Finite Group Databasebeta

← #31792308

Subgroups — #31792308

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×#17984107C211yesmaximal, characteristic, complemented
2#22578052C211yesmaximal, characteristic, complemented
3#22578047C211yesmaximal, characteristic, complemented
4#22577794C211yesmaximal, characteristic, complemented
5#22577661C211yesmaximal, characteristic, complemented
6#22577629C211yesmaximal, characteristic, complemented
7#22577616C211yesmaximal, characteristic, complemented
8#22577606C211yesmaximal, characteristic, complemented
9#22577565C211yesmaximal, characteristic, complemented
10#22577562C211yesmaximal, characteristic, complemented
11#22577554C211yesmaximal, characteristic, complemented
12#22577545C211yesmaximal, characteristic, complemented
13#22577515C211yesmaximal, characteristic, complemented
14#22577472C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic
17C2×#14555509C2211yescharacteristic, complemented
18#17984107—24no—
19C2×Dic548C2211yescharacteristic, complemented
20C2×#14555500C2211yescharacteristic, complemented
21C2×#14555498C2211yescharacteristic
22C2×#14555498C2211yescharacteristic
23C2×#14555498C2211yescharacteristic, complemented
24C2×C8×D137C2211yescharacteristic, complemented
25#17984086C2211yescharacteristic, complemented
26#17984075C2211yescharacteristic
27#17984071—21no—
28#17984069—21no—
29#17984069—21no—
30#17984067C2211yescharacteristic
31#17984067C2211yescharacteristic
32#17984064C2211yescharacteristic, complemented
33#17984060—21no—
34#17984060—21no—
35#17984059C2211yescharacteristic
36#17984057—21no—
37D4×Dic137C2211yescharacteristic
38D4×Dic137C2211yescharacteristic, complemented
39C4×C22⋊D137C2211yescharacteristic
40C2×#14555489—21no—
41C2×C22⋊Dic137—21no—
42#17984020C2211yescharacteristic, complemented
43#17984017C2211yescharacteristic, complemented
44#17984013C2211yescharacteristic, complemented
45#17984013C2211yescharacteristic, complemented
46#17984012C2211yescharacteristic, complemented
47#17984009C2211yescharacteristic
48#17984009C2211yescharacteristic
49#17984009C2211yescharacteristic, complemented
50C22⋊#11764040C2211yescharacteristic
51#17984000C2211yescharacteristic, complemented
52#17983998C2211yescharacteristic, complemented
53C2×#41×C137C2211yescharacteristic
54C22×D4×C137C2211yescharacteristic
55AGL(1,ℤ/8)×C137—24no—
56C2×SD16×C137C2211yescharacteristic
57C2×SD16×C137C2211yescharacteristic
58C2×D8×C137C2211yescharacteristic
59C2×D8×C137C2211yescharacteristic
60C2×M16×C137C2211yescharacteristic
61#14555509—22no—
62#14555509—22no—
63#14555509—41no—
64#14555509—41no—
65C2×Dic274C2311yescharacteristic
66C2×Dic274C2311yescharacteristic
67C2×C22⋊D137—21no—
68C2×C22⋊D137—21no—
69C2×C22⋊D137—21no—
70C2×C22⋊D137D411yescharacteristic
71C22×Dic137—21no—
72C22×Dic137—21no—
73C22×Dic137—21no—
74C22×Dic137—21no—
75C2×C4×D137C2311yescharacteristic
76Dic548—22no—
77#14555500—41no—
78#14555498—22no—
79#14555498—22no—
80#14555498—41no—
81C2×#11764040C2311yescharacteristic
82C2×#11764040C2311yescharacteristic
83C8×D137—22no—
84C4⋊Dic137C2311yescharacteristic
85C4⋊Dic137C2311yescharacteristic
86#14555489—21no—
87#14555489—21no—
88#14555489—21no—
89#14555489—21no—
90#14555489—21no—
91#14555489D411yescharacteristic
92C22⋊Dic137—21no—
93C22⋊Dic137—21no—
94C22⋊Dic137—21no—
95C22⋊Dic137—21no—
96C22⋊Dic137—21no—
97C22⋊Dic137—21no—
98C22⋊Dic137—21no—
99C22⋊Dic137D411yescharacteristic
