Finite Group Databasebeta

← #34058368

Subgroups — #34058368

Order 19072.

1522 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011959C211yesmaximal, characteristic, complemented
2C22×#14558737C211yesmaximal, characteristic, complemented
3Q8×C22⋊D149C211yesmaximal, characteristic, complemented
4#23750287C211yesmaximal, characteristic, complemented
5#23750286C211yesmaximal, characteristic, complemented
6#23750278C213yesmaximal, complemented
7#23750277C213yesmaximal, complemented
8#23750271C213yesmaximal, complemented
9#23750266C213yesmaximal, complemented
10#23750205C213yesmaximal, complemented
11#23750200C213yesmaximal, complemented
12#23750134C213yesmaximal, complemented
13C22⋊Q8×D149C213yesmaximal, complemented
14#23750107C211yesmaximal, characteristic, complemented
15D4×Q8×C149C211yesmaximal, characteristic, complemented
16#18011959—28no—
17C2×#14558737—28no—
18C2×#14558737C2213yes—
19C2×#14558737C2213yes—
20C2×#14558737C2211yescharacteristic
21C2×#14558737C2211yescharacteristic
22C2×#14558737C2213yescomplemented
23C2×#14558736C2213yescomplemented
24C2×#14558736C2213yescomplemented
25C2×Q8×D149C2213yescomplemented
26C2×Q8×D149C2211yescharacteristic, complemented
27C22×D596C2213yes—
28C22×C4×D149C2213yes—
29#18011909C2213yescomplemented
30#18011900C2213yescomplemented
31#18011899C2213yescomplemented
32#18011896C2216yescomplemented
33#18011896C2216yescomplemented
34#18011896C2213yescomplemented
35#18011893C2213yescomplemented
36#18011891C2213yescomplemented
37#18011890C2213yes—
38#18011889C2213yescomplemented
39#18011889C2216yescomplemented
40#18011888C2213yes—
41#18011887C2213yescomplemented
42#18011887C2213yescomplemented
43#18011886C2216yescomplemented
44#18011886C2213yescomplemented
45#18011885C2213yescomplemented
46#18011884C2213yescomplemented
47#18011884C2216yescomplemented
48#18011883C2211yescharacteristic
49#18011881C2213yes—
50#18011879C2213yes—
51Q8×Dic149C2211yescharacteristic, complemented
52C4×D596C2213yescomplemented
53C4×D596C2213yescomplemented
54#18011873C2213yescomplemented
55C4×C22⋊D149C2213yes—
56C4×C22⋊D149C2213yescomplemented
57#18011868C2213yescomplemented
58C4⋊C4×D149C2213yescomplemented
59C4⋊C4×D149C2216yescomplemented
60C22⋊C4×D149C2216yes—
61C22×Q8×C149C2211yescharacteristic
62C22×Q8×C149C2211yescharacteristic, complemented
63C4⋊Q8×C149C2213yescomplemented
64C22⋊Q8×C149C2213yes—
65C22⋊Q8×C149C2213yescomplemented
66C4×Q8×C149C2211yescharacteristic, complemented
67C4×D4×C149C2213yes—
68C23×D149C2313yes—
69#14558737—212no—
70#14558737—212no—
71#14558737—212no—
72#14558737—24no—
73#14558737D418yescomplemented
74#14558736—212no—
75#14558736—212no—
76C2×Dic298C2313yes—
77C2×Dic298C2313yes—
78Q8×D149—212no—
79Q8×D149—24no—
80C2×D596—212no—
81C2×D596C2316yes—
82C2×D596C2313yes—
83C2×D596C2313yes—
84C2×D596C2313yes—
85C2×D596C2313yes—
86C2×C22⋊D149C2313yes—
87C2×C22⋊D149C2311yescharacteristic
