Finite Group Databasebeta

← #28372335

Subgroups — #28372335

Order 13952.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812744C211yesmaximal, characteristic, complemented
2C2×#16812719C211yesmaximal, characteristic, complemented
3#20295604C211yesmaximal, characteristic, complemented
4#20295185C212yesmaximal, complemented
5#20295175C212yesmaximal, complemented
6#20295168C212yesmaximal, complemented
7#20295158C212yesmaximal, complemented
8#20295028C211yesmaximal, characteristic, complemented
9C2×#16812620C211yesmaximal, characteristic, complemented
10#20294994C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C109C211yesmaximal, characteristic, complemented
12#16812744—24no—
13#16812744—24no—
14C2×#13182029C2211yescharacteristic
15C2×#13182029C2211yescharacteristic
16C2×#13182028—21no—
17C2×#13182028C2212yes—
18C22×C22⋊D109—22no—
19C2×D4×D109—21no—
20C2×D4×D109C2212yes—
21#16812719—24no—
22C2×#13182019C2211yescharacteristic, complemented
23C2×#13182017C2211yescharacteristic, complemented
24C2×#13182016C2211yescharacteristic, complemented
25C2×#13182014C2211yescharacteristic, complemented
26C22×#11724784C2211yescharacteristic, complemented
27#16812678—21no—
28#16812675C2212yescomplemented
29#16812673—21no—
30#16812671—22no—
31#16812670C2212yescomplemented
32C2×C4⋊Dic109C2211yescharacteristic, complemented
33#16812624C2212yescomplemented
34#16812621C2212yescomplemented
35#16812620C2212yescomplemented
36#16812620C2212yescomplemented
37#16812611C2212yescomplemented
38#16812611C2212yescomplemented
39C2×#41×C109C2211yescharacteristic, complemented
40C22×D4×C109C2211yescharacteristic
41AGL(1,ℤ/8)×C109—24no—
42C2×SD16×C109C2212yescomplemented
43C2×D8×C109C2212yescomplemented
44C2×M16×C109C2211yescharacteristic, complemented
45C23×D109—22no—
46#13182029—22no—
47#13182029—22no—
48#13182029—22no—
49#13182029D414yescomplemented
50#13182028—28no—
51#13182028—24no—
52#13182028—24no—
53C2×Dic218D411yescharacteristic
54C2×Dic218C2311yescharacteristic
55C2×D436D411yescharacteristic
56C2×D436C2311yescharacteristic
57C2×C22⋊D109—216no—
58C2×C22⋊D109—21no—
59C2×C22⋊D109—22no—
60C2×C22⋊D109—22no—
61C2×C22⋊D109—22no—
62C2×C22⋊D109D412yes—
63D4×D109—28no—
64D4×D109—24no—
65D4×D109—24no—
66C22×Dic109—22no—
67C22×Dic109—21no—
68C2×C4×D109—21no—
69C2×C4×D109C2312yes—
70#13182019—22no—
71#13182017—22no—
72#13182016—22no—
73#13182014—22no—
74C2×#11724784—22no—
75C2×#11724784C2311yescharacteristic
76C2×#11724784C2311yescharacteristic
77C4⋊Dic109—21no—
78C4⋊Dic109C2312yes—
79#13182008—22no—
80C22⋊Dic109—22no—
81#13182006—22no—
82C24×C109—21no—
83#41×C109—22no—
84#41×C109—22no—
85C2×Q8×C109C2311yescharacteristic
86C2×D4×C109—21no—
87C2×D4×C109—21no—
88C2×D4×C109—22no—
89C2×D4×C109D414yescomplemented
90C2×D4×C109C2311yescharacteristic
91C2×D4×C109C2312yes—
92C22×C4×C109—21no—
93C22×C4×C109C2311yescharacteristic
94SD16×C109—24no—
95D8×C109—24no—
96M16×C109—22no—
97C2×C8×C109C2312yes—
98C23×C109—22no—
99C23×C109—24no—
100C23×C109—21no—
101C23×C109—81no—
102C23×C109—81no—
103C23×C109C2×D411yescharacteristic
104C22×D109—21no—
105C22×D109—28no—
