Finite Group Databasebeta

← #31792299

Subgroups — #31792299

Order 17536.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984117C211yesmaximal, characteristic, complemented
2Q8×C22⋊D137C211yesmaximal, characteristic, complemented
3#22578047C211yesmaximal, characteristic
4#22577794C211yesmaximal, characteristic, complemented
5#22577646C211yesmaximal, characteristic, complemented
6#22577644C211yesmaximal, characteristic, complemented
7#22577623C211yesmaximal, characteristic, complemented
8#22577606C211yesmaximal, characteristic, complemented
9#22577562C211yesmaximal, characteristic, complemented
10#22577559C211yesmaximal, characteristic, complemented
11#22577554C211yesmaximal, characteristic, complemented
12#22577540C211yesmaximal, characteristic, complemented
13#22577504C211yesmaximal, characteristic, complemented
14#22577474C211yesmaximal, characteristic, complemented
15C22×SD16×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic
17C2×Q8×D137C2211yescharacteristic, complemented
18#17984117—24no—
19C2×Dic548C2211yescharacteristic, complemented
20C2×#14555500C2211yescharacteristic, complemented
21C2×#14555500C2211yescharacteristic, complemented
22C2×#14555498C2211yescharacteristic
23C2×#14555498C2211yescharacteristic
24C2×#14555493C2211yescharacteristic, complemented
25#17984086—21no—
26#17984086C2211yescharacteristic, complemented
27#17984075C2211yescharacteristic
28#17984069—21no—
29#17984067—21no—
30#17984067C2211yescharacteristic
31#17984067C2211yescharacteristic
32#17984064—21no—
33#17984064C2211yescharacteristic, complemented
34#17984060—21no—
35#17984059C2211yescharacteristic
36#17984057—21no—
37Q8×Dic137C2211yescharacteristic, complemented
38D4×Dic137C2211yescharacteristic
39C4×C22⋊D137—21no—
40C4×C22⋊D137C2211yescharacteristic
41C2×C22⋊Dic137—21no—
42#17984022C2211yescharacteristic, complemented
43#17984017C2211yescharacteristic, complemented
44#17984013C2211yescharacteristic, complemented
45#17984013C2211yescharacteristic, complemented
46#17984012C2211yescharacteristic, complemented
47#17984012C2211yescharacteristic, complemented
48#17984009C2211yescharacteristic
49#17984009C2211yescharacteristic
50C22⋊#11764040C2211yescharacteristic
51#17984000C2211yescharacteristic, complemented
52#17983998C2211yescharacteristic, complemented
53C22×Q8×C137C2211yescharacteristic
54C22×D4×C137C2211yescharacteristic
55C2×SD16×C137—24no—
56C2×SD16×C137C2211yescharacteristic
57C2×SD16×C137C2211yescharacteristic
58C2×SD16×C137C2211yescharacteristic
59C2×SD16×C137C2211yescharacteristic
60C22×C8×C137C2211yescharacteristic
61#14555509—22no—
62#14555509—41no—
63C2×Dic274—21no—
64C2×Dic274C2311yescharacteristic
65C2×Dic274C2311yescharacteristic
66Q8×D137—22no—
67Q8×D137—41no—
68C2×C22⋊D137—21no—
69C2×C22⋊D137—21no—
70C2×C22⋊D137D411yescharacteristic
71C22×Dic137—21no—
72C22×Dic137—21no—
73C2×C4×D137—21no—
74C2×C4×D137C2311yescharacteristic
75Dic548—22no—
76#14555500—22no—
77#14555500—41no—
78#14555498—22no—
79#14555498—41no—
80#14555493—22no—
81C2×#11764040C2311yescharacteristic
82C2×#11764040C2311yescharacteristic
83C4⋊Dic137—21no—
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⋊Dic137D411yescharacteristic
99#14555487—21no—
100#14555487D411yescharacteristic
101C4×Dic137—21no—
102C4×Dic137C2311yescharacteristic
103C24×C137—21no—
104C2×Q8×C137—21no—
105C2×Q8×C137—22no—
106C2×Q8×C137—41no—
107C2×Q8×C137C2311yescharacteristic
108C2×Q8×C137C2311yescharacteristic
109C2×D4×C137—22no—
110C2×D4×C137—21no—
111C2×D4×C137—41no—
112C2×D4×C137C2311yescharacteristic