88C2×C22⋊D149C2311yescharacteristic
89C2×C22⋊D149C2313yes—
90C22×Dic149C2311yescharacteristic
91C2×C4×D149—212no—
92C2×C4×D149C2313yes—
93C2×C4×D149C2313yes—
94C2×C4×D149C2313yes—
95C2×C4×D149C2316yes—
96C2×C4×D149C2313yes—
97C2×C4×D149C2316yes—
98C2×C4×D149C2313yes—
99C4⋊Dic149C2313yes—
100C4⋊Dic149C2313yes—
101#14558715C2313yes—
102#14558715C2316yes—
103#14558715C2313yes—
104#14558715C2313yes—
105#14558715C2311yescharacteristic
106C22⋊Dic149C2313yes—
107C22⋊Dic149C2311yescharacteristic
108#14558713C2316yes—
109#14558713C2316yes—
110#14558713C2313yes—
111#14558713C2313yes—
112#14558713C2313yes—
113#14558713C2313yes—
114#14558713C2311yescharacteristic
115#14558713C2313yes—
116C4×Dic149C2313yes—
117C2×Q8×C149—24no—
118C2×Q8×C149—24no—
119C2×Q8×C149C2313yes—
120C2×Q8×C149C2313yes—
121C2×Q8×C149C2311yescharacteristic
122C2×D4×C149C2311yescharacteristic
123C22×C4×C149C2313yes—
124C22×C4×C149C2313yes—
125C4⋊C4×C149C2313yes—
126C4⋊C4×C149C2313yes—
127C4⋊C4×C149C2316yes—
128C22⋊C4×C149C2313yes—
129C22⋊C4×C149C2313yes—
130C42×C149C2313yes—
131C22×D149—26no—
132C22×D149—26no—
133C22×D149—43no—
134C22×D149C2411yescharacteristic
135C22×D149C2413yes—
136C22×D149C2413yes—
137Dic298—26no—
138Dic298—26no—
139D596—212no—
140D596—26no—
141D596—26no—
142D596—43no—
143D596C2×D4112yes—
144C22⋊D149—22no—
145C22⋊D149—26no—
146C22⋊D149—22no—
147C22⋊D149—43no—
148C2×Dic149—22no—
149C2×Dic149C2411yescharacteristic
150C2×Dic149C2413yes—
151C2×Dic149C2413yes—
152C2×Dic149C2411yescharacteristic
153C2×Dic149C2411yescharacteristic
154C4×D149—26no—
155C4×D149—212no—
156C4×D149—26no—
157C4×D149—26no—
158C4×D149—26no—
159C4×D149—46no—
160C4×D149C2×D4112yes—
161C23×C149C2411yescharacteristic
162C23×C149C2411yescharacteristic
163Q8×C149—26no—
164Q8×C149—26no—
165Q8×C149C2×D414yes—
166D4×C149—22no—
167C2×C4×C149—26no—
168C2×C4×C149—26no—
169C2×C4×C149C2413yes—
170C2×C4×C149C2413yes—
171C2×C4×C149C2413yes—
172C2×C4×C149C2411yescharacteristic
173C2×C4×C149C2413yes—
174C22×C149—21no—
175C22×C149—21no—
176C22×C149—21no—
177C22×C149—21no—
178C22×C149—21no—
179C22×C1492−1+412yescomplemented
180C22×C149C2511yescharacteristic
181C2×D149—23no—
182C2×D149—23no—
183C2×D149—21no—
184C2×D149—23no—
185C2×D149—21no—
186C2×D149—23no—
187C2×D149—23no—
188C2×D149—21no—
189C2×D149—412no—
190C2×D149C22×D416yes—
191C4×C149—23no—
192C4×C149—23no—
193C4×C149—21no—
194C4×C149—23no—
195C4×C149C22×D416yes—
196Dic149—21no—
197Dic149—23no—
198Dic149—21no—
199Dic149—23no—
200Dic149C22×D412yes—
201C2×C149—22no—
202C2×C149—41no—
203C2×C149C23×D411yescharacteristic
204C2×C149C2×2−1+411yescharacteristic
205C2×C149#58511yescharacteristic
206D149—26no—
207D149—41no—
208D149—43no—
209C149#348911yescharacteristic, complemented, Sylow
210#3489—1491nomaximal, Sylow
211C2×2−1+4—1491no—
212C22×#41—1491no—
213D4×Q8—1491no—
214D4×Q8D14911yescharacteristic, complemented
215#548—1491no—
