Finite Group Databasebeta

← #160640

Subgroups — #160640

Order 512.

824 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34335C211yesmaximal, characteristic
2#22828C211yesmaximal, characteristic
3#22018C211yesmaximal, characteristic
4#7604C212yesmaximal
5#7596C212yesmaximal
6#7591C211yesmaximal, characteristic
7#7588C212yesmaximal
8#7581C212yesmaximal
9#7577C211yesmaximal, characteristic
10#7570C211yesmaximal, characteristic
11#7562C211yesmaximal, characteristic
12C2×Q82C411yescharacteristic
13#2369C411yescharacteristic
14#2369C412yes—
15#2365—22no—
16#2365—21no—
17#2355C411yescharacteristic
18#2355C412yes—
19#2351—21no—
20Q8×C4⋊C4C411yescharacteristic
21#2331—22no—
22#2331—21no—
23C4×C4⋊Q8C2211yescharacteristic
24C4×C4⋊Q8C2212yes—
25#2295C2211yescharacteristic
26#2295C2212yes—
27#2276—21no—
28C42×Q8C2211yescharacteristic
29#1984—21no—
30#1984—21no—
31#1939—21no—
32#1916—22no—
33#1916—22no—
34#1839C2212yes—
35#1839C2212yes—
36#1831C2212yes—
37#1831C2211yescharacteristic
38C8×C4⋊C4—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1744C2211yescharacteristic
42#1744C2212yes—
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2212yes—
50#1271C2212yes—
51#1271C2212yes—
52#1271C2212yes—
53Q82—24no—
54Q82C818yescomplemented
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C412yes—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—22no—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8C2×C412yes—
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—22no—
72C2×C4×C8—22no—
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76#391—22no—
77#391—21no—
78#391—22no—
79#391—22no—
80#391—22no—
81#391C2×C411yescharacteristic
82#391C2×C412yes—
83#391C2311yescharacteristic
84#389—22no—
85#389—21no—
86#389—21no—
87#389—22no—
88#389—21no—
89#389—22no—
90#389C2×C411yescharacteristic
91#389C2×C412yes—
92#389C2312yes—
93#389C2311yescharacteristic
94#387—22no—
95#387—21no—
96#387—21no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#384C2×C411yescharacteristic
102#384C2×C412yes—
103#382—21no—
104#382—22no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4D412yes—
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D412yes—
123C4×C4⋊C4D411yescharacteristic
124#376—22no—
125#376—21no—
126C43C2312yes—
127C43C2311yescharacteristic
128#336—21no—
129#336—22no—
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×Q8—22no—
144C22×Q8C22⋊C411yescharacteristic
145C22×Q8C22×C411yescharacteristic
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C4⋊Q8—216no—
153C4⋊Q8—28no—
154C4⋊Q8—24no—
155C4⋊Q8—22no—
156C4⋊Q8C2×C818yes—
157C4⋊Q8C2×C814yes—
158C4×Q8—28no—
159C4×Q8—24no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—42no—
164C4×Q8—44no—
165C4×Q8—81no—
166C4×Q8—82no—
167C4×Q8—81no—
168C4×Q8—82no—
169C4×Q8C2×C818yes—
170C4×Q8C2×C814yes—
171C4×Q8C2×C814yes—
172C4×Q8M1614yes—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C42—22no—
197C2×C42—21no—
198C2×C42—22no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42C22×C412yes—
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C412yes—
206C2×C42C22×C411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2411yescharacteristic
216C4⋊C8—42no—
217C4⋊C8—42no—
218C4⋊C8—42no—
219C4⋊C8—82no—
220C4×C8—42no—
221C4×C8—44no—
222C4×C8—42no—
223C4×C8—81no—
224#95—22no—
225#95—21no—
226#95—22no—
227#95—21no—
228#95—21no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—21no—
234#95—22no—
235#95—21no—
236#95—22no—
237#95—21no—
238#95—21no—
239#95C22⋊C412yes—
240#95C22⋊C412yes—
241#95C22⋊C412yes—
242C2×Q8—21no—
243C2×Q8—24no—
244C2×Q8—28no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—81no—
250C2×Q8—82no—
251C2×Q8—82no—
252C2×Q8—81no—
253C2×Q8C22⋊C814yes—
254C2×Q8C22×C812yes—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C23×C411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C2×C8—22no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—41no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—41no—
289C2×C8—41no—
290C4⋊C4—22no—
291C4⋊C4—21no—
292C4⋊C4—28no—
293C4⋊C4—24no—
294C4⋊C4—28no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—44no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—82no—
310C4⋊C4—81no—
311C4⋊C4—81no—
312C4⋊C4—82no—
313C4⋊C4C22⋊C818yes—
314C4⋊C4C22⋊C814yes—
315C4⋊C4C22×C814yes—
316C4⋊C4C22×C812yes—
317C42—22no—
318C42—22no—
319C42—21no—
320C42—22no—
321C42—21no—
322C42—22no—
323C42—28no—
324C42—24no—
325C42—22no—
326C42—42no—
327C42—41no—
328C42—41no—
329C42—42no—
330C42—42no—
331C42—41no—
332C42—44no—
333C42—42no—
334C42—82no—
335C42—82no—
336C42—82no—
337C42C22×C814yes—
338C42C22×C812yes—
339C42C2×M1614yes—
340C42C2×M1612yes—
341C42#13712yes—
342C42#13712yes—
343C23C22×C22⋊C411yescharacteristic
344Q8—28no—
345Q8—42no—
346Q8—42no—
347Q8—84no—
348Q8—88no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—21no—
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—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4C2×C22⋊C814yes—
391C2×C4C2×C22⋊C814yes—
392C2×C4C2×C22⋊C812yes—
393C2×C4C2×C22⋊C812yes—
394C2×C4#41912yes—
395C2×C4#41912yes—
396C2×C4#42112yes—
397C2×C4C23×C811yescharacteristic
398C2×C4C22×M1611yescharacteristic
399C2×C4C2×#13711yescharacteristic
400C2×C4C2×#13711yescharacteristic
401C2×C4#57611yescharacteristic
402C2×C4#57711yescharacteristic
403C8—42no—
404C8—81no—
405C8—82no—
406C8—81no—
407C8—81no—
408C8—82no—
409C22C22×C22⋊C811yescharacteristic
410C22#288411yescharacteristic
411C22#288911yescharacteristic
412C22C2×#41911yescharacteristic
413C22#289711yescharacteristic
414C22C2×#42111yescharacteristic
415C22#290111yescharacteristic
416C4—24no—
417C4—22no—
418C4—24no—
419C4—22no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—41no—
427C4—42no—
428C4—44no—
429C4—42no—
430C4—42no—
431C4—42no—
432C4—81no—
433C4—82no—
434C4—81no—
435C4#147914yes—
436C2#1735211yescharacteristic
437C2C2×#147911yescharacteristic
438C2#1743811yescharacteristic
439C2#1760211yescharacteristic
440C2#1763211yescharacteristic
441C2#1768311yescharacteristic
442C2#1776111yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×C22⋊C4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
54. Q82C8semidirect

Maximal subgroups

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

subgroupindex
#343352
#228282
#220182
#76042
#75962
#75912
#75882
#75812
#75772
#75702
#75622