Finite Group Databasebeta

← #28374543

Subgroups — #28374543

Order 13952.

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×#16812730C211yesmaximal, characteristic, complemented
2C22×#13182016C211yesmaximal, characteristic, complemented
3#20295672C211yesmaximal, characteristic
4#20295418C211yesmaximal, characteristic, complemented
5#20295274C211yesmaximal, characteristic, complemented
6#20295265C211yesmaximal, characteristic, complemented
7#20295248C211yesmaximal, characteristic, complemented
8#20295228C211yesmaximal, characteristic, complemented
9#20295186C211yesmaximal, characteristic, complemented
10#20295182C211yesmaximal, characteristic, complemented
11#20295176C211yesmaximal, characteristic, complemented
12#20295165C211yesmaximal, characteristic, complemented
13#20295124C211yesmaximal, characteristic, complemented
14#20295098C211yesmaximal, characteristic, complemented
15D4×Q8×C109C211yesmaximal, characteristic, complemented
16C2×#13182029C2211yescharacteristic
17C22×D436C2211yescharacteristic
18#16812730—24no—
19C2×#13182019C2211yescharacteristic, complemented
20C2×#13182019C2211yescharacteristic, complemented
21C2×#13182019C2211yescharacteristic, complemented
22C2×#13182016—24no—
23C2×#13182016C2211yescharacteristic
24C2×#13182016C2211yescharacteristic
25C2×#13182016C2211yescharacteristic
26C2×#13182016C2211yescharacteristic
27C2×#13182016C2211yescharacteristic
28C2×#13182016C2211yescharacteristic
29C2×#13182013C2211yescharacteristic, complemented
30C22×#11724784C2211yescharacteristic
31#16812687C2211yescharacteristic
32#16812684—21no—
33#16812678C2211yescharacteristic
34#16812673C2211yescharacteristic
35#16812672—21no—
36#16812667—21no—
37C4×D436C2211yescharacteristic
38C22⋊C4×D109—21no—
39#16812632C2211yescharacteristic, complemented
40C4⋊#11724784C2211yescharacteristic, complemented
41#16812623C2211yescharacteristic, complemented
42#16812623C2211yescharacteristic, complemented
43#16812623C2211yescharacteristic, complemented
44#16812622C2211yescharacteristic, complemented
45#16812619C2211yescharacteristic
46#16812619C2211yescharacteristic
47C22⋊#11724784C2211yescharacteristic
48C22⋊#11724784C2211yescharacteristic, complemented
49#16812610C2211yescharacteristic, complemented
50C22×Q8×C109C2211yescharacteristic
51C22×Q8×C109C2211yescharacteristic, complemented
52C4⋊Q8×C109—21no—
53C4⋊Q8×C109C2211yescharacteristic, complemented
54C22⋊Q8×C109—21no—
55C22⋊Q8×C109—21no—
56C22⋊Q8×C109C2211yescharacteristic
57C22⋊Q8×C109C2211yescharacteristic, complemented
58C4×Q8×C109C2211yescharacteristic, complemented
59C4×D4×C109—21no—
60C4×D4×C109C2211yescharacteristic
61C23×D109—21no—
62#13182029—22no—
63#13182029—41no—
64C2×Dic218C2311yescharacteristic
65C2×D436—22no—
66C2×D436—21no—
67C2×D436—41no—
68C2×D436C2311yescharacteristic
69C2×D436C2311yescharacteristic
70C2×C22⋊D109—21no—
71C2×C22⋊D109—21no—
72C2×C4×D109—21no—
73C2×C4×D109—21no—
74#13182019—22no—
75#13182019—22no—
76#13182019—41no—
77#13182016—22no—
78#13182016—22no—
79#13182016—22no—
80#13182016—22no—
81#13182016—41no—
82#13182016D414yescomplemented
83#13182013—22no—
84C2×#11724784—22no—
85C2×#11724784C2311yescharacteristic
86C2×#11724784C2311yescharacteristic
87C2×#11724784C2311yescharacteristic
88C2×#11724784C2311yescharacteristic
89C4⋊Dic109C2311yescharacteristic
90#13182008—21no—
91C22⋊Dic109—21no—
92#13182006—21no—
93#13182006—21no—
94#13182006—21no—
95#13182006—21no—
96C2×Q8×C109—21no—
97C2×Q8×C109—21no—
98C2×Q8×C109—22no—
99C2×Q8×C109—22no—
100C2×Q8×C109—41no—
101C2×Q8×C109—41no—
102C2×Q8×C109C2311yescharacteristic
103C2×Q8×C109C2311yescharacteristic
104C2×Q8×C109C2311yescharacteristic
105C2×D4×C109D411yescharacteristic
