Finite Group Databasebeta

← #34078461

Subgroups — #34078461

Order 19328.

724 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23753048C211yesmaximal, characteristic, complemented
2#23753012C211yesmaximal, characteristic, complemented
3#23753010C211yesmaximal, characteristic, complemented
4#23752762C211yesmaximal, characteristic, complemented
5#23752396C212yesmaximal, complemented
6#23752391C212yesmaximal, complemented
7#23752386C212yesmaximal, complemented
8#23752381C212yesmaximal, complemented
9C2×#18012667C211yesmaximal, characteristic, complemented
10#23752194C211yesmaximal, characteristic, complemented
11C2×#104×C151C211yesmaximal, characteristic, complemented
12#18012787—21no—
13#18012786—21no—
14#18012786C2212yes—
15#18012785—21no—
16#18012785C2212yes—
17#18012784—21no—
18#18012784—22no—
19C2×#14559004—22no—
20C2×#14559004C2211yescharacteristic
21C2×#14559004C2211yescharacteristic
22C2×#14559004C2211yescharacteristic, complemented
23#41×D151—22no—
24#41×D151—22no—
25#18012772—21no—
26#18012772C2212yescomplemented
27#18012770—21no—
28#18012769C2212yescomplemented
29#18012767—21no—
30#18012767C2212yescomplemented
31#18012765—21no—
32#18012764C2212yescomplemented
33#18012761—22no—
34#18012759—22no—
35C2×#14558988C2211yescharacteristic, complemented
36#18012739C2211yescharacteristic, complemented
37#18012736C2211yescharacteristic, complemented
38#18012727C2211yescharacteristic, complemented
39#18012727C2211yescharacteristic, complemented
40#18012713—22no—
41C4×D604—22no—
42#18012667C2212yescomplemented
43#18012667C2212yescomplemented
44#18012657C2212yescomplemented
45#18012655C2212yescomplemented
46C2×#41×C151C2211yescharacteristic
47C2×M16×C151C2211yescharacteristic, complemented
48C2×C42×C151C2211yescharacteristic, complemented
49#104×C151C2212yescomplemented
50#104×C151C2212yescomplemented
51#14559005—22no—
52#14559005—22no—
53#14559004—24no—
54#14559004—24no—
55#14559004—22no—
56#14559004—22no—
57#14559004—21no—
58#14559004—21no—
59#14559004—21no—
60#14559004—21no—
61#14559004—21no—
62#14559004—21no—
63#14559004—21no—
64#14559004—21no—
65#14559004—21no—
66#14559004—21no—
67#14559004D412yescomplemented
68#14559004C2312yes—
69#14559003—24no—
70#14559003—22no—
71#14559003—24no—
72#14559003—22no—
73C2×Dic302—22no—
74C2×Dic302D411yescharacteristic, complemented
75C2×Dic302C2311yescharacteristic
76C2×Dic302C2311yescharacteristic
77Q8×D151—22no—
78Q8×D151—22no—
79C2×D604—22no—
80C2×D604—22no—
81C2×D604D411yescharacteristic, complemented
82C2×D604C2311yescharacteristic
83C2×D604C2311yescharacteristic
84C2×C22⋊D151—24no—
85C2×C22⋊D151—21no—
86C2×C22⋊D151—21no—
87C2×C22⋊D151D412yes—
88D4×D151—24no—
89D4×D151—22no—
90D4×D151—24no—
91D4×D151—22no—
92C2×C4×D151—24no—
93C2×C4×D151—21no—
94C2×C4×D151—21no—
