Finite Group Databasebeta

← #34058570

Subgroups — #34058570

Order 19072.

992 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750352C212yesmaximal, complemented
2#23750351C212yesmaximal, complemented
3#23750346C212yesmaximal, complemented
4#23750343C212yesmaximal, complemented
5#23750316C212yesmaximal, complemented
6#23750314C212yesmaximal, complemented
7#23750313C212yesmaximal, complemented
8#23750309C212yesmaximal, complemented
9#23750235C212yesmaximal, complemented
10#23750233C212yesmaximal, complemented
11#23750198C212yesmaximal
12#23750182C211yesmaximal, characteristic, complemented
13#23750180C211yesmaximal, characteristic, complemented
14#23750180C211yesmaximal, characteristic, complemented
15#23750180C211yesmaximal, characteristic, complemented
16C4⋊Q8×D149C211yesmaximal, characteristic, complemented
17#23750101C212yesmaximal, complemented
18C4×#14558736C211yesmaximal, characteristic
19#533×C149C211yesmaximal, characteristic, complemented
20C2×#14558736C2211yescharacteristic
21C2×Q8×D149C2212yes—
22#18011911C2211yescharacteristic
23#18011911C2211yescharacteristic
24#18011910C2212yes—
25#18011910C2212yes—
26#18011910C2214yes—
27#18011909C2212yes—
28#18011906C2214yes—
29#18011905C2214yes—
30#18011904C2214yes—
31#18011904C2214yes—
32#18011903C2212yes—
33#18011903C2214yes—
34#18011902C2212yes—
35#18011902C2212yes—
36#18011902C2212yes—
37#18011902C2214yes—
38#18011901C2214yes—
39#18011901C2214yes—
40#18011897C2214yes—
41#18011896C2214yes—
42#18011893C2214yes—
43#18011891C2212yes—
44#18011891C2212yes—
45#18011891C2212yes—
46#18011890C2212yes—
47#18011890C2212yes—
48#18011887C2212yes—
49#18011887C2212yes—
50#18011887C2212yes—
51Q8×Dic149C2212yes—
52Q8×Dic149C2212yes—
53Q8×Dic149C2212yes—
54C4×Dic298C2212yes—
55C4×Dic298C2212yes—
56C4×Dic298C2214yes—
57C4×Dic298C2211yescharacteristic
58C4×Dic298C2211yescharacteristic
59#18011876C2212yes—
60#18011876C2212yes—
61#18011876C2212yes—
62#18011876C2214yes—
63C4×D596C2211yescharacteristic
64C4×D596C2211yescharacteristic
65C4×C22⋊D149C2211yescharacteristic
66C4×C22⋊D149C2211yescharacteristic
67C4×C22⋊D149C2211yescharacteristic
68C4×C22⋊D149C2211yescharacteristic
69#18011870C2212yes—
70#18011870C2211yescharacteristic
71#18011869C2212yes—
72#18011869C2212yes—
73#18011869C2212yes—
74#18011869C2214yes—
75#18011868C2211yescharacteristic
76#18011868C2214yes—
77#18011868C2211yescharacteristic
78C4⋊C4×D149C2212yes—
79C4⋊C4×D149C2212yes—
80C42×D149C2211yescharacteristic
81C42×D149C2211yescharacteristic
82C4⋊Q8×C149C2211yescharacteristic
83C4⋊Q8×C149C2211yescharacteristic
84#125×C149C2212yes—
85C22⋊Q8×C149C2212yes—
86C22⋊Q8×C149C2212yes—
87C4×Q8×C149C2212yes—
88C4×Q8×C149C2212yes—
89#117×C149C2212yes—
90C2×C42×C149C2211yescharacteristic
91#14558736—24no—
92C2×Dic298C2312yes—
93C2×Dic298C2311yescharacteristic
94C2×Dic298C2314yes—
95C2×Dic298C2312yes—
96Q8×D149—28no—
97C2×D596C2311yescharacteristic
98C2×C22⋊D149C2311yescharacteristic
99C2×C22⋊D149C2311yescharacteristic