113C2×D4×C137C2311yescharacteristic
114C22×C4×C137—21no—
115C22×C4×C137C2311yescharacteristic
116SD16×C137—22no—
117SD16×C137—22no—
118SD16×C137—22no—
119SD16×C137D414yescomplemented
120C2×C8×C137—22no—
121C2×C8×C137C2311yescharacteristic
122C2×C8×C137C2311yescharacteristic
123C22×D137C2×D411yescharacteristic
124Dic274—21no—
125Dic274—21no—
126Dic274—21no—
127Dic274—21no—
128Dic274—42no—
129C22⋊D137—22no—
130C22⋊D137—42no—
131C22⋊D137—81no—
132C2×Dic137—21no—
133C2×Dic137—21no—
134C2×Dic137—21no—
135C2×Dic137—21no—
136C2×Dic137—42no—
137C2×Dic137—81no—
138C2×Dic137C2×D411yescharacteristic
139C2×Dic137C2×D411yescharacteristic
140C2×Dic137C2×D411yescharacteristic
141C4×D137—21no—
142C4×D137—21no—
143C4×D137—42no—
144#11764040—21no—
145#11764040—21no—
146C23×C137—21no—
147C23×C137—21no—
148C23×C137—42no—
149C23×C137—42no—
150C23×C137—81no—
151C23×C137C2×D411yescharacteristic
152Q8×C137—21no—
153Q8×C137—21no—
154Q8×C137—21no—
155Q8×C137—42no—
156Q8×C137C2×D412yes—
157D4×C137—21no—
158D4×C137—21no—
159D4×C137—21no—
160D4×C137—42no—
161D4×C137C2×D412yes—
162C2×C4×C137—21no—
163C2×C4×C137—21no—
164C2×C4×C137—21no—
165C2×C4×C137—21no—
166C2×C4×C137—42no—
167C2×C4×C137C2×D411yescharacteristic
168C2×C4×C137C2411yescharacteristic
169C8×C137—21no—
170C8×C137C2×D412yes—
171C22×C137—21no—
172C22×C137—21no—
173C22×C137—22no—
174C22×C137—41no—
175C22×C137—41no—
176C22×C137—41no—
177C22×C137—41no—
178C22×C137—41no—
179C22×C137—84no—
180C22×C137#13712yescomplemented
181C22×C137C22×D411yescharacteristic
182C2×D137—21no—
183C2×D137—21no—
184C2×D137—21no—
185C4×C137—21no—
186C4×C137—22no—
187C4×C137—41no—
188C4×C137C22×D411yescharacteristic
189C4×C137H(5,𝔽2)11yescharacteristic
190Dic137—21no—
191Dic137—21no—
192Dic137—21no—
193Dic137—41no—
194Dic137—41no—
195Dic137—41no—
196Dic137—41no—
197C2×C137—22no—
198C2×C137—42no—
199C2×C137—81no—
200C2×C137D4211yescharacteristic
201C2×C137C2×#13711yescharacteristic
202C2×C137#57711yescharacteristic
203D137—41no—
204C137#326611yescharacteristic, complemented, Sylow
205#3266—1371nomaximal, Sylow
206C2×#137—1371no—
207C22×SD16D13711yescharacteristic, complemented
208D4×Q8—1371no—
209#546—1371no—
210#497—1371no—
211#468—1371no—
212#467—1371no—
213#464—1371no—
214#461—1371no—
215#452—1371no—
216#451—1371no—
217#450—1371no—
218#448—1371no—
219#440—1371no—
220#435—1371no—
221C2×#41—1371no—
222C22×Q8—1371no—
223C22×Q8C2×D13711yescharacteristic
224C22×D4C2×D13711yescharacteristic
225#137—2744no—
226C2×Q16—1371no—
227C2×Q16—1371no—
228C2×Q16—1371no—
229C2×SD16—24no—
230C2×SD16—1371no—
231C2×SD16—1371no—
232C2×SD16C2×D13711yescharacteristic
233C2×SD16C2×D13711yescharacteristic
234C2×SD16C2×D13711yescharacteristic
235C2×SD16C2×D13711yescharacteristic
236C2×M16—1371no—
237C22×C8C2×D13711yescharacteristic
238C4⋊Q8—2741no—
239C4⋊Q8—1371no—
240#124—1371no—
241#123—2741no—
242C22⋊Q8—2741no—
243C22⋊Q8—2741no—
244C22⋊Q8—1371no—
245C22⋊Q8—1371no—
246C22⋊Q8—1371no—
247#121—2741no—
248#121—1371no—
249#120—2741no—
250C4×Q8—1371no—
251C4×D4—2741no—
252C4×D4—1371no—
253C4×D4—1371no—
254C2×C22⋊C4—2741no—
255#107—1371no—
256C4⋊C8—1371no—
257#103—1371no—
258#103—1371no—
259#103—1371no—
260#103—1371no—
261#102—1371no—
262#102—1371no—
263C22⋊C8—1371no—
264C22⋊C8—1371no—
265#97—1371no—
266C24—21no—
267#41—2742no—
268#41—5481no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—2741no—
273C2×Q8—2742no—
274C2×Q8—5481no—
275C2×Q8—1371no—
276C2×Q8—1371no—
277C2×Q8C22×D13711yescharacteristic
278C2×Q8C22×D13711yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—2741no—