95C2×C4×D151D412yes—
96Dic604—22no—
97#14558994—22no—
98#14558993—22no—
99#14558993—22no—
100#14558992—22no—
101#14558991—22no—
102D1208—22no—
103#14558989—22no—
104#14558988—21no—
105#14558988C2312yes—
106C2×C151⋊C8—21no—
107C4⋊Dic151—22no—
108(C2×D151)⋊C4—22no—
109(C2×D151)⋊C4—22no—
110#41×C151—21no—
111#41×C151—21no—
112#41×C151D412yescomplemented
113#41×C151C2312yes—
114C2×Q8×C151D411yescharacteristic, complemented
115C2×D4×C151—21no—
116C2×D4×C151D411yescharacteristic, complemented
117C22×C4×C151—21no—
118C22×C4×C151—21no—
119C22×C4×C151C2311yescharacteristic
120M16×C151—21no—
121M16×C151C2312yes—
122C2×C8×C151—21no—
123C42×C151—21no—
124C42×C151C2311yescharacteristic
125C42×C151C2311yescharacteristic
126C22×D151—24no—
127C22×D151—21no—
128C22×D151—21no—
129C22×D151C2×D412yes—
130Dic302—24no—
131Dic302—21no—
132Dic302—21no—
133Dic302—21no—
134Dic302—21no—
135Dic302—21no—
136Dic302C2×D412yes—
137D604—24no—
138D604—21no—
139D604—21no—
140D604—21no—
141D604—21no—
142D604—21no—
143D604—44no—
144D604C2×D412yes—
145C22⋊D151—24no—
146C22⋊D151—24no—
147C22⋊D151—22no—
148C22⋊D151—22no—
149C22⋊D151—22no—
150C22⋊D151—22no—
151C22⋊D151—41no—
152C22⋊D151—41no—
153C2×Dic151—24no—
154C2×Dic151—21no—
155C2×Dic151—21no—
156C2×Dic151C2×D412yes—
157C4×D151—24no—
158C4×D151—24no—
159C4×D151—22no—
160C4×D151—22no—
161C4×D151—22no—
162C4×D151—22no—
163C4×D151—41no—
164C4×D151—41no—
165C151⋊C8—22no—
166C23×C151—21no—
167C23×C151C2×D411yescharacteristic
168Q8×C151—21no—
169Q8×C151C2×D412yes—
170D4×C151—22no—
171D4×C151—22no—
172D4×C151—21no—
173D4×C151C2×D412yes—
174C2×C4×C151—21no—
175C2×C4×C151—21no—
176C2×C4×C151—22no—
177C2×C4×C151—22no—
178C2×C4×C151—21no—
179C2×C4×C151—42no—
180C2×C4×C151C2×D411yescharacteristic
181C2×C4×C151C2×D411yescharacteristic
182C2×C4×C151C2×D411yescharacteristic
183C2×C4×C151C2×D411yescharacteristic
184C2×C4×C151C2×D411yescharacteristic
185C2×C4×C151C2411yescharacteristic
186C8×C151—22no—
187C22×C151—22no—
188C22×C151—41no—
189C22×C151—81no—
190C22×C151#12011yescharacteristic
191C22×C151#12011yescharacteristic
192C22×C151C22×D411yescharacteristic
193C2×D151—22no—
194C2×D151—22no—
195C2×D151—42no—
196C2×D151—42no—
197C2×D151—41no—
198C2×D151—81no—
199C2×D151—84no—
200C4×C151—22no—
201C4×C151—22no—
202C4×C151—41no—
203C4×C151#12011yescharacteristic
204C4×C151#12011yescharacteristic
205C4×C151C22×D411yescharacteristic
206C4×C151C22×D411yescharacteristic
207Dic151—22no—
208Dic151—22no—
209Dic151—41no—
210C2×C151—21no—
211C2×C151—21no—
212C2×C151—21no—
213C2×C151—42no—
214C2×C151C2×#12011yescharacteristic
215D151—42no—
216D151—42no—
217D151—81no—
218C151#301811yescharacteristic, complemented, Sylow
219#3018—1511nomaximal, Sylow
220#585—1511no—
221#575—1511no—
222#575—1511no—
223#532—1511no—
