Finite Group Databasebeta

← #28374551

Subgroups — #28374551

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×#16812718C211yesmaximal, characteristic, complemented
2#20295675C211yesmaximal, characteristic, complemented
3#20295670C211yesmaximal, characteristic, complemented
4#20295417C211yesmaximal, characteristic, complemented
5#20295284C211yesmaximal, characteristic, complemented
6#20295252C211yesmaximal, characteristic, complemented
7#20295239C211yesmaximal, characteristic, complemented
8#20295229C211yesmaximal, characteristic, complemented
9#20295188C211yesmaximal, characteristic, complemented
10#20295185C211yesmaximal, characteristic, complemented
11#20295177C211yesmaximal, characteristic, complemented
12#20295168C211yesmaximal, characteristic, complemented
13#20295138C211yesmaximal, characteristic, complemented
14#20295095C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C109C211yesmaximal, characteristic, complemented
16C2×#13182028C2211yescharacteristic
17C2×#13182028C2211yescharacteristic, complemented
18#16812718—24no—
19C2×Dic436C2211yescharacteristic, complemented
20C2×#13182019C2211yescharacteristic, complemented
21C2×#13182017C2211yescharacteristic
22C2×#13182017C2211yescharacteristic
23C2×#13182017C2211yescharacteristic, complemented
24C2×C8×D109C2211yescharacteristic, complemented
25#16812697C2211yescharacteristic, complemented
26#16812686C2211yescharacteristic
27#16812682—21no—
28#16812680—21no—
29#16812680—21no—
30#16812678C2211yescharacteristic
31#16812678C2211yescharacteristic
32#16812675C2211yescharacteristic, complemented
33#16812671—21no—
34#16812671—21no—
35#16812670C2211yescharacteristic
36#16812668—21no—
37D4×Dic109C2211yescharacteristic
38D4×Dic109C2211yescharacteristic, complemented
39C4×C22⋊D109C2211yescharacteristic
40C2×#13182008—21no—
41C2×C22⋊Dic109—21no—
42#16812631C2211yescharacteristic, complemented
43#16812628C2211yescharacteristic, complemented
44#16812624C2211yescharacteristic, complemented
45#16812624C2211yescharacteristic, complemented
46#16812623C2211yescharacteristic, complemented
47#16812620C2211yescharacteristic
48#16812620C2211yescharacteristic
49#16812620C2211yescharacteristic, complemented
50C22⋊#11724784C2211yescharacteristic
51#16812611C2211yescharacteristic, complemented
52#16812609C2211yescharacteristic, complemented
53C2×#41×C109C2211yescharacteristic
54C22×D4×C109C2211yescharacteristic
55AGL(1,ℤ/8)×C109—24no—
56C2×SD16×C109C2211yescharacteristic
57C2×SD16×C109C2211yescharacteristic
58C2×D8×C109C2211yescharacteristic
59C2×D8×C109C2211yescharacteristic
60C2×M16×C109C2211yescharacteristic
61#13182028—22no—
62#13182028—22no—
63#13182028—41no—
64#13182028—41no—
65C2×Dic218C2311yescharacteristic
66C2×Dic218C2311yescharacteristic
67C2×C22⋊D109—21no—
68C2×C22⋊D109—21no—
69C2×C22⋊D109—21no—
70C2×C22⋊D109D411yescharacteristic
71C22×Dic109—21no—
72C22×Dic109—21no—
73C22×Dic109—21no—
74C22×Dic109—21no—
75C2×C4×D109C2311yescharacteristic
76Dic436—22no—
77#13182019—41no—
78#13182017—22no—
79#13182017—22no—
80#13182017—41no—
81C2×#11724784C2311yescharacteristic
82C2×#11724784C2311yescharacteristic
83C8×D109—22no—
84C4⋊Dic109C2311yescharacteristic
85C4⋊Dic109C2311yescharacteristic
86#13182008—21no—
87#13182008—21no—
88#13182008—21no—
89#13182008—21no—
90#13182008—21no—
91#13182008D411yescharacteristic
92C22⋊Dic109—21no—
93C22⋊Dic109—21no—
94C22⋊Dic109—21no—
95C22⋊Dic109—21no—
96C22⋊Dic109—21no—
97C22⋊Dic109—21no—