106C22×D109—82no—
107C22×D109—82no—
108C22×D109C2×D412yes—
109Q8×C109—21no—
110Q8×C109—21no—
111D4×C109—24no—
112D4×C109—22no—
113D4×C109—21no—
114D4×C109—21no—
115D4×C109—42no—
116D4×C109C2×D414yes—
117C2×C4×C109—21no—
118C2×C4×C109—42no—
119C2×C4×C109C2×D411yescharacteristic
120C2×C4×C109C2×D412yes—
121C2×C4×C109C2×D412yes—
122C2×C4×C109C2411yescharacteristic
123C22⋊D109—28no—
124C22⋊D109—28no—
125C22⋊D109—24no—
126C22⋊D109—24no—
127C22⋊D109—82no—
128C2×Dic109—28no—
129C2×Dic109—21no—
130C2×Dic109—22no—
131C2×Dic109—81no—
132C2×Dic109C2×D412yes—
133D436—22no—
134D436—21no—
135D436C2×D412yes—
136C4×D109—24no—
137C4×D109—22no—
138C4×D109—22no—
139Dic218—22no—
140Dic218—21no—
141Dic218C2×D412yes—
142C8×C109—22no—
143#11724784—21no—
144#11724784—21no—
145C22×C109—21no—
146C22×C109—21no—
147C22×C109—24no—
148C22×C109—42no—
149C22×C109—42no—
150C22×C109—41no—
151C22×C109—41no—
152C22×C109—41no—
153C22×C109—84no—
154C22×C109#12012yes—
155C22×C109C22×D411yescharacteristic
156C2×D109—22no—
157C2×D109—22no—
158C2×D109—22no—
159C2×D109—22no—
160C2×D109—41no—
161C2×D109—41no—
162C2×D109—88no—
163C4×C109—41no—
164C4×C109#12012yes—
165C4×C109C22×D411yescharacteristic
166C4×C109C22×D411yescharacteristic
167Dic109—22no—
168Dic109—22no—
169Dic109—42no—
170C2×C109—22no—
171C2×C109—44no—
172C2×C109—81no—
173C2×C109C2×#12011yescharacteristic
174C2×C109#57611yescharacteristic
175C2×C109#57711yescharacteristic
176D109—42no—
177D109—42no—
178#3012—1091nomaximal, Sylow
179C109#301211yescharacteristic, complemented, Sylow
180C2×H(5,𝔽2)—1091no—
181C2×AGL(1,ℤ/8)D10911yescharacteristic, complemented
182C2×#135—1091no—
183#537—1091no—
184#452—1092no—
185#450—1092no—
186#449—1092no—
187#447—1092no—
188#417—1091no—
189C2×#102—1091no—
190#408—1091no—
191H(5,𝔽2)—2184no—
192H(5,𝔽2)—2184no—
193C2×#41—1091no—
194C2×#41—1092no—
195C2×#41—1091no—
196C2×#41—2181no—
197C2×#41C2×D10911yescharacteristic, complemented
198C22×D4—1092no—
199C22×D4—2181no—
200C22×D4—2182no—
201C22×D4C2×D10911yescharacteristic
202AGL(1,ℤ/8)—24no—
203#135—2184no—
204C2×Q16—1091no—
205C2×SD16—1091no—
206C2×SD16—1091no—
207C2×SD16C2×D10912yescomplemented
208C2×D8—1091no—
209C2×D8C2×D10912yescomplemented
210C2×M16C2×D10911yescharacteristic, complemented
211C22×C8—1091no—
212C22⋊Q8—1092no—
213C22⋊Q8—2181no—
214#121—1092no—
215#121—2181no—
216#121—2182no—
217C2×C4⋊C4—1091no—
218#103—1092no—
219#102—1092no—
220#102—1092no—
221#102—1092no—
222C22⋊C8—1092no—
223C22⋊C8—1092no—
224C24—21no—
225C24—2182no—
226#41—22no—
227#41—22no—
228#41—1094no—
229#41—2184no—
230#41—2188no—
231#41—2184no—
232#41—2182no—
233#41—2182no—
234#41—2182no—
235C2×Q8—1091no—
236C2×Q8—1091no—
237C2×Q8C22×D10911yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—1092no—
242C2×D4—1091no—
243C2×D4—1091no—
244C2×D4—2182no—
245C2×D4—2184no—
246C2×D4—2188no—
247C2×D4—2182no—
248C2×D4—2182no—
249C2×D4—21816no—
250C2×D4—2181no—
251C2×D4—2184no—
252C2×D4C22⋊D10914yescomplemented
253C2×D4C22×D10911yescharacteristic
254C2×D4C22×D10912yes—
255C22×C4—21no—
256C22×C4—1092no—
257C22×C4—2181no—
258C22×C4—2182no—
259C22×C4—2181no—