283C2×D4—2741no—
284C2×D4—1371no—
285C2×D4C22×D13711yescharacteristic
286C2×D4C22×D13711yescharacteristic
287C22×C4—21no—
288C22×C4—2741no—
289C22×C4—2741no—
290C22×C4—2741no—
291C22×C4—1371no—
292C22×C4C22×D13711yescharacteristic
293Q16—2742no—
294Q16—2742no—
295Q16—5481no—
296SD16—22no—
297SD16—22no—
298SD16—22no—
299SD16—2742no—
300SD16—5481no—
301SD16C22⋊D13714yescomplemented
302M16—2742no—
303C2×C8—22no—
304C2×C8—1371no—
305C2×C8—1371no—
306C2×C8C22×D13711yescharacteristic
307C2×C8C22×D13711yescharacteristic
308C4⋊C4—2741no—
309C4⋊C4—2741no—
310C4⋊C4—2741no—
311C4⋊C4—2741no—
312C4⋊C4—2741no—
313C4⋊C4—2741no—
314C4⋊C4—1371no—
315C4⋊C4—1371no—
316C4⋊C4—1371no—
317C22⋊C4—2741no—
318C22⋊C4—2741no—
319C22⋊C4—2741no—
320C22⋊C4—2741no—
321C22⋊C4—2741no—
322C22⋊C4—2741no—
323C22⋊C4—2741no—
324C22⋊C4—1371no—
325C22⋊C4—1371no—
326C42—2741no—
327C42—1371no—
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—1371no—
334C23D4×D13711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—42no—
339Q8—2741no—
340Q8—2741no—
341Q8—2741no—
342Q8—2741no—
343Q8—5482no—
344Q8C2×C22⋊D13712yes—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—2742no—
350D4—5482no—
351D4—10961no—
352D4C2×C22⋊D13712yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—2741no—
359C2×C4—2741no—
360C2×C4—2741no—
361C2×C4—2741no—
362C2×C4—2741no—
363C2×C4—2741no—
364C2×C4—5482no—
365C2×C4—5482no—
366C2×C4—10961no—
367C2×C4—1371no—
368C2×C4—1371no—
369C2×C4—1371no—
370C2×C4D4×D13711yescharacteristic
371C2×C4C23×D13711yescharacteristic
372C8—21no—
373C8—2741no—
374C8—2741no—
375C8C2×C22⋊D13712yes—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—2741no—
386C22—2741no—
387C22—2741no—
388C22#1798411712yescomplemented
389C22C2×D4×D13711yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—2741no—
394C4—2741no—
395C4—2741no—
396C4—5481no—
397C4—5481no—
398C4—5481no—
399C4—5481no—
400C4C22×C22⋊D13711yescharacteristic
401C4#1798413311yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—5481no—
406C2D4×C22⋊D13711yescharacteristic
407C2C2×#1798411711yescharacteristic
408C2#2257820911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC22×SD16×C137C2
socleC22×C137C22×D4
prime radical#22578047C2
hypercenterC22×SD16D137
hypocenterC137#3266
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#17984117C2semidirect
2. Q8×C22⋊D137C2semidirect
4. #22577794C2semidirect
5. #22577646C2semidirect
6. #22577644C2semidirect
7. #22577623C2semidirect
8. #22577606C2semidirect
9. #22577562C2semidirect
10. #22577559C2semidirect
11. #22577554C2semidirect
12. #22577540C2semidirect
13. #22577504C2semidirect
14. #22577474C2semidirect
15. C22×SD16×C137C2semidirect
17. C2×Q8×D137C22semidirect
19. C2×Dic548C22semidirect
20. C2×#14555500C22semidirect
21. C2×#14555500C22semidirect
24. C2×#14555493C22semidirect
26. #17984086C22semidirect
33. #17984064C22semidirect
37. Q8×Dic137C22semidirect
42. #17984022C22semidirect
43. #17984017C22semidirect
44. #17984013C22semidirect
45. #17984013C22semidirect
46. #17984012C22semidirect
47. #17984012C22semidirect
51. #17984000C22semidirect
52. #17983998C22semidirect
119. SD16×C137D4semidirect
180. C22×C137#137semidirect
204. C137#3266semidirect
207. C22×SD16D137semidirect
301. SD16C22⋊D137semidirect
388. C22#17984117semidirect

Maximal subgroups

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

subgroupindex
C2×#179841172
Q8×C22⋊D1372
#225780472
#225777942
#225776462
#225776442
#225776232
#225776062
#225775622
#225775592
#225775542
#225775402
#225775042
#225774742
C22×SD16×C1372
#3266137