106C22×C4×C109—21no—
107C22×C4×C109—21no—
108C22×C4×C109C2311yescharacteristic
109C22×C4×C109C2311yescharacteristic
110C4⋊C4×C109—21no—
111C4⋊C4×C109—21no—
112C4⋊C4×C109—21no—
113C4⋊C4×C109—21no—
114C4⋊C4×C109—21no—
115C4⋊C4×C109D411yescharacteristic
116C4⋊C4×C109C2311yescharacteristic
117C22⋊C4×C109—21no—
118C22⋊C4×C109—21no—
119C22⋊C4×C109D411yescharacteristic
120C22⋊C4×C109D411yescharacteristic
121C42×C109—21no—
122C42×C109C2311yescharacteristic
123C23×C109C2×D411yescharacteristic
124C23×C109C2×D411yescharacteristic
125C22×D109—21no—
126C22×D109—21no—
127C22×D109—42no—
128C22×D109—42no—
129C22×D109—81no—
130Q8×C109—21no—
131Q8×C109—21no—
132Q8×C109—21no—
133Q8×C109—21no—
134Q8×C109—21no—
135Q8×C109—42no—
136Q8×C109—42no—
137Q8×C109C2×D412yes—
138D4×C109—22no—
139C2×C4×C109—21no—
140C2×C4×C109—21no—
141C2×C4×C109—21no—
142C2×C4×C109—21no—
143C2×C4×C109—21no—
144C2×C4×C109—21no—
145C2×C4×C109—21no—
146C2×C4×C109—21no—
147C2×C4×C109—42no—
148C2×C4×C109—42no—
149C2×C4×C109C2×D411yescharacteristic
150C2×C4×C109C2×D411yescharacteristic
151C2×C4×C109C2×D411yescharacteristic
152C2×C4×C109C2×D411yescharacteristic
153C2×C4×C109C2411yescharacteristic
154C22⋊D109—42no—
155C22⋊D109—81no—
156C2×Dic109—21no—
157C2×Dic109—21no—
158D436—21no—
159D436—21no—
160D436—21no—
161D436—42no—
162D436C2×D412yes—
163C4×D109—42no—
164C4×D109—81no—
165Dic218—21no—
166Dic218—21no—
167#11724784—21no—
168#11724784—21no—
169#11724784—21no—
170#11724784C2×D412yes—
171C22×C109—21no—
172C22×C109—21no—
173C22×C109—21no—
174C22×C109—21no—
175C22×C109—21no—
176C22×C109#13712yescomplemented
177C22×C109C22×D411yescharacteristic
178C2×D109—22no—
179C2×D109—41no—
180C2×D109—41no—
181C2×D109—41no—
182C2×D109—41no—
183C2×D109—41no—
184C2×D109—84no—
185C4×C109—21no—
186C4×C109—21no—
187C4×C109—22no—
188C4×C109—21no—
189C4×C109—21no—
190C4×C109—41no—
191C4×C109—41no—
192C4×C109—41no—
193C4×C109C22×D411yescharacteristic
194C4×C109H(5,𝔽2)11yescharacteristic
195Dic109—41no—
196Dic109—41no—
197C2×C109—22no—
198C2×C109—41no—
199C2×C109D4211yescharacteristic
200C2×C109C2×#13711yescharacteristic
201C2×C109#57711yescharacteristic
202D109—42no—
203D109—81no—
204#3266—1091nomaximal, Sylow
205C109#326611yescharacteristic, complemented, Sylow
206C2×#137—1091no—
207C22×SD16—1091no—
208D4×Q8D10911yescharacteristic, complemented
209#546—1091no—
210#497—1091no—
211#468—1091no—
212#467—1091no—
213#464—1091no—
214#461—1091no—
215#452—1091no—
216#451—1091no—
217#450—1091no—
218#448—1091no—
219#440—1091no—
220#435—1091no—
221C2×#41—1091no—
222C22×Q8C2×D10911yescharacteristic
223C22×Q8C2×D10911yescharacteristic, complemented
224C22×D4—1091no—
225#137—2184no—
226C2×Q16—1091no—
227C2×Q16—1091no—
228C2×Q16—1091no—
229C2×SD16—1091no—
230C2×SD16—1091no—
231C2×SD16—1091no—
232C2×SD16—1091no—
233C2×SD16—1091no—
234C2×SD16—1091no—
235C2×SD16—2184no—
236C2×M16—1091no—
237C22×C8—1091no—
238C4⋊Q8—21no—
239C4⋊Q8C2×D10911yescharacteristic, complemented
240#124—1091no—
241#123—2181no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—1091no—
245C22⋊Q8C2×D10911yescharacteristic
246C22⋊Q8C2×D10911yescharacteristic, complemented
247#121—1091no—
248#121—2181no—
249#120—2181no—
250C4×Q8C2×D10911yescharacteristic, complemented
251C4×D4—21no—
252C4×D4—1091no—
253C4×D4C2×D10911yescharacteristic
254C2×C22⋊C4—2181no—
255#107—1091no—
256C4⋊C8—1091no—
257#103—1091no—
258#103—1091no—
259#103—1091no—
260#103—1091no—
261#102—1091no—
262#102—1091no—
263C22⋊C8—1091no—
264C22⋊C8—1091no—
265#97—1091no—
266C24—2181no—
267#41—4361no—
268#41—2182no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—1091no—
276C2×Q8C22×D10911yescharacteristic
277C2×Q8C22×D10911yescharacteristic
278C2×Q8C22×D10911yescharacteristic
