Finite Group Databasebeta

← #31792310

Subgroups — #31792310

Order 17536.

500 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×#17984114C211yesmaximal, characteristic, complemented
2C2×#17984108C211yesmaximal, characteristic, complemented
3#22578058C211yesmaximal, characteristic, complemented
4#22577798C211yesmaximal, characteristic, complemented
5#22577659C211yesmaximal, characteristic, complemented
6#22577630C211yesmaximal, characteristic, complemented
7#22577615C211yesmaximal, characteristic, complemented
8#22577608C211yesmaximal, characteristic, complemented
9#22577565C211yesmaximal, characteristic, complemented
10#22577563C211yesmaximal, characteristic, complemented
11#22577554C211yesmaximal, characteristic, complemented
12#22577546C211yesmaximal, characteristic, complemented
13#22577513C211yesmaximal, characteristic, complemented
14#22577475C211yesmaximal, characteristic, complemented
15#546×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C2×#14555510C2211yescharacteristic, complemented
18#17984114—24no—
19#17984108—24no—
20C2×#14555500C2211yescharacteristic, complemented
21C2×#14555500C2211yescharacteristic, complemented
22C2×#14555498C2211yescharacteristic
23C2×#14555498C2211yescharacteristic
24C2×#14555498C2211yescharacteristic
25C2×#14555498C2211yescharacteristic
26C2×#14555497C2211yescharacteristic, complemented
27C2×#14555495C2211yescharacteristic
28C2×#14555495C2211yescharacteristic
29C2×#14555494C2211yescharacteristic
30C22×#11764040C2211yescharacteristic, complemented
31#17984088C2211yescharacteristic, complemented
32#17984070—21no—
33#17984070—21no—
34#17984067C2211yescharacteristic
35#17984067C2211yescharacteristic, complemented
36#17984059—21no—
37C4×D548C2211yescharacteristic, complemented
38C4⋊C4×D137—21no—
39#17984019C2211yescharacteristic, complemented
40C4⋊#11764040C2211yescharacteristic, complemented
41#17984012C2211yescharacteristic, complemented
42#17984011C2211yescharacteristic, complemented
43#17984011C2211yescharacteristic, complemented
44#17984009C2211yescharacteristic
45#17984009C2211yescharacteristic
46#17984008C2211yescharacteristic, complemented
47C22⋊#11764040C2211yescharacteristic
48C22⋊#11764040C2211yescharacteristic, complemented
49#17983999C2211yescharacteristic, complemented
50C2×#41×C137C2211yescharacteristic
51C22×D4×C137C2211yescharacteristic
52#124×C137C2211yescharacteristic
53#123×C137—21no—
54C22⋊Q8×C137C2211yescharacteristic
55#121×C137—21no—
56#121×C137C2211yescharacteristic
57#120×C137—21no—
58C4×D4×C137C2211yescharacteristic
59C4×D4×C137C2211yescharacteristic
60C2×C22⋊C4×C137—21no—
61#14555510—22no—
62#14555510—22no—
63#14555510—41no—
64#14555510—41no—
65C2×Dic274C2311yescharacteristic
66C2×Dic274C2311yescharacteristic
67C2×D548C2311yescharacteristic
68C2×C22⋊D137—21no—
69C2×C22⋊D137—21no—
70C2×C4×D137—21no—
71C2×C4×D137—21no—
72C2×C4×D137—21no—
73#14555500—22no—
74#14555500—41no—
75#14555498—22no—
76#14555498—22no—
77#14555498—22no—
78#14555498—41no—
79#14555497—22no—
80#14555495—22no—
81#14555495D414yescomplemented
82#14555494—22no—
83C2×#11764040—22no—
84C2×#11764040C2311yescharacteristic
85C2×#11764040C2311yescharacteristic
86C2×#11764040C2311yescharacteristic
87C2×#11764040C2311yescharacteristic
88C4⋊Dic137—21no—
89C4⋊Dic137—21no—
90C4⋊Dic137C2311yescharacteristic
91#14555489—21no—
92#14555489—21no—
93C22⋊Dic137—21no—
94C22⋊Dic137—21no—
95#14555487—21no—
96C24×C137—21no—
97#41×C137—22no—
98#41×C137—41no—
99C2×Q8×C137C2311yescharacteristic
100C2×D4×C137—22no—
101C2×D4×C137—21no—
102C2×D4×C137—21no—
103C2×D4×C137—21no—
104C2×D4×C137—41no—
105C2×D4×C137D411yescharacteristic