98C22⋊Dic109—21no—
99C22⋊Dic109D411yescharacteristic
100#13182006—21no—
101#13182006D411yescharacteristic
102C4×Dic109C2311yescharacteristic
103C24×C109—21no—
104#41×C109—22no—
105#41×C109—41no—
106C2×Q8×C109C2311yescharacteristic
107C2×D4×C109—22no—
108C2×D4×C109—21no—
109C2×D4×C109—21no—
110C2×D4×C109—41no—
111C2×D4×C109C2311yescharacteristic
112C2×D4×C109C2311yescharacteristic
113C2×D4×C109C2311yescharacteristic
114C22×C4×C109—21no—
115C22×C4×C109C2311yescharacteristic
116SD16×C109—22no—
117SD16×C109—22no—
118D8×C109—22no—
119D8×C109D414yescomplemented
120M16×C109—22no—
121C2×C8×C109C2311yescharacteristic
122C2×C8×C109C2311yescharacteristic
123C23×C109—21no—
124C23×C109—21no—
125C23×C109—21no—
126C23×C109—42no—
127C23×C109—42no—
128C23×C109—81no—
129C23×C109C2×D411yescharacteristic
130C22×D109C2×D411yescharacteristic
131Q8×C109—21no—
132Q8×C109—21no—
133D4×C109—21no—
134D4×C109—21no—
135D4×C109—21no—
136D4×C109—21no—
137D4×C109—42no—
138D4×C109—42no—
139D4×C109C2×D412yes—
140D4×C109C2×D412yes—
141C2×C4×C109—21no—
142C2×C4×C109—21no—
143C2×C4×C109—21no—
144C2×C4×C109—42no—
145C2×C4×C109C2×D411yescharacteristic
146C2×C4×C109C2411yescharacteristic
147C22⋊D109—41no—
148C22⋊D109—42no—
149C22⋊D109—42no—
150C22⋊D109—81no—
151C2×Dic109—21no—
152C2×Dic109—21no—
153C2×Dic109—21no—
154C2×Dic109—21no—
155C2×Dic109—42no—
156C2×Dic109—42no—
157C2×Dic109—42no—
158C2×Dic109—81no—
159C2×Dic109C2×D411yescharacteristic
160C2×Dic109C2×D411yescharacteristic
161C2×Dic109C2×D411yescharacteristic
162C4×D109—21no—
163C4×D109—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—21no—
175C22×C109—21no—
176C22×C109—21no—
177C22×C109—41no—
178C22×C109—41no—
179C22×C109—41no—
180C22×C109—41no—
181C22×C109—41no—
182C22×C109—41no—
183C22×C109—41no—
184C22×C109—84no—
185C22×C109C22×D411yescharacteristic
186C2×D109—21no—
187C2×D109—21no—
188C2×D109—21no—
189C4×C109—21no—
190C4×C109—41no—
191C4×C109C22×D411yescharacteristic
192C4×C109H(5,𝔽2)11yescharacteristic
193Dic109—21no—
194Dic109—21no—
195Dic109—21no—
196Dic109—41no—
197Dic109—41no—
198Dic109—41no—
199Dic109—41no—
200C2×C109—41no—
201C2×C109—42no—
202C2×C109—42no—
203C2×C109—81no—
204C2×C109D4211yescharacteristic
205C2×C109#57511yescharacteristic
206C2×C109#57711yescharacteristic
207D109—41no—
208#3267—1091nomaximal, Sylow
209C109#326711yescharacteristic, complemented, Sylow
210C2×AGL(1,ℤ/8)D10911yescharacteristic, complemented
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#442—1091no—
224#435—1091no—
225C2×#41—1091no—
226C2×#41—1091no—
227C2×#41C2×D10911yescharacteristic
228C22×D4C2×D10911yescharacteristic
229AGL(1,ℤ/8)—24no—
230#135—2184no—
231C2×Q16—1091no—
232C2×Q16—1091no—
233C2×SD16—1091no—
234C2×SD16—1091no—
235C2×SD16—1091no—
236C2×SD16C2×D10911yescharacteristic
237C2×SD16C2×D10911yescharacteristic
238C2×D8C2×D10911yescharacteristic
239C2×D8C2×D10911yescharacteristic
240C2×M16C2×D10911yescharacteristic
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×D4—1091no—
256C4×D4—1091no—
257C2×C4⋊C4—2181no—
258C2×C22⋊C4—2181no—
259#106—1091no—
260C4⋊C8—1091no—
261#103—1091no—
262#103—1091no—
263#103—1091no—
264#102—1091no—
265#102—1091no—
266#102—1091no—
267C22⋊C8—1091no—
268C22⋊C8—1091no—
269#97—1091no—
270C24—21no—
271#41—22no—
272#41—41no—
273#41—4361no—
274#41—4361no—
275#41—2182no—
276#41—2182no—
277C2×Q8—1091no—
278C2×Q8—1091no—
279C2×Q8C22×D10911yescharacteristic
280C2×D4—22no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4—1091no—
