Finite Group Databasebeta

← #28374559

Subgroups — #28374559

Order 13952.

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×#16812723C211yesmaximal, characteristic, complemented
2C2×#16812718C211yesmaximal, characteristic, complemented
3#20295680C211yesmaximal, characteristic, complemented
4#20295671C211yesmaximal, characteristic, complemented
5#20295420C211yesmaximal, characteristic, complemented
6#20295285C211yesmaximal, characteristic, complemented
7#20295256C211yesmaximal, characteristic, complemented
8#20295241C211yesmaximal, characteristic, complemented
9#20295231C211yesmaximal, characteristic, complemented
10#20295189C211yesmaximal, characteristic, complemented
11#20295189C211yesmaximal, characteristic, complemented
12#20295178C211yesmaximal, characteristic, complemented
13#20295172C211yesmaximal, characteristic, complemented
14#20295099C211yesmaximal, characteristic, complemented
15#442×C109C211yesmaximal, characteristic, complemented
16C2×#13182028C2211yescharacteristic
17C2×#13182028C2211yescharacteristic
18C2×#13182028C2211yescharacteristic, complemented
19C2×D4×D109C2211yescharacteristic
20#16812723—24no—
21#16812718—24no—
22C2×Dic436C2211yescharacteristic, complemented
23C2×Dic436C2211yescharacteristic, complemented
24C2×#13182018C2211yescharacteristic
25C2×#13182018C2211yescharacteristic
26C2×#13182017C2211yescharacteristic
27C2×#13182017C2211yescharacteristic
28C2×#13182017C2211yescharacteristic, complemented
29C2×#13182014C2211yescharacteristic
30C2×#13182012C2211yescharacteristic
31C2×C8×D109C2211yescharacteristic, complemented
32#16812699C2211yescharacteristic, complemented
33#16812687C2211yescharacteristic
34#16812683—21no—
35#16812683—21no—
36#16812683—21no—
37#16812679C2211yescharacteristic
38#16812679C2211yescharacteristic
39#16812679C2211yescharacteristic, complemented
40#16812674C2211yescharacteristic
41#16812673—21no—
42#16812673—21no—
43#16812669—21no—
44C4×D436C2211yescharacteristic
45C2×C4⋊Dic109—21no—
46C2×#13182006—21no—
47C4⋊#11724784C2211yescharacteristic, complemented
48#16812622C2211yescharacteristic, complemented
49#16812622C2211yescharacteristic, complemented
50#16812622C2211yescharacteristic, complemented
51#16812619C2211yescharacteristic
52#16812611C2211yescharacteristic
53#16812611C2211yescharacteristic, complemented
54C2×D8×C109C2211yescharacteristic
55C4×D4×C109C2211yescharacteristic
56C4×D4×C109C2211yescharacteristic, complemented
57#106×C109C2211yescharacteristic, complemented
58#102×C109C2211yescharacteristic
59#102×C109C2211yescharacteristic, complemented
60#97×C109C2211yescharacteristic, complemented
61C23×D109—21no—
62#13182028—22no—
63#13182028—22no—
64#13182028—22no—
65#13182028—41no—
66#13182028—41no—
67#13182028—41no—
68C2×Dic218C2311yescharacteristic
69C2×Dic218C2311yescharacteristic
70C2×Dic218C2311yescharacteristic
71C2×D436—21no—
72C2×D436D411yescharacteristic
73C2×D436C2311yescharacteristic
74C2×C22⋊D109—21no—
75C2×C22⋊D109—21no—
76C2×C22⋊D109—21no—
77C2×C22⋊D109—21no—
78D4×D109—22no—
79D4×D109—41no—
80C22×Dic109—21no—
81C22×Dic109—21no—
82C22×Dic109—21no—
83C2×C4×D109C2311yescharacteristic
84C2×C4×D109C2311yescharacteristic
85Dic436—22no—
86Dic436—41no—
87#13182018—22no—
88#13182018—41no—
89#13182017—22no—
90#13182017—22no—
91#13182017—22no—
92#13182014—22no—
93#13182012—22no—
94C2×#11724784C2311yescharacteristic
95C2×#11724784C2311yescharacteristic
96C8×D109—22no—
97C4⋊Dic109—21no—
98C4⋊Dic109—21no—
99C4⋊Dic109—21no—
100C4⋊Dic109—21no—
101C4⋊Dic109—21no—
102C4⋊Dic109D411yescharacteristic
103#13182006—21no—
104#13182006—21no—
105#13182006—21no—
106#13182006—21no—
107#13182006—21no—
108#13182006—21no—
109#13182006D411yescharacteristic
110#13182006D411yescharacteristic