106C2×D4×C137C2311yescharacteristic
107C2×D4×C137C2311yescharacteristic
108C22×C4×C137—21no—
109C22×C4×C137—21no—
110C22×C4×C137C2311yescharacteristic
111C22×C4×C137C2311yescharacteristic
112C4⋊C4×C137—21no—
113C4⋊C4×C137D411yescharacteristic
114C4⋊C4×C137C2311yescharacteristic
115C22⋊C4×C137—21no—
116C22⋊C4×C137—21no—
117C22⋊C4×C137—21no—
118C22⋊C4×C137—21no—
119C22⋊C4×C137—21no—
120C22⋊C4×C137D411yescharacteristic
121C22⋊C4×C137D411yescharacteristic
122C42×C137C2311yescharacteristic
123C22×D137—21no—
124Dic274—21no—
125Dic274—21no—
126Dic274—21no—
127Dic274—21no—
128D548—21no—
129D548C2×D412yes—
130C22⋊D137—42no—
131C22⋊D137—42no—
132C2×Dic137—21no—
133C2×Dic137—21no—
134C2×Dic137—21no—
135C4×D137—42no—
136C4×D137—42no—
137C4×D137—42no—
138#11764040—21no—
139#11764040—21no—
140#11764040—21no—
141#11764040C2×D412yes—
142C23×C137—21no—
143C23×C137—21no—
144C23×C137—42no—
145C23×C137—42no—
146C23×C137—81no—
147C23×C137C2×D411yescharacteristic
148C23×C137C2×D411yescharacteristic
149Q8×C137—21no—
150Q8×C137—21no—
151D4×C137—21no—
152D4×C137—21no—
153D4×C137—21no—
154D4×C137—41no—
155D4×C137—42no—
156D4×C137—42no—
157D4×C137—81no—
158D4×C137C2×D412yes—
159C2×C4×C137—21no—
160C2×C4×C137—21no—
161C2×C4×C137—21no—
162C2×C4×C137—21no—
163C2×C4×C137—21no—
164C2×C4×C137—21no—
165C2×C4×C137—42no—
166C2×C4×C137—81no—
167C2×C4×C137C2×D411yescharacteristic
168C2×C4×C137C2×D411yescharacteristic
169C2×C4×C137C2×D411yescharacteristic
170C2×C4×C137C2×D411yescharacteristic
171C2×C4×C137C2411yescharacteristic
172C22×C137—21no—
173C22×C137—21no—
174C22×C137—22no—
175C22×C137—21no—
176C22×C137—21no—
177C22×C137—21no—
178C22×C137—21no—
179C22×C137—41no—
180C22×C137—41no—
181C22×C137—41no—
182C22×C137—41no—
183C22×C137—41no—
184C22×C137—84no—
185C22×C137C22×D411yescharacteristic
186C2×D137—22no—
187C2×D137—41no—
188C2×D137—41no—
189C4×C137—21no—
190C4×C137—21no—
191C4×C137—21no—
192C4×C137—21no—
193C4×C137—41no—
194C4×C137—41no—
195C4×C137C22×D411yescharacteristic
196C4×C137H(5,𝔽2)11yescharacteristic
197Dic137—41no—
198Dic137—41no—
199Dic137—41no—
200C2×C137—41no—
201C2×C137—41no—
202C2×C137—42no—
203C2×C137—81no—
204C2×C137D4211yescharacteristic
205C2×C137#57511yescharacteristic
206C2×C137#57711yescharacteristic
207D137—42no—
208C137#326711yescharacteristic, complemented, Sylow
209#3267—1371nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1371no—
211C2×#135—1371no—
212#547—1371no—
213#546D13711yescharacteristic, complemented
214#497—1371no—
215#470—1371no—
216#465—1371no—
217#463—1371no—
218#461—1371no—
219#452—1371no—
220#452—1371no—
221#450—1371no—
222#449—1371no—
223#442—1371no—
224#435—1371no—
225C2×#41—1371no—
226C2×#41—1371no—
227C2×#41C2×D13711yescharacteristic
228C22×D4C2×D13711yescharacteristic
229AGL(1,ℤ/8)—2744no—
230#135—2744no—
231C2×Q16—1371no—
232C2×Q16—1371no—
233C2×SD16—1371no—
234C2×SD16—1371no—
235C2×SD16—1371no—
236C2×SD16—1371no—
237C2×SD16—1371no—
238C2×D8—1371no—
239C2×D8—1371no—
240C2×M16—1371no—
241C22×C8—1371no—
242C4⋊Q8—1371no—
243#124C2×D13711yescharacteristic
244#123—21no—
245#123—2741no—
246#123—2741no—
247C22⋊Q8—1371no—
248C22⋊Q8—1371no—
249C22⋊Q8C2×D13711yescharacteristic
250#121—21no—
251#121—2741no—
252#121C2×D13711yescharacteristic
253#120—21no—
254C4×D4—1371no—
255C4×D4C2×D13711yescharacteristic
256C4×D4C2×D13711yescharacteristic
257C2×C4⋊C4—2741no—
258C2×C22⋊C4—21no—
259#106—1371no—
260C4⋊C8—1371no—
261#103—1371no—
262#103—1371no—
263#103—1371no—
264#102—1371no—
265#102—1371no—
266#102—1371no—
267C22⋊C8—1371no—
268C22⋊C8—1371no—
269#97—1371no—
270C24—21no—
271#41—22no—
272#41—41no—
273#41—2742no—
274#41—2742no—
275#41—5481no—
276#41—5481no—
277C2×Q8—1371no—
