Finite Group Databasebeta

← #34058459

Subgroups — #34058459

Order 19072.

1492 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011959C211yesmaximal, characteristic, complemented
2C2×#18011958C211yesmaximal, characteristic, complemented
3#23750294C211yesmaximal, characteristic, complemented
4#23750293C213yesmaximal, complemented
5Q8×C22⋊D149C211yesmaximal, characteristic, complemented
6#23750288C213yesmaximal, complemented
7#23750287C211yesmaximal, characteristic, complemented
8#23750284C213yesmaximal, complemented
9#23750283C213yesmaximal, complemented
10#23750205C213yesmaximal, complemented
11#23750204C213yesmaximal, complemented
12#23750174C213yesmaximal, complemented
13#23750172C213yesmaximal, complemented
14C4×Q8×D149C211yesmaximal, characteristic, complemented
15#548×C149C211yesmaximal, characteristic, complemented
16#18011959—28no—
17#18011958—28no—
18C2×#14558737C2213yes—
19C2×#14558737C2211yescharacteristic
20C2×#14558736C2213yes—
21C2×#14558736C2213yes—
22C2×#14558736C2213yes—
23C2×#14558735C2213yes—
24C2×#14558735C2213yes—
25C22×Dic298C2213yes—
26C2×Q8×D149C2211yescharacteristic
27C2×Q8×D149C2213yes—
28C2×Q8×D149C2211yescharacteristic
29#18011910C2213yes—
30#18011909C2213yes—
31#18011907C2213yes—
32#18011907C2213yes—
33#18011899C2213yes—
34#18011897C2213yes—
35#18011897C2213yes—
36#18011891C2213yes—
37#18011890C2213yes—
38#18011889C2216yes—
39#18011888C2216yes—
40#18011887C2213yes—
41#18011886C2213yes—
42#18011886C2213yes—
43#18011885C2213yes—
44#18011884C2211yescharacteristic
45#18011884C2213yes—
46#18011884C2216yes—
47#18011884C2216yes—
48#18011883C2216yes—
49#18011882C2213yes—
50#18011882C2213yes—
51Q8×Dic149C2211yescharacteristic
52C4×Dic298C2213yes—
53#18011876C2213yes—
54C4×D596C2213yes—
55C4×C22⋊D149C2213yes—
56C4×C22⋊D149C2213yes—
57#18011871C2216yes—
58#18011871C2216yes—
59C4⋊C4×D149C2213yes—
60C42×D149C2213yes—
61C2×#41×C149C2211yescharacteristic
62C22×Q8×C149C2211yescharacteristic
63#124×C149C2213yes—
64C22⋊Q8×C149C2213yes—
65#121×C149C2213yes—
66C4×Q8×C149C2211yescharacteristic
67C4×D4×C149C2213yes—
68#14558737—212no—
69#14558737—24no—
70#14558736—212no—
71#14558736—212no—
72#14558736—212no—
73#14558735—212no—
74#14558735—212no—
75C2×Dic298—212no—
76C2×Dic298C2313yes—
77C2×Dic298C2313yes—
78C2×Dic298C2313yes—
79C2×Dic298C2313yes—
80Q8×D149—212no—
81Q8×D149—24no—
82Q8×D149D418yescomplemented
83C2×D596C2313yes—
84C2×D596C2311yescharacteristic
85C2×D596C2313yes—
86C2×D596C2313yes—
87C2×C22⋊D149C2313yes—
88C2×C22⋊D149C2311yescharacteristic
89C2×C22⋊D149C2313yes—
90C2×C22⋊D149C2313yes—
91C2×C22⋊D149C2313yes—
92C2×C22⋊D149C2311yescharacteristic
93C2×C22⋊D149C2316yes—
94C22×Dic149C2316yes—
95C2×C4×D149C2313yes—
96C2×C4×D149C2316yes—
97C2×C4×D149C2313yes—
98C2×C4×D149C2311yescharacteristic
99C2×C4×D149C2313yes—
100C2×C4×D149C2313yes—
101C4⋊Dic149C2313yes—
102C4⋊Dic149C2313yes—
103#14558715C2313yes—
104#14558715C2316yes—
105#14558715C2311yescharacteristic
106#14558715C2316yes—
107C22⋊Dic149C2313yes—
108C22⋊Dic149C2313yes—
109#14558713C2313yes—
110#14558713C2316yes—
111#14558713C2313yes—
112#14558713C2313yes—
113#14558713C2311yescharacteristic
114C4×Dic149C2313yes—
115C4×Dic149C2313yes—
116C4×Dic149C2313yes—
117#41×C149—24no—
118C2×Q8×C149—24no—
119C2×Q8×C149C2313yes—
120C2×Q8×C149C2311yescharacteristic