216#548—1491no—
217#547—1493no—
218#547—1493no—
219#546—1493no—
220#546—1493no—
221#537—1493no—
222#536—1493no—
223#525—1493no—
224C2×C22⋊Q8—1493no—
225#520—1491no—
2262−1+4—2988no—
227C2×#41—2988no—
228C2×#41—1493no—
229C2×#41—1493no—
230C2×#41—1493no—
231C2×#41—1491no—
232C2×#41—1491no—
233C2×#41—1493no—
234C2×#41—1493no—
235C22×Q8—1493no—
236C22×Q8—1491no—
237C22×Q8C2×D14911yescharacteristic
238C22×Q8C2×D14911yescharacteristic, complemented
239C22×D4—1493no—
240C23×C4—1493no—
241C4⋊Q8—1493no—
242C4⋊Q8C2×D14913yescomplemented
243#124—1493no—
244#124—1493no—
245#123—1493no—
246#123—1496no—
247#123—1496no—
248#123—1493no—
249C22⋊Q8—1493no—
250C22⋊Q8—1493no—
251C22⋊Q8—1493no—
252C22⋊Q8—1496no—
253C22⋊Q8—1493no—
254C22⋊Q8—1493no—
255C22⋊Q8—1493no—
256C22⋊Q8C2×D14913yes—
257C22⋊Q8C2×D14913yescomplemented
258#121—1491no—
259#121—1493no—
260#121—1496no—
261#121—1493no—
262#121—1493no—
263#121—1496no—
264#120—1493no—
265#120—1493no—
266C4×Q8—1491no—
267C4×Q8C2×D14911yescharacteristic, complemented
268C4×D4—1493no—
269C4×D4—1493no—
270C4×D4—1493no—
271C4×D4—1493no—
272C4×D4—1493no—
273C4×D4C2×D14913yes—
274#117—1493no—
275C2×C4⋊C4—1493no—
276C2×C4⋊C4—1496no—
277C2×C22⋊C4—1496no—
278C24—1493no—
279#41—29812no—
280#41—29812no—
281#41—2984no—
282#41—29812no—
283#41—29812no—
284#41—29812no—
285#41—1498no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—2984no—
289C2×Q8—29812no—
290C2×Q8—1493no—
291C2×Q8—1493no—
292C2×Q8C22×D14913yes—
293C2×Q8C22×D14913yes—
294C2×Q8C22×D14911yescharacteristic
295C2×D4—29812no—
296C2×D4—1491no—
297C2×D4—1493no—
298C2×D4—1491no—
299C2×D4—1493no—
300C2×D4—1493no—
301C2×D4—1496no—
302C2×D4—1493no—
303C2×D4—1493no—
304C2×D4—1493no—
305C2×D4C22×D14911yescharacteristic
306C22×C4—29812no—
307C22×C4—1491no—
308C22×C4—1496no—
309C22×C4—1493no—
310C22×C4—1493no—
311C22×C4—1493no—
312C22×C4—1493no—
313C22×C4—1496no—
314C22×C4—1493no—
315C22×C4C22×D14913yes—
316C22×C4C22×D14913yes—
317C4⋊C4—1493no—
318C4⋊C4—1493no—
319C4⋊C4—1493no—
320C4⋊C4—1493no—
321C4⋊C4—1496no—
322C4⋊C4—1491no—
323C4⋊C4—1493no—
324C4⋊C4C22×D14913yes—
325C4⋊C4C22×D14916yes—
326C4⋊C4C22×D14913yes—
327C22⋊C4—1493no—
328C22⋊C4—1491no—
329C22⋊C4—1496no—
330C22⋊C4—1496no—
331C22⋊C4—1493no—
332C22⋊C4—1493no—
333C22⋊C4—1491no—
334C22⋊C4—1493no—
335C22⋊C4—1493no—
336C22⋊C4—1493no—
337C22⋊C4C22×D14913yes—
338C22⋊C4C22×D14913yes—
339C42—1493no—
340C42C22×D14913yes—
341C23—2986no—
342C23—2986no—
343C23—5963no—
344C23—1493no—
345C23—1491no—
346C23—1493no—
347C23C23×D14911yescharacteristic
348C23C23×D14911yescharacteristic
349Q8—26no—
350Q8—26no—
351Q8—2986no—
352Q8—2986no—
353Q8D4×D14914yes—
354D4—22no—
355D4—29812no—
356D4—2986no—
357D4—2986no—
358D4—2982no—
359D4—2982no—
360D4—2986no—
361D4—5963no—
362D4—5963no—
363D4—14912no—
364C2×C4—26no—
365C2×C4—26no—
366C2×C4—2986no—
367C2×C4—29812no—
368C2×C4—2986no—
369C2×C4—2982no—
370C2×C4—2986no—
371C2×C4—2986no—