111C2×D4×C109C2311yescharacteristic
112C2×D4×C109C2311yescharacteristic
113C22×C4×C109—21no—
114C22×C4×C109—21no—
115D8×C109D414yescomplemented
116C2×C8×C109C2311yescharacteristic
117C2×C8×C109C2311yescharacteristic
118C4⋊C4×C109C2311yescharacteristic
119C4⋊C4×C109C2311yescharacteristic
120C22⋊C4×C109—21no—
121C22⋊C4×C109—21no—
122C42×C109C2311yescharacteristic
123C23×C109—21no—
124C23×C109—21no—
125C22×D109—21no—
126C22×D109—42no—
127C22×D109—42no—
128C22×D109—81no—
129C22×D109C2×D411yescharacteristic
130C22×D109C2×D411yescharacteristic
131D4×C109—21no—
132D4×C109—21no—
133D4×C109C2×D412yes—
134D4×C109C2×D412yes—
135C2×C4×C109—21no—
136C2×C4×C109—21no—
137C2×C4×C109—42no—
138C2×C4×C109—81no—
139C2×C4×C109C2×D411yescharacteristic
140C2×C4×C109C2×D411yescharacteristic
141C2×C4×C109C2411yescharacteristic
142C22⋊D109—42no—
143C22⋊D109—42no—
144C22⋊D109—42no—
145C22⋊D109—42no—
146C2×Dic109—21no—
147C2×Dic109—21no—
148C2×Dic109—21no—
149C2×Dic109—42no—
150C2×Dic109—42no—
151C2×Dic109—42no—
152C2×Dic109C2×D411yescharacteristic
153C2×Dic109C2×D411yescharacteristic
154D436—21no—
155D436—21no—
156D436—41no—
157D436—81no—
158C4×D109—21no—
159C4×D109—21no—
160C4×D109—21no—
161C4×D109—21no—
162Dic218—21no—
163Dic218—21no—
164Dic218—21no—
165Dic218—21no—
166Dic218—21no—
167Dic218—21no—
168C8×C109—21no—
169C8×C109C2×D412yes—
170#11724784—21no—
171#11724784—21no—
172C22×C109—22no—
173C22×C109—22no—
174C22×C109—41no—
175C22×C109—41no—
176C22×C109—41no—
177C22×C109—41no—
178C22×C109C22×D411yescharacteristic
179C2×D109—21no—
180C2×D109—21no—
181C2×D109—21no—
182C2×D109—21no—
183C2×D109—21no—
184C2×D109—21no—
185C2×D109—41no—
186C2×D109—41no—
187C2×D109—41no—
188C2×D109—84no—
189C4×C109—21no—
190C4×C109—21no—
191C4×C109—41no—
192C4×C109—41no—
193C4×C109C22×D411yescharacteristic
194C4×C109H(5,𝔽2)11yescharacteristic
195Dic109—21no—
196Dic109—21no—
197Dic109—41no—
198Dic109—41no—
199Dic109—41no—
200C2×C109—42no—
201C2×C109—42no—
202C2×C109D4211yescharacteristic
203C2×C109#57511yescharacteristic
204C2×C109#57711yescharacteristic
205D109—41no—
206D109—41no—
207D109—81no—
208#3267—1091nomaximal, Sylow
209C109#326711yescharacteristic, complemented, Sylow
210C2×AGL(1,ℤ/8)—1091no—
211C2×#135—1091no—
212#547—1091no—
213#546—1091no—
214#497—1091no—
215#470—1091no—
216#465—1091no—
217#463—1091no—
218#461—1091no—
219#452—1091no—
220#452—1091no—
221#450—1091no—
222#449—1091no—
223#442D10911yescharacteristic, complemented
224#435—1091no—
225C2×#41—1091no—
226C2×#41—1091no—
227C2×#41—1091no—
228C22×D4—1091no—
229AGL(1,ℤ/8)—2184no—
230#135—2184no—
231C2×Q16—1091no—
232C2×Q16—1091no—
233C2×SD16—1091no—
234C2×SD16—1091no—
235C2×SD16—1091no—
236C2×SD16—1091no—
237C2×SD16—1091no—
238C2×D8—1091no—
239C2×D8C2×D10911yescharacteristic
240C2×M16—1091no—
241C22×C8—1091no—
242C4⋊Q8—1091no—
243#124—1091no—
244#123—2181no—
245#123—2181no—
246#123—2181no—
247C22⋊Q8—1091no—
248C22⋊Q8—1091no—
249C22⋊Q8—1091no—
250#121—1091no—
251#121—2181no—
252#121—2181no—
253#120—2181no—
254C4×D4—1091no—
255C4×D4C2×D10911yescharacteristic
256C4×D4C2×D10911yescharacteristic, complemented
257C2×C4⋊C4—2181no—
258C2×C22⋊C4—2181no—
259#106C2×D10911yescharacteristic, complemented
260C4⋊C8—1091no—
261#103—1091no—
262#103—1091no—
263#103—1091no—
264#102—1091no—
265#102C2×D10911yescharacteristic
266#102C2×D10911yescharacteristic, complemented
267C22⋊C8—1091no—
268C22⋊C8—1091no—
269#97C2×D10911yescharacteristic, complemented
270C24—2181no—
271#41—4361no—
272#41—4361no—
273#41—4361no—
274#41—2182no—
275#41—2182no—
276#41—2182no—
277C2×Q8—1091no—
278C2×Q8—1091no—
279C2×Q8—1091no—
280C2×D4—1091no—
281C2×D4—1091no—
282C2×D4—4361no—
283C2×D4—2181no—
284C2×D4—2181no—
285C2×D4—2181no—
286C2×D4—2181no—
