Finite Group Databasebeta

← #160914

Subgroups — #160914

Order 512.

834 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34328C211yesmaximal, characteristic
2#22736C211yesmaximal, characteristic
3#20570C211yesmaximal, characteristic
4#7636C212yesmaximal
5#7630C212yesmaximal
6#7619C212yesmaximal
7#7615C212yesmaximal
8#7588C211yesmaximal, characteristic
9#7577C211yesmaximal, characteristic
10#7570C211yesmaximal, characteristic
11#7562C211yesmaximal, characteristic
12C2×#557C411yescharacteristic
13C2×#446—21no—
14C2×#446—21no—
15#2371—22no—
16#2370C412yes—
17#2365—21no—
18#2365C411yescharacteristic
19#2356—21no—
20#2356C412yes—
21#2355C411yescharacteristic
22#2355C411yescharacteristic
23#2336—22no—
24#2327—21no—
25C4×C4⋊Q8C2211yescharacteristic
26C4×#125C2212yes—
27#2295C2211yescharacteristic
28#2295C2212yes—
29#2276—21no—
30C42×Q8C2211yescharacteristic
31#1940—22no—
32#1939—21no—
33#1831C2211yescharacteristic
34#1831C2212yes—
35#1771—22no—
36#1770—21no—
37#1767C2212yes—
38#1767C2212yes—
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
67#446—28no—
68#446—44no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—21no—
75C2×#97—22no—
76C2×#97—22no—
77C2×#97—22no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81#391—21no—
82#391—21no—
83#391—21no—
84#391C2×C412yes—
85#391C2×C411yescharacteristic
86#391C2311yescharacteristic
87#389—22no—
88#389—21no—
89#389C2×C412yes—
90#389C2×C411yescharacteristic
91#389C2311yescharacteristic
92#389C2312yes—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—21no—
101#387—22no—
102#387—21no—
103#387—22no—
104#387—22no—
105#387—22no—
106#384C2×C411yescharacteristic
107#383—22no—
108#382—21no—
109#382C2×C412yes—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D412yes—
128C4×C4⋊C4D411yescharacteristic
129#376—21no—
130#376—22no—
131#376—22no—
132C43C2312yes—
133C43C2311yescharacteristic
134#336—22no—
135#336—21no—
136#336—21no—
137#336—22no—
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2312yes—
143#336C2311yescharacteristic
144C22×Q8—21no—
145C22×Q8C22⋊C411yescharacteristic
146C22×Q8C22×C411yescharacteristic
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C4⋊Q8—28no—
154C4⋊Q8—22no—
155C4⋊Q8M1614yes—
156#125—216no—
157#125—24no—
158#125—24no—
159C4×Q8—28no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—44no—
169C4×Q8—42no—
170C4×Q8—42no—
171C4×Q8M1614yes—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C42—22no—
195C2×C42—21no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42C22×C412yes—
202C2×C42C22×C412yes—
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2411yescharacteristic
214C4⋊C8—24no—
215C4⋊C8—28no—
216C4⋊C8—44no—
217C4⋊C8—44no—
218C4⋊C8—42no—
219C4⋊C8—42no—
220#97—28no—
221#97—44no—
222#97—82no—
223C4×C8—24no—
224C4×C8—42no—
225C4×C8—81no—
226#95—21no—
227#95—22no—
228#95—21no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—21no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—21no—
238#95—22no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—22no—
243#95C22⋊C412yes—
244#95C22⋊C412yes—
245#95C22⋊C412yes—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—24no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8C2×M1612yes—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C23×C411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—24no—
276C2×C8—22no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—42no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—42no—
291C4⋊C4—28no—
292C4⋊C4—28no—
293C4⋊C4—28no—
294C4⋊C4—21no—
295C4⋊C4—22no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—81no—
313C4⋊C4—81no—
314C4⋊C4#10414yes—
315C4⋊C4C2×M1614yes—
316C4⋊C4C2×M1612yes—
317C42—28no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—21no—
323C42—22no—
324C42—21no—
325C42—22no—
326C42—24no—
327C42—22no—
328C42—24no—
329C42—42no—
330C42—41no—
331C42—41no—
332C42—42no—
333C42—41no—
334C42—44no—
335C42—44no—
336C42—42no—
337C42—82no—
338C42—82no—
339C42C2×M1612yes—
340C42C2×M1612yes—
341C42AGL(1,ℤ/8)12yes—
342C42#13712yes—
343C23C22×C22⋊C411yescharacteristic
344Q8—42no—
345Q8—42no—
346Q8—44no—
347Q8—48no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4#40712yes—
391C2×C4#40712yes—
392C2×C4#41712yes—
393C2×C4#41712yes—
394C2×C4C2×#10414yes—
395C2×C4C2×#10412yes—
396C2×C4C22×M1611yescharacteristic
397C2×C4C22×M1611yescharacteristic
398C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
399C2×C4C2×#13711yescharacteristic
400C2×C4#57711yescharacteristic
401C2×C4#57811yescharacteristic
402C8—42no—
403C8—44no—
404C8—42no—
405C8—42no—
406C8—81no—
407C8—82no—
408C22C2×#40711yescharacteristic
409C22#288411yescharacteristic
410C22#288911yescharacteristic
411C22C2×#41711yescharacteristic
412C22#289811yescharacteristic
413C22C22×#10411yescharacteristic
414C22#290111yescharacteristic
415C4—22no—
416C4—22no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—24no—
424C4—42no—
425C4—44no—
426C4—42no—
427C4—42no—
428C4—42no—
429C4—42no—
430C4—41no—
431C4—81no—
432C4—82no—
433C4—81no—
434C4#148914yes—
435C2#1737111yescharacteristic
436C2C2×#148911yescharacteristic
437C2#1745411yescharacteristic
438C2#1754911yescharacteristic
439C2#1762811yescharacteristic
440C2#1765311yescharacteristic
441C2#1776311yescharacteristic
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
#227362
#205702
#76362
#76302
#76192
#76152
#75882
#75772
#75702
#75622