100#14555487—21no—
101#14555487D411yescharacteristic
102C4×Dic137C2311yescharacteristic
103C24×C137—21no—
104#41×C137—22no—
105#41×C137—41no—
106C2×Q8×C137C2311yescharacteristic
107C2×D4×C137—22no—
108C2×D4×C137—21no—
109C2×D4×C137—21no—
110C2×D4×C137—41no—
111C2×D4×C137C2311yescharacteristic
112C2×D4×C137C2311yescharacteristic
113C2×D4×C137C2311yescharacteristic
114C22×C4×C137—21no—
115C22×C4×C137C2311yescharacteristic
116SD16×C137—22no—
117SD16×C137—22no—
118D8×C137—22no—
119D8×C137D414yescomplemented
120M16×C137—22no—
121C2×C8×C137C2311yescharacteristic
122C2×C8×C137C2311yescharacteristic
123C22×D137C2×D411yescharacteristic
124Dic274—21no—
125Dic274—21no—
126Dic274—21no—
127Dic274—21no—
128C22⋊D137—41no—
129C22⋊D137—42no—
130C22⋊D137—42no—
131C22⋊D137—81no—
132C2×Dic137—21no—
133C2×Dic137—21no—
134C2×Dic137—21no—
135C2×Dic137—21no—
136C2×Dic137—42no—
137C2×Dic137—42no—
138C2×Dic137—42no—
139C2×Dic137—81no—
140C2×Dic137C2×D411yescharacteristic
141C2×Dic137C2×D411yescharacteristic
142C2×Dic137C2×D411yescharacteristic
143C4×D137—21no—
144C4×D137—21no—
145#11764040—21no—
146#11764040—21no—
147C23×C137—21no—
148C23×C137—21no—
149C23×C137—21no—
150C23×C137—42no—
151C23×C137—42no—
152C23×C137—81no—
153C23×C137C2×D411yescharacteristic
154Q8×C137—21no—
155Q8×C137—21no—
156D4×C137—21no—
157D4×C137—21no—
158D4×C137—21no—
159D4×C137—21no—
160D4×C137—42no—
161D4×C137—42no—
162D4×C137C2×D412yes—
163D4×C137C2×D412yes—
164C2×C4×C137—21no—
165C2×C4×C137—21no—
166C2×C4×C137—21no—
167C2×C4×C137—42no—
168C2×C4×C137C2×D411yescharacteristic
169C2×C4×C137C2411yescharacteristic
170C8×C137—21no—
171C8×C137C2×D412yes—
172C22×C137—21no—
173C22×C137—22no—
174C22×C137—22no—
175C22×C137—21no—
176C22×C137—21no—
177C22×C137—41no—
178C22×C137—41no—
179C22×C137—41no—
180C22×C137—41no—
181C22×C137—41no—
182C22×C137—41no—
183C22×C137—41no—
184C22×C137—84no—
185C22×C137C22×D411yescharacteristic
186C2×D137—21no—
187C2×D137—21no—
188C2×D137—21no—
189C4×C137—21no—
190C4×C137—41no—
191C4×C137C22×D411yescharacteristic
192C4×C137H(5,𝔽2)11yescharacteristic
193Dic137—21no—
194Dic137—21no—
195Dic137—21no—
196Dic137—41no—
197Dic137—41no—
198Dic137—41no—
199Dic137—41no—
200C2×C137—41no—
201C2×C137—42no—
202C2×C137—42no—
203C2×C137—81no—
204C2×C137D4211yescharacteristic
205C2×C137#57511yescharacteristic
206C2×C137#57711yescharacteristic
207D137—41no—
208C137#326711yescharacteristic, complemented, Sylow
209#3267—1371nomaximal, Sylow
210C2×AGL(1,ℤ/8)D13711yescharacteristic, complemented
211C2×#135—1371no—
212#547—1371no—
213#546—1371no—
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)—24no—
230#135—2744no—
231C2×Q16—1371no—
232C2×Q16—1371no—
233C2×SD16—1371no—
234C2×SD16—1371no—
235C2×SD16—1371no—
236C2×SD16C2×D13711yescharacteristic
237C2×SD16C2×D13711yescharacteristic
238C2×D8C2×D13711yescharacteristic
239C2×D8C2×D13711yescharacteristic
240C2×M16C2×D13711yescharacteristic
241C22×C8—1371no—
242C4⋊Q8—1371no—
243#124—1371no—
244#123—2741no—
245#123—2741no—
246#123—2741no—
247C22⋊Q8—1371no—
248C22⋊Q8—1371no—
249C22⋊Q8—1371no—
250#121—2741no—
251#121—2741no—
252#121—1371no—
253#120—2741no—
254C4×D4—1371no—
255C4×D4—1371no—
256C4×D4—1371no—
257C2×C4⋊C4—2741no—
258C2×C22⋊C4—2741no—
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—41no—
284C2×D4—2741no—
285C2×D4—2741no—
286C2×D4—2741no—
287C2×D4—1371no—
288C2×D4C22×D13711yescharacteristic