100C2×C4×D149C2314yes—
101C2×C4×D149C2311yescharacteristic
102C2×C4×D149C2311yescharacteristic
103C2×C4×D149C2311yescharacteristic
104C2×C4×D149C2311yescharacteristic
105C2×C4×D149C2311yescharacteristic
106C2×C4×D149C2311yescharacteristic
107C4⋊Dic149C2314yes—
108C4⋊Dic149C2314yes—
109C4⋊Dic149C2312yes—
110C4⋊Dic149C2312yes—
111C4⋊Dic149C2314yes—
112C4⋊Dic149C2312yes—
113C4⋊Dic149C2312yes—
114C4⋊Dic149C2311yescharacteristic
115C4⋊Dic149C2311yescharacteristic
116#14558715C2314yes—
117#14558715C2314yes—
118#14558715C2314yes—
119#14558715C2314yes—
120#14558715C2314yes—
121#14558715C2314yes—
122#14558715C2314yes—
123#14558715C2311yescharacteristic
124#14558715C2311yescharacteristic
125#14558715C2311yescharacteristic
126#14558715C2311yescharacteristic
127C22⋊Dic149C2311yescharacteristic
128C22⋊Dic149C2311yescharacteristic
129C22⋊Dic149C2314yes—
130#14558713C2311yescharacteristic
131#14558713C2314yes—
132#14558713C2314yes—
133#14558713C2311yescharacteristic
134#14558713C2311yescharacteristic
135#14558713C2311yescharacteristic
136#14558713C2314yes—
137C4×Dic149C2314yes—
138C4×Dic149C2314yes—
139C4×Dic149C2314yes—
140C4×Dic149C2311yescharacteristic
141C4×Dic149C2311yescharacteristic
142C4×Dic149C2311yescharacteristic
143C4×Dic149C2314yes—
144C4×Dic149C2311yescharacteristic
145C4×Dic149C2311yescharacteristic
146C4×Dic149C2311yescharacteristic
147C2×Q8×C149C2312yes—
148C2×Q8×C149C2312yes—
149C22×C4×C149C2311yescharacteristic
150C22×C4×C149C2311yescharacteristic
151C22×C4×C149C2311yescharacteristic
152C4⋊C4×C149C2314yes—
153C4⋊C4×C149C2312yes—
154C4⋊C4×C149C2314yes—
155C4⋊C4×C149C2312yes—
156C4⋊C4×C149C2312yes—
157C4⋊C4×C149C2312yes—
158C22⋊C4×C149C2314yes—
159C42×C149C2314yes—
160C42×C149C2311yescharacteristic
161C42×C149C2311yescharacteristic
162C42×C149C2311yescharacteristic
163C42×C149C2311yescharacteristic
164C22×D149C2411yescharacteristic
165C22×D149C2411yescharacteristic
166Dic298—24no—
167Dic298—24no—
168Dic298—28no—
169Dic298—22no—
170D596—22no—
171C22⋊D149—22no—
172C22⋊D149—22no—
173C2×Dic149C2411yescharacteristic
174C2×Dic149C2414yes—
175C2×Dic149C2411yescharacteristic
176C2×Dic149C2411yescharacteristic
177C2×Dic149C2411yescharacteristic
178C2×Dic149C2411yescharacteristic
179C2×Dic149C2414yes—
180C2×Dic149C2411yescharacteristic
181C4×D149—28no—
182C4×D149—22no—
183C4×D149—22no—
184C4×D149—22no—
185C4×D149—22no—
186C4×D149—22no—
187C4×D149#4114yes—
188C23×C149C2411yescharacteristic
189Q8×C149—24no—
190Q8×C149—24no—
191C2×C4×C149—22no—
192C2×C4×C149—22no—
193C2×C4×C149—22no—
194C2×C4×C149C2414yes—
195C2×C4×C149C2411yescharacteristic
196C2×C4×C149C2411yescharacteristic
197C2×C4×C149C2411yescharacteristic
198C2×C4×C149C2414yes—
199C2×C4×C149C2411yescharacteristic
200C2×C4×C149C2411yescharacteristic
201C2×C4×C149C2411yescharacteristic
202C22×C149—21no—
203C22×C149—21no—
204C22×C149—21no—
205C22×C149C2511yescharacteristic
206C2×D149—21no—
207C2×D149—21no—
208C2×D149—21no—
209C2×D149—21no—
210C2×D149—21no—
211C2×D149C2×#4112yes—