224#456—1512no—
225#455—1512no—
226#454—1512no—
227#453—1512no—
228C2×#104—1511no—
229C2×#104D15111yescharacteristic, complemented
230#413—1511no—
2312−1+4—3021no—
2322−1+4—3021no—
2332−1+4—1512no—
234H(5,𝔽2)—3022no—
235H(5,𝔽2)—3021no—
236H(5,𝔽2)—3021no—
237H(5,𝔽2)—1512no—
238C2×#41—3022no—
239C2×#41—3022no—
240C2×#41—3022no—
241C2×#41—1511no—
242C2×#41—1511no—
243C2×#41—1511no—
244C2×#41C2×D15111yescharacteristic
245#137—3021no—
246#137—3021no—
247#137—1512no—
248#137—1512no—
249AGL(1,ℤ/8)—3021no—
250AGL(1,ℤ/8)—3021no—
251AGL(1,ℤ/8)—1512no—
252AGL(1,ℤ/8)—1512no—
253#135—3022no—
254#135—3022no—
255C2×M16—1511no—
256C2×M16C2×D15111yescharacteristic, complemented
257C4⋊Q8—1511no—
258#127—1511no—
259#124—1511no—
260#124—1511no—
261#121—3022no—
262C4×D4—3022no—
263C2×C42C2×D15111yescharacteristic, complemented
264#104—1512no—
265#104—1512no—
266#104C2×D15112yescomplemented
267#104C2×D15112yescomplemented
268#101—1512no—
269#100—1512no—
270#41—21no—
271#41—21no—
272#41—3024no—
273#41—3024no—
274#41—3022no—
275#41—3022no—
276#41—3024no—
277#41—3022no—
278#41—3022no—
279#41—3022no—
280#41—3022no—
281#41—3024no—
282#41—3021no—
283#41—3021no—
284#41—3021no—
285#41—3021no—
286#41—3021no—
287#41—3021no—
288#41—3021no—
289#41—3021no—
290#41—3021no—
291#41—3021no—
292#41—1512no—
293#41—1512no—
294#41D60412yescomplemented
295#41C22×D15112yes—
296C2×Q8—3022no—
297C2×Q8—3022no—
298C2×Q8—3022no—
299C2×Q8—1511no—
300C2×Q8—1511no—
301C2×Q8—1511no—
302C2×Q8D60411yescharacteristic, complemented
303C2×D4—21no—
304C2×D4—3022no—
305C2×D4—3022no—
306C2×D4—3024no—
307C2×D4—3022no—
308C2×D4—3024no—
309C2×D4—3024no—
310C2×D4—3021no—
311C2×D4—3022no—
312C2×D4—3021no—
313C2×D4—1512no—
314C2×D4—1511no—
315C2×D4—1511no—
316C2×D4—1511no—
317C2×D4D60411yescharacteristic, complemented
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—3024no—
321C22×C4—3021no—
322C22×C4—3021no—
323C22×C4—1512no—
324C22×C4C22×D15111yescharacteristic
325Q16—3022no—
326Q16—3022no—
327SD16—3022no—
328SD16—3022no—
329SD16—3022no—
330SD16—3022no—
331D8—3022no—
332D8—3022no—
333M16—21no—
334M16—3021no—
335M16—1512no—
336M16C22×D15112yes—
337C2×C8—21no—
338C2×C8—3021no—
339C4⋊C4—3022no—
340C22⋊C4—3022no—
341C22⋊C4—3022no—
342C42—21no—
343C42C22×D15111yescharacteristic
344C42C22×D15111yescharacteristic
345C23—21no—
346C23—3024no—
347C23—3021no—
348C23—3021no—
349C23—1512no—
350C23D4×D15111yescharacteristic
351Q8—21no—
352Q8—3024no—
353Q8—3021no—
354Q8—3021no—
355Q8—3021no—
356Q8—3021no—
357Q8—3021no—
358Q8—1512no—
359Q8C2×D60412yes—
360D4—22no—
361D4—22no—
362D4—21no—
363D4—3024no—
364D4—3024no—
365D4—3024no—
366D4—3022no—
367D4—3022no—
368D4—3022no—
369D4—3022no—
370D4—3021no—
371D4—3021no—
372D4—3021no—
373D4—3021no—
374D4—3021no—
375D4—6044no—
376D4—6041no—
377D4—6041no—
