Finite Group Databasebeta

← #160641

Subgroups — #160641

Order 512.

820 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34337C211yesmaximal, characteristic
2#22828C211yesmaximal, characteristic
3#22018C211yesmaximal, characteristic
4#7605C212yesmaximal
5#7596C212yesmaximal
6#7590C211yesmaximal, characteristic
7#7590C212yesmaximal
8#7581C212yesmaximal
9#7577C211yesmaximal, characteristic
10#7571C211yesmaximal, characteristic
11#7562C211yesmaximal, characteristic
12C2×#557C411yescharacteristic
13#2370—22no—
14#2370—21no—
15#2365C412yes—
16#2365C411yescharacteristic
17Q8×C4⋊C4—21no—
18Q8×C4⋊C4C411yescharacteristic
19Q8×C4⋊C4C411yescharacteristic
20Q8×C4⋊C4C412yes—
21#2330—22no—
22#2330—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#1272C2212yes—
47#1272C2212yes—
48#1272C2212yes—
49#1272C2212yes—
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53#557—24no—
54#557C818yescomplemented
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C4⋊Q8C2×C412yes—
57C2×#125—22no—
58C2×#125—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2×C412yes—
63C2×C4×Q8C2×C412yes—
64C2×C4×Q8C2×C412yes—
65C2×C4×Q8C2×C412yes—
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—22no—
73C2×C4×C8—22no—
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77#391C2×C411yescharacteristic
78#391C2×C412yes—
79#391C2312yes—
80#391C2311yescharacteristic
81#389—21no—
82#389—22no—
83#389—22no—
84#389—21no—
85#389—22no—
86#389—21no—
87#389—21no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389C2×C411yescharacteristic
93#389C2×C412yes—
94#389C2311yescharacteristic
95#387—22no—
96#387—21no—
97#387—21no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#384—21no—
103#384—22no—
104#384C2×C411yescharacteristic
105#384C2×C412yes—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—22no—
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×Q8C22⋊C411yescharacteristic
143C22×Q8C22×C411yescharacteristic
144C22×Q8C22×C412yes—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C4⋊Q8—24no—
152C4⋊Q8—22no—
153C4⋊Q8C2×C818yes—
154C4⋊Q8C2×C814yes—
155#125—216no—
156#125—28no—
157C4×Q8—28no—
158C4×Q8—24no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—42no—
163C4×Q8—44no—
164C4×Q8—82no—
165C4×Q8—82no—
166C4×Q8—82no—
167C4×Q8C2×C818yes—
168C4×Q8C2×C814yes—
169C4×Q8C2×C814yes—
170C4×Q8M1614yes—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
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—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22⋊C411yescharacteristic
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×C411yescharacteristic
206C2×C42C22×C412yes—
207C2×C42C22×C411yescharacteristic
208C2×C42C2×D412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2411yescharacteristic
217C4⋊C8—42no—
218C4⋊C8—42no—
219C4⋊C8—42no—
220C4⋊C8—82no—
221C4×C8—42no—
222C4×C8—44no—
223C4×C8—42no—
224C4×C8—81no—
225#95—22no—
226#95—21no—
227#95—22no—
228#95—21no—
229#95—21no—
230#95—22no—
231#95—21no—
232#95—22no—
233#95—21no—
234#95—22no—
235#95—21no—
236#95—22no—
237#95—21no—
238#95C22⋊C412yes—
239#95C22⋊C412yes—
240#95C22⋊C412yes—
241C2×Q8—22no—
242C2×Q8—21no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—42no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—42no—
249C2×Q8C22⋊C814yes—
250C2×Q8C22×C814yes—
251C2×Q8C22×C812yes—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C23×C411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C2×C8—22no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—42no—
280C2×C8—41no—
281C2×C8—42no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—41no—
286C2×C8—41no—
287C4⋊C4—21no—
288C4⋊C4—28no—
289C4⋊C4—24no—
290C4⋊C4—28no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—28no—
294C4⋊C4—24no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—44no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—82no—
306C4⋊C4—81no—
307C4⋊C4—82no—
308C4⋊C4—81no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4C22⋊C818yes—
313C4⋊C4C22⋊C814yes—
314C4⋊C4C22×C812yes—
315C42—22no—
316C42—22no—
317C42—21no—
318C42—22no—
319C42—21no—
320C42—22no—
321C42—28no—
322C42—24no—
323C42—22no—
324C42—42no—
325C42—41no—
326C42—41no—
327C42—42no—
328C42—42no—
329C42—41no—
330C42—44no—
331C42—42no—
332C42—82no—
333C42—82no—
334C42—82no—
335C42C22×C814yes—
336C42C22×C812yes—
337C42C2×M1614yes—
338C42C2×M1612yes—
339C42#13712yes—
340C42#13712yes—
341C23C22×C22⋊C411yescharacteristic
342Q8—28no—
343Q8—44no—
344Q8—42no—
345Q8—44no—
346Q8—42no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
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—24no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4C2×C22⋊C814yes—
389C2×C4C2×C22⋊C814yes—
390C2×C4C2×C22⋊C812yes—
391C2×C4C2×C22⋊C812yes—
392C2×C4#41912yes—
393C2×C4#41912yes—
394C2×C4#42112yes—
395C2×C4C23×C811yescharacteristic
396C2×C4C22×M1611yescharacteristic
397C2×C4C2×#13711yescharacteristic
398C2×C4C2×#13711yescharacteristic
399C2×C4#57711yescharacteristic
400C2×C4#57811yescharacteristic
401C8—42no—
402C8—81no—
403C8—82no—
404C8—81no—
405C8—81no—
406C8—82no—
407C22C22×C22⋊C811yescharacteristic
408C22#288411yescharacteristic
409C22#288811yescharacteristic
410C22C2×#41911yescharacteristic
411C22#289811yescharacteristic
412C22C2×#42111yescharacteristic
413C22#290111yescharacteristic
414C4—24no—
415C4—22no—
416C4—24no—
417C4—22no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—41no—
425C4—42no—
426C4—42no—
427C4—42no—
428C4—44no—
429C4—42no—
430C4—82no—
431C4—81no—
432C4—81no—
433C4#147914yes—
434C2#1735311yescharacteristic
435C2C2×#147911yescharacteristic
436C2#1743911yescharacteristic
437C2#1760111yescharacteristic
438C2#1763211yescharacteristic
439C2#1768411yescharacteristic
440C2#1776011yescharacteristic
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. #557C8semidirect

Maximal subgroups

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

subgroupindex
#343372
#228282
#220182
#76052
#75962
#75902
#75812
#75772
#75712
#75622