121C2×D4×C149C2313yes—
122C2×D4×C149C2313yes—
123C22×C4×C149C2313yes—
124C22×C4×C149C2313yes—
125C4⋊C4×C149C2313yes—
126C4⋊C4×C149C2313yes—
127C22⋊C4×C149C2311yescharacteristic
128C22⋊C4×C149C2313yes—
129C22⋊C4×C149C2316yes—
130C42×C149C2313yes—
131C22×D149C2411yescharacteristic
132C22×D149C2413yes—
133C22×D149C2411yescharacteristic
134Dic298—26no—
135Dic298—26no—
136Dic298—26no—
137Dic298C2×D4112yes—
138D596—22no—
139D596—26no—
140D596—26no—
141D596—43no—
142C22⋊D149—26no—
143C22⋊D149—22no—
144C22⋊D149—26no—
145C22⋊D149—26no—
146C22⋊D149—26no—
147C22⋊D149—22no—
148C22⋊D149—46no—
149C2×Dic149—212no—
150C2×Dic149C2411yescharacteristic
151C2×Dic149C2413yes—
152C2×Dic149C2413yes—
153C2×Dic149C2413yes—
154C2×Dic149C2411yescharacteristic
155C4×D149—26no—
156C4×D149—26no—
157C4×D149—212no—
158C4×D149—22no—
159C4×D149—43no—
160C4×D149C2×D4112yes—
161C23×C149C2413yes—
162C23×C149C2411yescharacteristic
163Q8×C149—26no—
164Q8×C149C2×D414yes—
165D4×C149—26no—
166D4×C149—43no—
167C2×C4×C149—26no—
168C2×C4×C149—26no—
169C2×C4×C149C2413yes—
170C2×C4×C149C2413yes—
171C2×C4×C149C2411yescharacteristic
172C2×C4×C149C2411yescharacteristic
173C2×C4×C149C2413yes—
174C22×C149—21no—
175C22×C149—23no—
176C22×C149—23no—
177C22×C149—21no—
178C22×C149—21no—
179C22×C149—23no—
180C22×C149C2511yescharacteristic
181C2×D149—21no—
182C2×D149—21no—
183C2×D149—21no—
184C2×D149—23no—
185C2×D149—21no—
186C2×D149—23no—
187C2×D149—23no—
188C2×D149—21no—
189C2×D149C22×D412yes—
190C4×C149—21no—
191C4×C149—23no—
192C4×C149—21no—
193C4×C149—23no—
194C4×C149C22×D416yes—
195Dic149—21no—
196Dic149—23no—
197Dic149—23no—
198Dic149C22×D416yes—
199Dic1492−1+412yescomplemented
200C2×C149—41no—
201C2×C149—43no—
202C2×C149C23×D411yescharacteristic
203C2×C149C2×2−1+411yescharacteristic
204C2×C149#58511yescharacteristic
205D149—22no—
206D149—41no—
207D149—43no—
208C149#349711yescharacteristic, complemented, Sylow
209#3497—1491nomaximal, Sylow
210C2×2−1+4—1491no—
211C2×2−1+4—1491no—
212#550—1491no—
213#550—1493no—
214D4×Q8—1491no—
215D4×Q8—1493no—
216#548—1493no—
217#548—1491no—
218#548—1493no—
219#548D14911yescharacteristic, complemented
220#537—1493no—
221#537—1493no—
222#532—1493no—
223#532—1493no—
224C2×C4×Q8—1491no—
2252−1+4—2988no—
2262−1+4—2988no—
227C2×#41—1493no—
228C2×#41—1491no—
229C2×#41—1493no—
230C2×#41—1493no—
231C2×#41—1493no—
232C2×#41—1493no—
233C2×#41—1493no—
234C2×#41C2×D14911yescharacteristic
235C22×Q8—1493no—
236C22×Q8—1491no—
237C22×Q8—1491no—
238C22×Q8—1493no—
239C22×Q8C2×D14911yescharacteristic
240C4⋊Q8—1493no—
241C4⋊Q8—1493no—
242#127—1493no—
243#127—1493no—
244#124—1493no—
245#124—1493no—
246#124—1493no—
247#124C2×D14913yes—
248C22⋊Q8—1493no—
249C22⋊Q8—1493no—
250C22⋊Q8—1496no—
251C22⋊Q8—1493no—
252C22⋊Q8—1496no—
253C22⋊Q8C2×D14913yes—
254#121—1493no—
255#121—1493no—
256#121—1496no—
257#121—1493no—
258#121—1493no—
259#121—1493no—
260#121—1493no—
261#121—1491no—
262#121—1496no—
263#121—1496no—
264#121C2×D14913yes—
265C4×Q8—1491no—
266C4×Q8—1493no—
267C4×Q8—1493no—
268C4×Q8C2×D14911yescharacteristic
269C4×D4—1493no—
270C4×D4—1493no—
271C4×D4—1496no—
272C4×D4—1496no—
273C4×D4—1493no—
274C4×D4C2×D14913yes—
275C2×C4⋊C4—1493no—
276C2×C42—1493no—
277#41—24no—
278#41—29812no—
279#41—29812no—
280#41—29812no—
281#41—29812no—
282#41—2984no—
