1745C2212yes—
42#1744C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2212yes—
50#1271C2212yes—
51#1271C2212yes—
52#1271C2212yes—
53#557—24no—
54#557—24no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#125—22no—
58C2×#125C2×C412yes—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8C2×C412yes—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—22no—
71C2×#97—22no—
72C2×C4×C8—21no—
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75#391—21no—
76#391—21no—
77#391—21no—
78#391C2×C412yes—
79#391C2×C411yescharacteristic
80#391C2311yescharacteristic
81#389—22no—
82#389—21no—
83#389C2×C412yes—
84#389C2×C411yescharacteristic
85#389C2311yescharacteristic
86#389C2312yes—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—21no—
95#387—22no—
96#387—21no—
97#387—22no—
98#387—22no—
99#387—22no—
100#384C2×C411yescharacteristic
101#383—22no—
102#382—21no—
103#382C2×C412yes—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4D412yes—
120C4×C4⋊C4D411yescharacteristic
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D411yescharacteristic
123#376—21no—
124#376—22no—
125#376—22no—
126C43C2312yes—
127C43C2311yescharacteristic
128#336—22no—
129#336—21no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#336—21no—
135#336—22no—
136#336C2311yescharacteristic
137#336C2312yes—
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142C22×Q8—21no—
143C22×Q8C22⋊C411yescharacteristic
144C22×Q8C22×C411yescharacteristic
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—22no—
151C4⋊Q8—28no—
152C4⋊Q8—22no—
153C4⋊Q8M1614yes—
154#125—216no—
155#125—24no—
156#125—24no—
157C4×Q8—28no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—42no—
166C4×Q8—44no—
167C4×Q8—42no—
168C4×Q8—42no—
169C4×Q8M1614yes—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C42—22no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42C22×C412yes—
200C2×C42C22×C412yes—
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C411yescharacteristic
203C2×C42C2×D412yes—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2411yescharacteristic
212C4⋊C8—42no—
213C4⋊C8—42no—
214C4⋊C8—42no—
215C4⋊C8—82no—
216#97—44no—
217C4×C8—42no—
218C4×C8—42no—
219C4×C8—81no—
220#95—21no—
221#95—22no—
222#95—21no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—21no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—21no—
232#95—22no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—22no—
237#95C22⋊C412yes—
238#95C22⋊C412yes—
239#95C22⋊C412yes—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—24no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8C2×M1612yes—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C23×C411yescharacteristic
265C22×C4C22×D411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C2×C8—22no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—42no—
278C2×C8—41no—
279C2×C8—42no—
280C2×C8—41no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—41no—
285C4⋊C4—28no—
286C4⋊C4—28no—
287C4⋊C4—28no—
288C4⋊C4—21no—
289C4⋊C4—22no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—22no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—44no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—81no—
308C4⋊C4—81no—
309C4⋊C4C2×M1614yes—
310C4⋊C4C2×M1612yes—
311C42—28no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—21no—
317C42—22no—
318C42—21no—
319C42—22no—
320C42—24no—
321C42—22no—
322C42—42no—
323C42—41no—
324C42—41no—
325C42—42no—
326C42—42no—
327C42—41no—
328C42—44no—
329C42—44no—
330C42—42no—
331C42—82no—
332C42—82no—
333C42C2×M1612yes—
334C42C2×M1612yes—
335C42#13512yes—
336C42#13512yes—
337C23C22×C22⋊C411yescharacteristic
338Q8—42no—
339Q8—42no—
340Q8—44no—
341Q8—48no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4#40712yes—
386C2×C4#40712yes—
387C2×C4#41612yes—
388C2×C4#41612yes—
389C2×C4#42112yes—
390C2×C4C22×M1611yescharacteristic
391C2×C4C22×M1611yescharacteristic
392C2×C4C2×#13511yescharacteristic
393C2×C4C2×#13511yescharacteristic
394C2×C4#57711yescharacteristic
395C2×C4#57811yescharacteristic
396C8—42no—
397C8—81no—
398C8—82no—
399C8—81no—
400C8—81no—
401C8—82no—
402C22C2×#40711yescharacteristic
403C22#288411yescharacteristic
404C22#288911yescharacteristic
405C22C2×#41611yescharacteristic
406C22#289811yescharacteristic
407C22C2×#42111yescharacteristic
408C22#290111yescharacteristic
409C4—22no—
410C4—22no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—41no—
418C4—42no—
419C4—42no—
420C4—44no—
421C4—42no—
422C4—42no—
423C4—42no—
424C4—42no—
425C4—81no—
426C4—81no—
427C4—82no—
428C4#148814yes—
429C2#1737111yescharacteristic
430C2C2×#148811yescharacteristic
431C2#1745411yescharacteristic
432C2#1760111yescharacteristic
433C2#1761611yescharacteristic
434C2#1768211yescharacteristic
435C2#1772511yescharacteristic
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
#343282
#228322
#220182
#76502
#76442
#76192
#76112
#75882
#75772
#75702
#75622