212C4×C149—21no—
213C4×C149—24no—
214C4×C149—21no—
215C4×C149—21no—
216C4×C149—24no—
217C4×C149C2×#4112yes—
218C4×C1492−1+412yes—
219C4×C1492−1+412yes—
220Dic149—21no—
221Dic149—24no—
222Dic149—21no—
223Dic149—21no—
224Dic149—21no—
225Dic149—21no—
226Dic149—24no—
227Dic149C2×#4112yes—
228C2×C149—41no—
229C2×C149C22×#4111yescharacteristic
230C2×C149C2×2−1+411yescharacteristic
231C2×C149C2×2−1+411yescharacteristic
232D149—22no—
233D149—41no—
234C149#350911yescharacteristic, complemented, Sylow
235#3509—1491nomaximal, Sylow
236#557—1492no—
237#557—1492no—
238#556—1492no—
239#556—1492no—
240#552—1492no—
241#552—1492no—
242#552—1492no—
243#552—1492no—
244#541—1492no—
245#541—1492no—
246#536—1492no—
247#533—1491no—
248#533—1491no—
249#533—1491no—
250#533—1491no—
251#533D14911yescharacteristic, complemented
252C2×C4⋊Q8—1491no—
253#519—1492no—
254C4×#41—1491no—
255C2×#41—1491no—
256C22×Q8—1492no—
257C4⋊Q8—1492no—
258C4⋊Q8—1492no—
259C4⋊Q8—1492no—
260C4⋊Q8—1494no—
261C4⋊Q8—1491no—
262C4⋊Q8—1491no—
263C4⋊Q8C2×D14911yescharacteristic
264C4⋊Q8C2×D14911yescharacteristic
265#126—1494no—
266#126—1494no—
267#126—1494no—
268#126—1494no—
269#125—1492no—
270#125—1492no—
271#125—1492no—
272#125—1494no—
273#125—1494no—
274#125—1494no—
275#125—1492no—
276#125—1494no—
277#125C2×D14912yes—
278#124—1494no—
279#123—1494no—
280#123—1494no—
281C22⋊Q8—1492no—
282C22⋊Q8—1492no—
283C22⋊Q8—1492no—
284C22⋊Q8—1492no—
285C22⋊Q8—1492no—
286C22⋊Q8—1492no—
287C22⋊Q8—1492no—
288C22⋊Q8—1492no—
289C22⋊Q8C2×D14912yes—
290C22⋊Q8C2×D14912yes—
291C4×Q8—1492no—
292C4×Q8—1492no—
293C4×Q8—1492no—
294C4×Q8—1492no—
295C4×Q8—1494no—
296C4×Q8—1492no—
297C4×Q8—1492no—
298C4×Q8—1491no—
299C4×Q8—1491no—
300C4×Q8—1492no—
301C4×Q8—1494no—
302C4×Q8—1492no—
303C4×Q8C2×D14912yes—
304C4×Q8C2×D14912yes—
305C4×D4—1491no—
306C4×D4—1491no—
307C4×D4—1491no—
308C4×D4—1491no—
309C4×D4—1491no—
310C4×D4—1491no—
311#117—1491no—
312#117—1492no—
313#117—1492no—
314#117—1494no—
315#117—1492no—
316#117—1492no—
317#117—1491no—
318#117—1491no—
319#117—1494no—
320#117C2×D14912yes—
321C2×C4⋊C4—1492no—
322C2×C4⋊C4—1492no—
323C2×C42—1491no—
324C2×C42—1491no—
325C2×C42C2×D14911yescharacteristic
326#41—2984no—
327C2×Q8—2988no—
328C2×Q8—1492no—
329C2×Q8—1492no—
330C2×Q8—1494no—
331C2×Q8—1491no—
332C2×Q8C22×D14912yes—
333C2×Q8C22×D14912yes—
334C2×D4—1491no—
335C2×D4—1491no—
336C2×D4—1491no—
337C22×C4—1491no—
338C22×C4—1491no—
339C22×C4—1491no—
340C22×C4—1491no—
341C22×C4—1491no—
342C22×C4—1494no—
343C22×C4—1491no—
344C22×C4C22×D14911yescharacteristic
345C22×C4C22×D14911yescharacteristic
346C22×C4C22×D14911yescharacteristic
347C4⋊C4—1492no—
348C4⋊C4—1492no—
349C4⋊C4—1494no—
350C4⋊C4—1494no—
351C4⋊C4—1492no—
352C4⋊C4—1491no—
353C4⋊C4—1491no—
354C4⋊C4—1491no—
355C4⋊C4—1491no—
356C4⋊C4—1491no—
357C4⋊C4—1491no—
358C4⋊C4—1492no—
359C4⋊C4—1494no—
360C4⋊C4—1494no—
361C4⋊C4—1494no—
362C4⋊C4—1494no—
363C4⋊C4—1494no—
364C4⋊C4—1494no—
365C4⋊C4—1494no—
366C4⋊C4—1494no—
367C4⋊C4C22×D14912yes—
368C4⋊C4C22×D14912yes—