378D4—1512no—
379D4C2×D60412yes—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—42no—
386C2×C4—3024no—
387C2×C4—3024no—
388C2×C4—3024no—
389C2×C4—3021no—
390C2×C4—3022no—
391C2×C4—3022no—
392C2×C4—3022no—
393C2×C4—3022no—
394C2×C4—3021no—
395C2×C4—6041no—
396C2×C4—6041no—
397C2×C4—1512no—
398C2×C4D4×D15111yescharacteristic
399C2×C4D4×D15111yescharacteristic
400C2×C4D4×D15111yescharacteristic
401C2×C4C2×D60411yescharacteristic
402C2×C4C2×D60411yescharacteristic
403C2×C4C23×D15111yescharacteristic
404C8—22no—
405C8—3022no—
406C22—22no—
407C22—41no—
408C22—81no—
409C22—3022no—
410C22—3022no—
411C22—6042no—
412C22—6042no—
413C22—6041no—
414C22—12081no—
415C22—12084no—
416C22#1801270911yescharacteristic
417C22#1801270911yescharacteristic
418C22C2×D4×D15111yescharacteristic
419C4—22no—
420C4—22no—
421C4—41no—
422C4—3022no—
423C4—3022no—
424C4—6041no—
425C4#1801270911yescharacteristic
426C4#1801270911yescharacteristic
427C4C2×D4×D15111yescharacteristic
428C4C22×D60411yescharacteristic
429C2—21no—
430C2—21no—
431C2—21no—
432C2—42no—
433C2—6042no—
434C2—6042no—
435C2—12081no—
436C2C2×#1801270911yescharacteristic
subgroupG/H
centerC4#18012709
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
FittingC2×#104×C151C2
socleC2×C151C2×#120
hypercenterC2×#104D151
hypocenterC151#3018
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
1. #23753048C2semidirect
2. #23753012C2semidirect
3. #23753010C2semidirect
4. #23752762C2semidirect
5. #23752396C2semidirect
6. #23752391C2semidirect
7. #23752386C2semidirect
8. #23752381C2semidirect
9. C2×#18012667C2semidirect
10. #23752194C2semidirect
11. C2×#104×C151C2semidirect
22. C2×#14559004C22semidirect
26. #18012772C22semidirect
28. #18012769C22semidirect
30. #18012767C22semidirect
32. #18012764C22semidirect
35. C2×#14558988C22semidirect
36. #18012739C22semidirect
37. #18012736C22semidirect
38. #18012727C22semidirect
39. #18012727C22semidirect
42. #18012667C22semidirect
43. #18012667C22semidirect
44. #18012657C22semidirect
45. #18012655C22semidirect
47. C2×M16×C151C22semidirect
48. C2×C42×C151C22semidirect
49. #104×C151C22semidirect
50. #104×C151C22semidirect
67. #14559004D4semidirect
74. C2×Dic302D4semidirect
81. C2×D604D4semidirect
112. #41×C151D4semidirect
114. C2×Q8×C151D4semidirect
116. C2×D4×C151D4semidirect
218. C151#3018semidirect
229. C2×#104D151semidirect
256. C2×M16C2×D151semidirect
263. C2×C42C2×D151semidirect
266. #104C2×D151semidirect
267. #104C2×D151semidirect
294. #41D604semidirect
302. C2×Q8D604semidirect
317. C2×D4D604semidirect

Maximal subgroups

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

subgroupindex
#237530482
#237530122
#237530102
#237527622
#237523962
#237523912
#237523862
#237523812
C2×#180126672
#237521942
C2×#104×C1512
#3018151