260C22×C4C22×D10911yescharacteristic
261Q16—2182no—
262SD16—24no—
263SD16—2182no—
264SD16—2182no—
265D8—24no—
266D8—2182no—
267M16—22no—
268C2×C8—1091no—
269C2×C8—1091no—
270C2×C8—2182no—
271C2×C8C22×D10912yes—
272C4⋊C4—1092no—
273C4⋊C4—2182no—
274C4⋊C4—2181no—
275C22⋊C4—2182no—
276C22⋊C4—2182no—
277C23—22no—
278C23—24no—
279C23—21no—
280C23—81no—
281C23—81no—
282C23—8722no—
283C23—8722no—
284C23—1092no—
285C23—2188no—
286C23—2181no—
287C23D4×D10911yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—1092no—
291Q8—2182no—
292Q8—2181no—
293D4—24no—
294D4—22no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—8722no—
299D4—1092no—
300D4—2188no—
301D4—2188no—
302D4—2184no—
303D4—2184no—
304D4—2182no—
305D4—2181no—
306D4C2×C22⋊D10914yes—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—8721no—
310C2×C4—1092no—
311C2×C4—2182no—
312C2×C4—2188no—
313C2×C4—2181no—
314C2×C4—2184no—
315C2×C4—2182no—
316C2×C4—2182no—
317C2×C4D4×D10911yescharacteristic
318C2×C4D4×D10912yes—
319C2×C4C2×C22⋊D10912yes—
320C2×C4C23×D10911yescharacteristic
321C8—22no—
322C8—2181no—
323C8—2181no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—84no—
333C22—8728no—
334C22—4361no—
335C22—4361no—
336C22—2182no—
337C22—2182no—
338C22—2182no—
339C22—2182no—
340C22#1681266712yes—
341C22C2×D4×D10911yescharacteristic
342C4—41no—
343C4—4362no—
344C4—2182no—
345C4—2182no—
346C4#1681266712yes—
347C4C2×D4×D10911yescharacteristic
348C4C22×C22⋊D10911yescharacteristic
349C2—22no—
350C2—44no—
351C2—81no—
352C2—4362no—
353C2—4362no—
354C2C2×#1681266711yescharacteristic
355C2#2029583011yescharacteristic
356C2#2029583611yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC2×AGL(1,ℤ/8)×C109C2
socleC22×C109C22×D4
hypercenterC2×AGL(1,ℤ/8)D109
hypocenterC109#3012
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
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×#16812744C2semidirect
2. C2×#16812719C2semidirect
3. #20295604C2semidirect
4. #20295185C2semidirect
5. #20295175C2semidirect
6. #20295168C2semidirect
7. #20295158C2semidirect
8. #20295028C2semidirect
9. C2×#16812620C2semidirect
10. #20294994C2semidirect
11. C2×AGL(1,ℤ/8)×C109C2semidirect
22. C2×#13182019C22semidirect
23. C2×#13182017C22semidirect
24. C2×#13182016C22semidirect
25. C2×#13182014C22semidirect
26. C22×#11724784C22semidirect
28. #16812675C22semidirect
31. #16812670C22semidirect
32. C2×C4⋊Dic109C22semidirect
33. #16812624C22semidirect
34. #16812621C22semidirect
35. #16812620C22semidirect
36. #16812620C22semidirect
37. #16812611C22semidirect
38. #16812611C22semidirect
39. C2×#41×C109C22semidirect
42. C2×SD16×C109C22semidirect
43. C2×D8×C109C22semidirect
44. C2×M16×C109C22semidirect
49. #13182029D4semidirect
89. C2×D4×C109D4semidirect
179. C109#3012semidirect
181. C2×AGL(1,ℤ/8)D109semidirect
197. C2×#41C2×D109semidirect
207. C2×SD16C2×D109semidirect
209. C2×D8C2×D109semidirect
210. C2×M16C2×D109semidirect
252. C2×D4C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127442
C2×#168127192
#202956042
#202951852
#202951752
#202951682
#202951582
#202950282
C2×#168126202
#202949942
C2×AGL(1,ℤ/8)×C1092
#3012109