372C2×C4—5966no—
373C2×C4—1491no—
374C2×C4—1493no—
375C2×C4—1493no—
376C2×C4—1491no—
377C2×C4—1491no—
378C2×C4—14912no—
379C2×C4C23×D14911yescharacteristic
380C2×C4C23×D14913yes—
381C2×C4C23×D14913yes—
382C2×C4C23×D14913yes—
383C2×C4C23×D14913yes—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—2983no—
390C22—2983no—
391C22—2981no—
392C22—2983no—
393C22—2981no—
394C22—2983no—
395C22—2981no—
396C22—2983no—
397C22—59612no—
398C22—1496no—
399C22#1801195912yescomplemented
400C22C24×D14911yescharacteristic
401C4—23no—
402C4—23no—
403C4—21no—
404C4—23no—
405C4—2981no—
406C4—2983no—
407C4—2981no—
408C4—2983no—
409C4—1492no—
410C4C2×D4×D14916yes—
411C2—22no—
412C2—41no—
413C2—2986no—
414C2—5963no—
415C2—5961no—
416C2C22×D4×D14911yescharacteristic
417C2C2×#1801195911yescharacteristic
418C2#2375046211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D149
commutator, socle, cosocleC22×C149C25
FittingD4×Q8×C149C2
hypercenterD4×Q8D149
hypocenterC149#3489
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#18011959C2semidirect
2. C22×#14558737C2semidirect
3. Q8×C22⋊D149C2semidirect
4. #23750287C2semidirect
5. #23750286C2semidirect
6. #23750278C2semidirect
7. #23750277C2semidirect
8. #23750271C2semidirect
9. #23750266C2semidirect
10. #23750205C2semidirect
11. #23750200C2semidirect
12. #23750134C2semidirect
13. C22⋊Q8×D149C2semidirect
14. #23750107C2semidirect
15. D4×Q8×C149C2semidirect
22. C2×#14558737C22semidirect
23. C2×#14558736C22semidirect
24. C2×#14558736C22semidirect
25. C2×Q8×D149C22semidirect
26. C2×Q8×D149C22semidirect
29. #18011909C22semidirect
30. #18011900C22semidirect
31. #18011899C22semidirect
32. #18011896C22semidirect
33. #18011896C22semidirect
34. #18011896C22semidirect
35. #18011893C22semidirect
36. #18011891C22semidirect
38. #18011889C22semidirect
39. #18011889C22semidirect
41. #18011887C22semidirect
42. #18011887C22semidirect
43. #18011886C22semidirect
44. #18011886C22semidirect
45. #18011885C22semidirect
46. #18011884C22semidirect
47. #18011884C22semidirect
51. Q8×Dic149C22semidirect
52. C4×D596C22semidirect
53. C4×D596C22semidirect
54. #18011873C22semidirect
56. C4×C22⋊D149C22semidirect
57. #18011868C22semidirect
58. C4⋊C4×D149C22semidirect
59. C4⋊C4×D149C22semidirect
62. C22×Q8×C149C22semidirect
63. C4⋊Q8×C149C22semidirect
65. C22⋊Q8×C149C22semidirect
66. C4×Q8×C149C22semidirect
73. #14558737D4semidirect
179. C22×C1492−1+4semidirect
209. C149#3489semidirect
214. D4×Q8D149semidirect
238. C22×Q8C2×D149semidirect
242. C4⋊Q8C2×D149semidirect
257. C22⋊Q8C2×D149semidirect
267. C4×Q8C2×D149semidirect
399. C22#18011959semidirect

Maximal subgroups

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

subgroupindex
C2×#180119592
C22×#145587372
Q8×C22⋊D1492
#237502872
#237502862
#237502782
#237502772
#237502712
#237502662
#237502052
#237502002
#237501342
C22⋊Q8×D1492
#237501072
D4×Q8×C1492
#3489149