285C2×D4—2181no—
286C2×D4—2181no—
287C2×D4—2181no—
288C2×D4C22×D10911yescharacteristic
289C2×D4C22×D10911yescharacteristic
290C2×D4C22×D10911yescharacteristic
291C22×C4—21no—
292C22×C4—1091no—
293C22×C4—2181no—
294C22×C4—2181no—
295C22×C4—2181no—
296C22×C4—2181no—
297C22×C4C22×D10911yescharacteristic
298Q16—4361no—
299Q16—2182no—
300SD16—22no—
301SD16—22no—
302SD16—4361no—
303SD16—2182no—
304SD16—2182no—
305D8—22no—
306D8C22⋊D10914yescomplemented
307M16—22no—
308C2×C8—1091no—
309C2×C8—1091no—
310C2×C8—2182no—
311C2×C8C22×D10911yescharacteristic
312C2×C8C22×D10911yescharacteristic
313C4⋊C4—1091no—
314C4⋊C4—1091no—
315C4⋊C4—1091no—
316C4⋊C4—2181no—
317C4⋊C4—2181no—
318C4⋊C4—2181no—
319C4⋊C4—2181no—
320C4⋊C4—2181no—
321C22⋊C4—1091no—
322C22⋊C4—1091no—
323C22⋊C4—2181no—
324C22⋊C4—2181no—
325C22⋊C4—2181no—
326C22⋊C4—2181no—
327C22⋊C4—2181no—
328C22⋊C4—2181no—
329C22⋊C4—2181no—
330C22⋊C4—2181no—
331C42—1091no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23—1091no—
339C23D4×D10911yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—2181no—
343Q8—2181no—
344Q8—2181no—
345Q8—2181no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—42no—
351D4—42no—
352D4—8721no—
353D4—4362no—
354D4—4362no—
355D4—4361no—
356D4C2×C22⋊D10912yes—
357D4C2×C22⋊D10912yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—8721no—
363C2×C4—1091no—
364C2×C4—1091no—
365C2×C4—1091no—
366C2×C4—4362no—
367C2×C4—4362no—
368C2×C4—4362no—
369C2×C4—2181no—
370C2×C4—2181no—
371C2×C4—2181no—
372C2×C4—2181no—
373C2×C4—2181no—
374C2×C4—2181no—
375C2×C4D4×D10911yescharacteristic
376C2×C4C23×D10911yescharacteristic
377C8—21no—
378C8—2181no—
379C8—2181no—
380C8C2×C22⋊D10912yes—
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—2181no—
395C22—2181no—
396C22—2181no—
397C22C2×D4×D10911yescharacteristic
398C4—21no—
399C4—41no—
400C4—4361no—
401C4—4361no—
402C4—4361no—
403C4—4361no—
404C4—2181no—
405C4—2181no—
406C4—2181no—
407C4C22×C22⋊D10911yescharacteristic
408C4#1681274411yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—81no—
413C2—4361no—
414C2D4×C22⋊D10911yescharacteristic
415C2#2029582211yescharacteristic
416C2#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#3267
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×#16812718C2semidirect
2. #20295675C2semidirect
3. #20295670C2semidirect
4. #20295417C2semidirect
5. #20295284C2semidirect
6. #20295252C2semidirect
7. #20295239C2semidirect
8. #20295229C2semidirect
9. #20295188C2semidirect
10. #20295185C2semidirect
11. #20295177C2semidirect
12. #20295168C2semidirect
13. #20295138C2semidirect
14. #20295095C2semidirect
15. C2×AGL(1,ℤ/8)×C109C2semidirect
17. C2×#13182028C22semidirect
19. C2×Dic436C22semidirect
20. C2×#13182019C22semidirect
23. C2×#13182017C22semidirect
24. C2×C8×D109C22semidirect
25. #16812697C22semidirect
32. #16812675C22semidirect
38. D4×Dic109C22semidirect
42. #16812631C22semidirect
43. #16812628C22semidirect
44. #16812624C22semidirect
45. #16812624C22semidirect
46. #16812623C22semidirect
49. #16812620C22semidirect
51. #16812611C22semidirect
52. #16812609C22semidirect
119. D8×C109D4semidirect
209. C109#3267semidirect
210. C2×AGL(1,ℤ/8)D109semidirect
306. D8C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127182
#202956752
#202956702
#202954172
#202952842
#202952522
#202952392
#202952292
#202951882
#202951852
#202951772
#202951682
#202951382
#202950952
C2×AGL(1,ℤ/8)×C1092
#3267109