278C2×Q8—1371no—
279C2×Q8C22×D13711yescharacteristic
280C2×D4—22no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—2741no—
286C2×D4—2741no—
287C2×D4—1371no—
288C2×D4C22⋊D13711yescharacteristic
289C2×D4C22×D13711yescharacteristic
290C2×D4C22×D13711yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—2741no—
294C22×C4—2741no—
295C22×C4—2741no—
296C22×C4C22×D13711yescharacteristic
297C22×C4C22×D13711yescharacteristic
298Q16—2742no—
299Q16—5481no—
300SD16—2742no—
301SD16—2742no—
302SD16—2742no—
303SD16—2742no—
304SD16—5481no—
305D8—2742no—
306D8—1374no—
307M16—2742no—
308C2×C8—2742no—
309C2×C8—1371no—
310C2×C8—1371no—
311C2×C8—1371no—
312C2×C8—1371no—
313C4⋊C4—21no—
314C4⋊C4—2741no—
315C4⋊C4—2741no—
316C4⋊C4—2741no—
317C4⋊C4—2741no—
318C4⋊C4—1371no—
319C4⋊C4C22⋊D13711yescharacteristic
320C4⋊C4C22×D13711yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—2741no—
327C22⋊C4—2741no—
328C22⋊C4—2741no—
329C22⋊C4C22⋊D13711yescharacteristic
330C22⋊C4C22⋊D13711yescharacteristic
331C42C22×D13711yescharacteristic
332C23—21no—
333C23—21no—
334C23—42no—
335C23—42no—
336C23—81no—
337C23—2741no—
338C23C2×C22⋊D13711yescharacteristic
339C23C2×C22⋊D13711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—2741no—
343Q8—2741no—
344Q8—2741no—
345Q8—2741no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—41no—
350D4—42no—
351D4—42no—
352D4—81no—
353D4—2741no—
354D4—5482no—
355D4—5482no—
356D4—1372no—
357D4D4×D13712yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—81no—
366C2×C4—2741no—
367C2×C4—2741no—
368C2×C4—2741no—
369C2×C4—5482no—
370C2×C4—5482no—
371C2×C4—5482no—
372C2×C4C2×C22⋊D13711yescharacteristic
373C2×C4C2×C22⋊D13711yescharacteristic
374C2×C4C2×C22⋊D13711yescharacteristic
375C2×C4C2×C22⋊D13711yescharacteristic
376C2×C4C23×D13711yescharacteristic
377C8—2741no—
378C8—2741no—
379C8—2741no—
380C8—1372no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—22no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—2742no—
395C22—5481no—
396C22—5481no—
397C22C22×C22⋊D13711yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—5481no—
405C4—5481no—
406C4—5481no—
407C4C2×D4×D13711yescharacteristic
408C4#1798413311yescharacteristic
409C2—41no—
410C2—41no—
411C2—42no—
412C2—81no—
413C2—5482no—
414C2D4×C22⋊D13711yescharacteristic
415C2#2257820111yescharacteristic
416C2#2257821411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#546×C137C2
socleC22×C137C22×D4
hypercenter#546D137
hypocenterC137#3267
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#17984114C2semidirect
2. C2×#17984108C2semidirect
3. #22578058C2semidirect
4. #22577798C2semidirect
5. #22577659C2semidirect
6. #22577630C2semidirect
7. #22577615C2semidirect
8. #22577608C2semidirect
9. #22577565C2semidirect
10. #22577563C2semidirect
11. #22577554C2semidirect
12. #22577546C2semidirect
13. #22577513C2semidirect
14. #22577475C2semidirect
15. #546×C137C2semidirect
17. C2×#14555510C22semidirect
20. C2×#14555500C22semidirect
21. C2×#14555500C22semidirect
26. C2×#14555497C22semidirect
30. C22×#11764040C22semidirect
31. #17984088C22semidirect
35. #17984067C22semidirect
37. C4×D548C22semidirect
39. #17984019C22semidirect
40. C4⋊#11764040C22semidirect
41. #17984012C22semidirect
42. #17984011C22semidirect
43. #17984011C22semidirect
46. #17984008C22semidirect
48. C22⋊#11764040C22semidirect
49. #17983999C22semidirect
81. #14555495D4semidirect
208. C137#3267semidirect
213. #546D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841142
C2×#179841082
#225780582
#225777982
#225776592
#225776302
#225776152
#225776082
#225775652
#225775632
#225775542
#225775462
#225775132
#225774752
#546×C1372
#3267137