283#41—29812no—
284#41—29812no—
285C2×Q8—24no—
286C2×Q8—29812no—
287C2×Q8—29812no—
288C2×Q8—2984no—
289C2×Q8—1498no—
290C2×Q8—1493no—
291C2×Q8—1493no—
292C2×Q8—1493no—
293C2×Q8—1493no—
294C2×Q8C22×D14913yes—
295C2×Q8C22×D14911yescharacteristic
296C2×D4—1491no—
297C2×D4—1493no—
298C2×D4—1491no—
299C2×D4—1493no—
300C2×D4—1493no—
301C2×D4—1493no—
302C2×D4—1496no—
303C2×D4—1491no—
304C2×D4—1493no—
305C2×D4—1493no—
306C2×D4—1493no—
307C2×D4C22×D14913yes—
308C2×D4C22×D14913yes—
309C22×C4—1496no—
310C22×C4—1491no—
311C22×C4—1496no—
312C22×C4—1493no—
313C22×C4—1493no—
314C22×C4—1493no—
315C22×C4—1493no—
316C22×C4C22×D14913yes—
317C22×C4C22×D14913yes—
318C4⋊C4—1496no—
319C4⋊C4—1493no—
320C4⋊C4—1491no—
321C4⋊C4—1493no—
322C4⋊C4—1496no—
323C4⋊C4—1493no—
324C4⋊C4C22×D14913yes—
325C4⋊C4C22×D14913yes—
326C22⋊C4—1493no—
327C22⋊C4—1493no—
328C22⋊C4—1493no—
329C22⋊C4—1491no—
330C22⋊C4—1493no—
331C22⋊C4—1493no—
332C22⋊C4—1496no—
333C22⋊C4C22×D14913yes—
334C22⋊C4C22×D14916yes—
335C22⋊C4C22×D14911yescharacteristic
336C42—1493no—
337C42—1493no—
338C42—1493no—
339C42C22×D14913yes—
340C23—1491no—
341C23—1493no—
342C23—1491no—
343C23C23×D14911yescharacteristic
344C23C23×D14913yes—
345Q8—26no—
346Q8—2986no—
347Q8—2986no—
348Q8—2986no—
349Q8—14912no—
350Q8D4×D14914yes—
351D4—26no—
352D4—43no—
353D4—2982no—
354D4—2986no—
355D4—2982no—
356D4—2986no—
357D4—2982no—
358D4—2986no—
359D4—2986no—
360D4—2986no—
361D4—2986no—
362D4—5963no—
363D4—5966no—
364C2×C4—26no—
365C2×C4—26no—
366C2×C4—2982no—
367C2×C4—29812no—
368C2×C4—29812no—
369C2×C4—2986no—
370C2×C4—2986no—
371C2×C4—5963no—
372C2×C4—1493no—
373C2×C4—1491no—
374C2×C4—1493no—
375C2×C4—1491no—
376C2×C4—1493no—
377C2×C4—14912no—
378C2×C4C23×D14911yescharacteristic
379C2×C4C23×D14913yes—
380C2×C4C23×D14911yescharacteristic
381C2×C4C23×D14913yes—
382C2×C4C23×D14913yes—
383C22—21no—
384C22—23no—
385C22—21no—
386C22—23no—
387C22—21no—
388C22—23no—
389C22—2981no—
390C22—2981no—
391C22—2983no—
392C22—2981no—
393C22—2981no—
394C22—2983no—
395C22—2981no—
396C22—2983no—
397C22—1492no—
398C22C24×D14911yescharacteristic
399C4—21no—
400C4—23no—
401C4—21no—
402C4—23no—
403C4—2983no—
404C4—2981no—
405C4—2983no—
406C4—1496no—
407C4—1492no—
408C4C2×D4×D14916yes—
409C2—41no—
410C2—43no—
411C2—2982no—
412C2—5963no—
413C2—5961no—
414C2C22×D4×D14911yescharacteristic
415C22−1+4×D14911yescharacteristic
416C2#2375046111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D149
commutator, socle, cosocleC22×C149C25
Fitting#548×C149C2
hypercenter#548D149
hypocenterC149#3497
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#18011959C2semidirect
2. C2×#18011958C2semidirect
3. #23750294C2semidirect
4. #23750293C2semidirect
5. Q8×C22⋊D149C2semidirect
6. #23750288C2semidirect
7. #23750287C2semidirect
8. #23750284C2semidirect
9. #23750283C2semidirect
10. #23750205C2semidirect
11. #23750204C2semidirect
12. #23750174C2semidirect
13. #23750172C2semidirect
14. C4×Q8×D149C2semidirect
15. #548×C149C2semidirect
82. Q8×D149D4semidirect
199. Dic1492−1+4semidirect
208. C149#3497semidirect
219. #548D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119592
C2×#180119582
#237502942
#237502932
Q8×C22⋊D1492
#237502882
#237502872
#237502842
#237502832
#237502052
#237502042
#237501742
#237501722
C4×Q8×D1492
#548×C1492
#3497149