289C2×D4C22×D13711yescharacteristic
290C2×D4C22×D13711yescharacteristic
291C22×C4—21no—
292C22×C4—2741no—
293C22×C4—2741no—
294C22×C4—2741no—
295C22×C4—2741no—
296C22×C4—1371no—
297C22×C4C22×D13711yescharacteristic
298Q16—2742no—
299Q16—5481no—
300SD16—22no—
301SD16—22no—
302SD16—2742no—
303SD16—2742no—
304SD16—5481no—
305D8—22no—
306D8C22⋊D13714yescomplemented
307M16—22no—
308C2×C8—2742no—
309C2×C8—1371no—
310C2×C8—1371no—
311C2×C8C22×D13711yescharacteristic
312C2×C8C22×D13711yescharacteristic
313C4⋊C4—2741no—
314C4⋊C4—2741no—
315C4⋊C4—2741no—
316C4⋊C4—2741no—
317C4⋊C4—2741no—
318C4⋊C4—1371no—
319C4⋊C4—1371no—
320C4⋊C4—1371no—
321C22⋊C4—2741no—
322C22⋊C4—2741no—
323C22⋊C4—2741no—
324C22⋊C4—2741no—
325C22⋊C4—2741no—
326C22⋊C4—2741no—
327C22⋊C4—2741no—
328C22⋊C4—2741no—
329C22⋊C4—1371no—
330C22⋊C4—1371no—
331C42—1371no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23—1371no—
339C23D4×D13711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—2741no—
343Q8—2741no—
344Q8—2741no—
345Q8—2741no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—42no—
351D4—42no—
352D4—5481no—
353D4—5482no—
354D4—5482no—
355D4—10961no—
356D4C2×C22⋊D13712yes—
357D4C2×C22⋊D13712yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—2741no—
363C2×C4—2741no—
364C2×C4—2741no—
365C2×C4—2741no—
366C2×C4—2741no—
367C2×C4—2741no—
368C2×C4—5482no—
369C2×C4—5482no—
370C2×C4—5482no—
371C2×C4—10961no—
372C2×C4—1371no—
373C2×C4—1371no—
374C2×C4—1371no—
375C2×C4D4×D13711yescharacteristic
376C2×C4C23×D13711yescharacteristic
377C8—21no—
378C8—2741no—
379C8—2741no—
380C8C2×C22⋊D13712yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—2741no—
395C22—2741no—
396C22—2741no—
397C22C2×D4×D13711yescharacteristic
398C4—21no—
399C4—41no—
400C4—2741no—
401C4—2741no—
402C4—2741no—
403C4—5481no—
404C4—5481no—
405C4—5481no—
406C4—5481no—
407C4C22×C22⋊D13711yescharacteristic
408C4#1798413311yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—81no—
413C2—5481no—
414C2D4×C22⋊D13711yescharacteristic
415C2#2257819911yescharacteristic
416C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC2×AGL(1,ℤ/8)×C137C2
socleC22×C137C22×D4
hypercenterC2×AGL(1,ℤ/8)D137
hypocenterC137#3267
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#17984107C2semidirect
2. #22578052C2semidirect
3. #22578047C2semidirect
4. #22577794C2semidirect
5. #22577661C2semidirect
6. #22577629C2semidirect
7. #22577616C2semidirect
8. #22577606C2semidirect
9. #22577565C2semidirect
10. #22577562C2semidirect
11. #22577554C2semidirect
12. #22577545C2semidirect
13. #22577515C2semidirect
14. #22577472C2semidirect
15. C2×AGL(1,ℤ/8)×C137C2semidirect
17. C2×#14555509C22semidirect
19. C2×Dic548C22semidirect
20. C2×#14555500C22semidirect
23. C2×#14555498C22semidirect
24. C2×C8×D137C22semidirect
25. #17984086C22semidirect
32. #17984064C22semidirect
38. D4×Dic137C22semidirect
42. #17984020C22semidirect
43. #17984017C22semidirect
44. #17984013C22semidirect
45. #17984013C22semidirect
46. #17984012C22semidirect
49. #17984009C22semidirect
51. #17984000C22semidirect
52. #17983998C22semidirect
119. D8×C137D4semidirect
208. C137#3267semidirect
210. C2×AGL(1,ℤ/8)D137semidirect
306. D8C22⋊D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841072
#225780522
#225780472
#225777942
#225776612
#225776292
#225776162
#225776062
#225775652
#225775622
#225775542
#225775452
#225775152
#225774722
C2×AGL(1,ℤ/8)×C1372
#3267137