Finite Group Databasebeta

← #160902

Subgroups — #160902

Order 512.

836 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34335C211yesmaximal, characteristic
2#22832C211yesmaximal, characteristic
3#20570C211yesmaximal, characteristic
4#7638C212yesmaximal
5#7630C212yesmaximal
6#7625C212yesmaximal
7#7611C212yesmaximal
8#7591C211yesmaximal, characteristic
9#7577C211yesmaximal, characteristic
10#7570C211yesmaximal, characteristic
11#7562C211yesmaximal, characteristic
12C2×Q82C411yescharacteristic
13C2×#446—21no—
14#2369C412yes—
15#2369C411yescharacteristic
16#2365—21no—
17#2365—22no—
18#2355C412yes—
19#2355C411yescharacteristic
20#2351—21no—
21Q8×C4⋊C4C411yescharacteristic
22#2331—22no—
23#2331—21no—
24C4×C4⋊Q8C2211yescharacteristic
25C4×C4⋊Q8C2212yes—
26#2295C2211yescharacteristic
27#2295C2212yes—
28#2276—21no—
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1917—22no—
32#1839C2212yes—
33#1831C2211yescharacteristic
34#1771—22no—
35#1770—21no—
36#1770—21no—
37#1767C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40#1745C2212yes—
41#1745C2212yes—
42#1744C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2212yes—
50#1271C2212yes—
51#1271C2212yes—
52#1271C2212yes—
53Q82—24no—
54Q82—24no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8C2×C412yes—
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—22no—
63C2×C4×Q8C2×C412yes—
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68#446—28no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×#97—22no—
74C2×#97—22no—
75C2×#97—22no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79#391—22no—
80#391—21no—
81#391—22no—
82#391—22no—
83#391—22no—
84#391C2×C411yescharacteristic
85#391C2×C412yes—
86#391C2311yescharacteristic
87#389—21no—
88#389—22no—
89#389—22no—
90#389—21no—
91#389—21no—
92#389—22no—
93#389C2×C411yescharacteristic
94#389C2×C412yes—
95#389C2312yes—
96#389C2311yescharacteristic
97#387—22no—
98#387—21no—
99#387—21no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#384C2×C412yes—
105#384C2×C411yescharacteristic
106#382—22no—
107#382—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D412yes—
127#376—22no—
128#376—21no—
129C43C2311yescharacteristic
130C43C2312yes—
131#336—21no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—21no—
136#336—22no—
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2312yes—
143C22×Q8—22no—
144C22×Q8—21no—
145C22×Q8C22⋊C411yescharacteristic
146C22×Q8C22×C411yescharacteristic
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—22no—
153C4⋊Q8—216no—
154C4⋊Q8—28no—
155C4⋊Q8—24no—
156C4⋊Q8—24no—
157C4⋊Q8—22no—
158C4⋊Q8M1614yes—
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—81no—
170C4×Q8—82no—
171C4×Q8—81no—
172C4×Q8—82no—
173C4×Q8M1614yes—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42C22×C412yes—
205C2×C42C22×C412yes—
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2411yescharacteristic
217C4⋊C8—24no—
218C4⋊C8—28no—
219C4⋊C8—42no—
220C4⋊C8—42no—
221#97—28no—
222#97—44no—
223#97—82no—
224C4×C8—24no—
225C4×C8—42no—
226C4×C8—81no—
227#95—22no—
228#95—21no—
229#95—22no—
230#95—21no—
231#95—21no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—21no—
238#95—21no—
239#95—22no—
240#95—21no—
241#95—21no—
242#95C22⋊C412yes—
243#95C22⋊C412yes—
244#95C22⋊C412yes—
245C2×Q8—28no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—24no—
249C2×Q8—22no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—81no—
253C2×Q8—82no—
254C2×Q8—81no—
255C2×Q8—82no—
256C2×Q8#10414yes—
257C2×Q8C2×M1612yes—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C23×C411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—24no—
278C2×C8—22no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—42no—
293C4⋊C4—28no—
294C4⋊C4—28no—
295C4⋊C4—22no—
296C4⋊C4—21no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—41no—
312C4⋊C4—44no—
313C4⋊C4—42no—
314C4⋊C4—82no—
315C4⋊C4—81no—
316C4⋊C4—81no—
317C4⋊C4—82no—
318C4⋊C4C2×M1614yes—
319C4⋊C4C2×M1612yes—
320C42—28no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—21no—
326C42—22no—
327C42—21no—
328C42—22no—
329C42—24no—
330C42—22no—
331C42—24no—
332C42—42no—
333C42—41no—
334C42—41no—
335C42—42no—
336C42—41no—
337C42—44no—
338C42—42no—
339C42—82no—
340C42—82no—
341C42—82no—
342C42C2×M1612yes—
343C42C2×M1612yes—
344C42#13712yes—
345C42#13712yes—
346C23C22×C22⋊C411yescharacteristic
347Q8—28no—
348Q8—42no—
349Q8—42no—
350Q8—84no—
351Q8—88no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4#40712yes—
395C2×C4#40712yes—
396C2×C4#41912yes—
397C2×C4#41912yes—
398C2×C4C2×#10414yes—
399C2×C4C2×#10412yes—
400C2×C4C22×M1611yescharacteristic
401C2×C4C22×M1611yescharacteristic
402C2×C4C2×#13711yescharacteristic
403C2×C4C2×#13711yescharacteristic
404C2×C4#57611yescharacteristic
405C2×C4#57711yescharacteristic
406C8—42no—
407C8—44no—
408C8—42no—
409C8—42no—
410C8—81no—
411C8—82no—
412C22C2×#40711yescharacteristic
413C22#288411yescharacteristic
414C22#288911yescharacteristic
415C22C2×#41911yescharacteristic
416C22#289711yescharacteristic
417C22C22×#10411yescharacteristic
418C22#290111yescharacteristic
419C4—24no—
420C4—22no—
421C4—22no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—42no—
429C4—44no—
430C4—42no—
431C4—42no—
432C4—42no—
433C4—41no—
434C4—42no—
435C4—82no—
436C4—81no—
437C4—81no—
438C4#149114yes—
439C2#1737111yescharacteristic
440C2C2×#149111yescharacteristic
441C2#1745311yescharacteristic
442C2#1755011yescharacteristic
443C2#1763211yescharacteristic
444C2#1765411yescharacteristic
445C2#1776111yescharacteristic
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
#343352
#228322
#205702
#76382
#76302
#76252
#76112
#75912
#75772
#75702
#75622