369C4⋊C4C22×D14912yes—
370C4⋊C4C22×D14912yes—
371C4⋊C4C22×D14914yes—
372C4⋊C4C22×D14914yes—
373C22⋊C4—1491no—
374C22⋊C4—1491no—
375C22⋊C4—1494no—
376C22⋊C4—1491no—
377C22⋊C4—1491no—
378C22⋊C4—1494no—
379C22⋊C4—1491no—
380C22⋊C4—1491no—
381C22⋊C4—1494no—
382C22⋊C4—1494no—
383C22⋊C4C22×D14914yes—
384C42—1491no—
385C42—1491no—
386C42—1491no—
387C42—1491no—
388C42—1491no—
389C42—1491no—
390C42—1494no—
391C42—1494no—
392C42—1494no—
393C42—1494no—
394C42C22×D14911yescharacteristic
395C42C22×D14911yescharacteristic
396C42C22×D14911yescharacteristic
397C42C22×D14911yescharacteristic
398C42C22×D14914yes—
399C23—1491no—
400C23—1491no—
401C23C23×D14911yescharacteristic
402Q8—24no—
403Q8—24no—
404Q8—2988no—
405Q8—2982no—
406Q8—2984no—
407Q8—2984no—
408D4—2982no—
409D4—2982no—
410D4—2982no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—2988no—
415C2×C4—2982no—
416C2×C4—2982no—
417C2×C4—2982no—
418C2×C4—2982no—
419C2×C4—2982no—
420C2×C4—1491no—
421C2×C4—1491no—
422C2×C4—1494no—
423C2×C4—1491no—
424C2×C4—1491no—
425C2×C4—1494no—
426C2×C4—1491no—
427C2×C4—1491no—
428C2×C4—1494no—
429C2×C4C23×D14911yescharacteristic
430C2×C4C23×D14914yes—
431C2×C4C23×D14914yes—
432C2×C4C23×D14911yescharacteristic
433C2×C4C23×D14911yescharacteristic
434C2×C4C23×D14911yescharacteristic
435C2×C4C23×D14911yescharacteristic
436C2×C4C23×D14911yescharacteristic
437C22—21no—
438C22—21no—
439C22—21no—
440C22—2981no—
441C22—2981no—
442C22—2981no—
443C22—2981no—
444C22—2981no—
445C22—1492no—
446C22C24×D14911yescharacteristic
447C4—21no—
448C4—24no—
449C4—21no—
450C4—24no—
451C4—21no—
452C4—2981no—
453C4—2984no—
454C4—2981no—
455C4—2981no—
456C4—2981no—
457C4—2981no—
458C4—2984no—
459C4—1492no—
460C4#41×D14912yes—
461C4#1801195812yes—
462C4#1801195912yes—
463C2—41no—
464C2—2982no—
465C2—5961no—
466C2C2×#41×D14911yescharacteristic
467C2C2×#1801195811yescharacteristic
468C2C2×#1801195911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D149
commutator, socle, cosocleC22×C149C25
Fitting#533×C149C2
prime radicalC2×#14558736C22
hypercenter#533D149
hypocenterC149#3509
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ #533

Complements

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

NH ≅ G/Ntype
1. #23750352C2semidirect
2. #23750351C2semidirect
3. #23750346C2semidirect
4. #23750343C2semidirect
5. #23750316C2semidirect
6. #23750314C2semidirect
7. #23750313C2semidirect
8. #23750309C2semidirect
9. #23750235C2semidirect
10. #23750233C2semidirect
12. #23750182C2semidirect
13. #23750180C2semidirect
14. #23750180C2semidirect
15. #23750180C2semidirect
16. C4⋊Q8×D149C2semidirect
17. #23750101C2semidirect
19. #533×C149C2semidirect
234. C149#3509semidirect
251. #533D149semidirect

Maximal subgroups

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

subgroupindex
#237503522
#237503512
#237503462
#237503432
#237503162
#237503142
#237503132
#237503092
#237502352
#237502332
#237501982
#237501822
#237501802
C4⋊Q8×D1492
#237501012
C4×#145587362
#533×C1492
#3509149