279C2×D4—1091no—
280C2×D4—1091no—
281C2×D4—4361no—
282C2×D4—2182no—
283C2×D4—2181no—
284C2×D4—2181no—
285C2×D4—2181no—
286C2×D4C22⋊D10911yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—2181no—
290C22×C4—2181no—
291C22×C4C22×D10911yescharacteristic
292C22×C4C22×D10911yescharacteristic
293Q16—4361no—
294Q16—2182no—
295Q16—2182no—
296SD16—1094no—
297SD16—4361no—
298SD16—2182no—
299SD16—2182no—
300SD16—2182no—
301SD16—2182no—
302M16—2182no—
303C2×C8—1091no—
304C2×C8—1091no—
305C2×C8—1091no—
306C2×C8—1091no—
307C2×C8—2182no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—1091no—
314C4⋊C4—2181no—
315C4⋊C4C22⋊D10911yescharacteristic
316C4⋊C4C22×D10911yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—2181no—
320C22⋊C4—2181no—
321C22⋊C4—2181no—
322C22⋊C4—2181no—
323C22⋊C4—2181no—
324C22⋊C4C22⋊D10911yescharacteristic
325C22⋊C4C22⋊D10911yescharacteristic
326C42—21no—
327C42C22×D10911yescharacteristic
328C23—8721no—
329C23—4362no—
330C23—4362no—
331C23—2181no—
332C23—2181no—
333C23C2×C22⋊D10911yescharacteristic
334C23C2×C22⋊D10911yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—2181no—
343Q8—2181no—
344Q8D4×D10912yes—
345D4—22no—
346D4—8721no—
347D4—1092no—
348D4—4362no—
349D4—4362no—
350D4—2181no—
351D4—2181no—
352D4—2181no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—8721no—
364C2×C4—4362no—
365C2×C4—2181no—
366C2×C4—2181no—
367C2×C4C2×C22⋊D10911yescharacteristic
368C2×C4C2×C22⋊D10911yescharacteristic
369C2×C4C2×C22⋊D10911yescharacteristic
370C2×C4C2×C22⋊D10911yescharacteristic
371C2×C4C23×D10911yescharacteristic
372C8—1092no—
373C8—2181no—
374C8—2181no—
375C8—2181no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—8724no—
382C22—4361no—
383C22—4361no—
384C22—4361no—
385C22—4361no—
386C22—4361no—
387C22—2182no—
388C22#1681273012yescomplemented
389C22C22×C22⋊D10911yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—4361no—
399C4—4361no—
400C4C2×D4×D10911yescharacteristic
401C4#1681274411yescharacteristic
402C2—22no—
403C2—41no—
404C2—8721no—
405C2—4362no—
406C2D4×C22⋊D10911yescharacteristic
407C2C2×#1681273011yescharacteristic
408C2#2029583411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingD4×Q8×C109C2
socleC22×C109C22×D4
prime radical#20295672C2
hypercenterD4×Q8D109
hypocenterC109#3266
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16812730C2semidirect
2. C22×#13182016C2semidirect
4. #20295418C2semidirect
5. #20295274C2semidirect
6. #20295265C2semidirect
7. #20295248C2semidirect
8. #20295228C2semidirect
9. #20295186C2semidirect
10. #20295182C2semidirect
11. #20295176C2semidirect
12. #20295165C2semidirect
13. #20295124C2semidirect
14. #20295098C2semidirect
15. D4×Q8×C109C2semidirect
19. C2×#13182019C22semidirect
20. C2×#13182019C22semidirect
21. C2×#13182019C22semidirect
29. C2×#13182013C22semidirect
39. #16812632C22semidirect
40. C4⋊#11724784C22semidirect
41. #16812623C22semidirect
42. #16812623C22semidirect
43. #16812623C22semidirect
44. #16812622C22semidirect
48. C22⋊#11724784C22semidirect
49. #16812610C22semidirect
51. C22×Q8×C109C22semidirect
53. C4⋊Q8×C109C22semidirect
57. C22⋊Q8×C109C22semidirect
58. C4×Q8×C109C22semidirect
82. #13182016D4semidirect
176. C22×C109#137semidirect
205. C109#3266semidirect
208. D4×Q8D109semidirect
223. C22×Q8C2×D109semidirect
239. C4⋊Q8C2×D109semidirect
246. C22⋊Q8C2×D109semidirect
250. C4×Q8C2×D109semidirect
388. C22#16812730semidirect

Maximal subgroups

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

subgroupindex
C2×#168127302
C22×#131820162
#202956722
#202954182
#202952742
#202952652
#202952482
#202952282
#202951862
#202951822
#202951762
#202951652
#202951242
#202950982
D4×Q8×C1092
#3266109