287C2×D4—2181no—
288C2×D4—2182no—
289C2×D4C22×D10911yescharacteristic
290C2×D4C22×D10911yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—1091no—
294C22×C4—1091no—
295C22×C4—2181no—
296C22×C4—2181no—
297C22×C4—2181no—
298Q16—4361no—
299Q16—2182no—
300SD16—4361no—
301SD16—2182no—
302SD16—2182no—
303SD16—2182no—
304SD16—2182no—
305D8—2182no—
306D8D43614yescomplemented
307M16—2182no—
308C2×C8—1091no—
309C2×C8—1091no—
310C2×C8—2182no—
311C2×C8C22×D10911yescharacteristic
312C2×C8C22×D10911yescharacteristic
313C4⋊C4—1091no—
314C4⋊C4—2181no—
315C4⋊C4—2181no—
316C4⋊C4—2181no—
317C4⋊C4—2181no—
318C4⋊C4—2181no—
319C4⋊C4C22×D10911yescharacteristic
320C4⋊C4C22×D10911yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—1091no—
324C22⋊C4—1091no—
325C22⋊C4—2181no—
326C22⋊C4—2181no—
327C22⋊C4—2181no—
328C22⋊C4—2181no—
329C22⋊C4—2181no—
330C22⋊C4—2181no—
331C42C22×D10911yescharacteristic
332C23—21no—
333C23—21no—
334C23—8721no—
335C23—1091no—
336C23—1091no—
337C23—4362no—
338C23—4362no—
339C23—2181no—
340Q8—2181no—
341Q8—2181no—
342Q8—2181no—
343Q8—2181no—
344Q8—2181no—
345Q8—2181no—
346D4—21no—
347D4—21no—
348D4—8721no—
349D4—4361no—
350D4—4362no—
351D4—4362no—
352D4—4362no—
353D4—4362no—
354D4—2181no—
355D4—2181no—
356D4C2×D43612yes—
357D4C2×D43612yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—1091no—
363C2×C4—1091no—
364C2×C4—4362no—
365C2×C4—4362no—
366C2×C4—4362no—
367C2×C4—2181no—
368C2×C4—2181no—
369C2×C4—2181no—
370C2×C4—2181no—
371C2×C4—2181no—
372C2×C4—2181no—
373C2×C4—2181no—
374C2×C4D4×D10911yescharacteristic
375C2×C4D4×D10911yescharacteristic
376C2×C4C23×D10911yescharacteristic
377C8—21no—
378C8—2181no—
379C8—2181no—
380C8C2×D43612yes—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—8724no—
388C22—4361no—
389C22—4361no—
390C22—4361no—
391C22—2181no—
392C22—2181no—
393C22—2181no—
394C22—2181no—
395C22—2181no—
396C22—2181no—
397C22C2×D4×D10911yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—4361no—
403C4—4361no—
404C4—4361no—
405C4—2181no—
406C4—2181no—
407C4C22×D43611yescharacteristic
408C4#1681274511yescharacteristic
409C2—42no—
410C2—42no—
411C2—8721no—
412C2—4361no—
413C2—4361no—
414C2D4×D43611yescharacteristic
415C2#2029582211yescharacteristic
416C2#2029583611yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#442×C109C2
socleC22×C109C22×D4
hypercenter#442D109
hypocenterC109#3267
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#16812723C2semidirect
2. C2×#16812718C2semidirect
3. #20295680C2semidirect
4. #20295671C2semidirect
5. #20295420C2semidirect
6. #20295285C2semidirect
7. #20295256C2semidirect
8. #20295241C2semidirect
9. #20295231C2semidirect
10. #20295189C2semidirect
11. #20295189C2semidirect
12. #20295178C2semidirect
13. #20295172C2semidirect
14. #20295099C2semidirect
15. #442×C109C2semidirect
18. C2×#13182028C22semidirect
22. C2×Dic436C22semidirect
23. C2×Dic436C22semidirect
28. C2×#13182017C22semidirect
31. C2×C8×D109C22semidirect
32. #16812699C22semidirect
39. #16812679C22semidirect
47. C4⋊#11724784C22semidirect
48. #16812622C22semidirect
49. #16812622C22semidirect
50. #16812622C22semidirect
53. #16812611C22semidirect
56. C4×D4×C109C22semidirect
57. #106×C109C22semidirect
59. #102×C109C22semidirect
60. #97×C109C22semidirect
115. D8×C109D4semidirect
209. C109#3267semidirect
223. #442D109semidirect
256. C4×D4C2×D109semidirect
259. #106C2×D109semidirect
266. #102C2×D109semidirect
269. #97C2×D109semidirect
306. D8D436semidirect

Maximal subgroups

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

subgroupindex
C2×#168127232
C2×#168127182
#202956802
#202956712
#202954202
#202952852
#202952562
#202952412
#202952312
#202951892
#202951782
#202951722
#202950992
#442×C1092
#3267109