Finite Group Databasebeta

← #28374558

Subgroups — #28374558

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×#16812724C211yesmaximal, characteristic, complemented
2C2×#16812719C211yesmaximal, characteristic, complemented
3#20295672C211yesmaximal, characteristic, complemented
4#20295418C211yesmaximal, characteristic, complemented
5#20295282C211yesmaximal, characteristic, complemented
6#20295257C211yesmaximal, characteristic, complemented
7#20295241C211yesmaximal, characteristic, complemented
8#20295228C211yesmaximal, characteristic, complemented
9#20295188C211yesmaximal, characteristic, complemented
10#20295186C211yesmaximal, characteristic, complemented
11#20295176C211yesmaximal, characteristic, complemented
12#20295171C211yesmaximal, characteristic, complemented
13#20295136C211yesmaximal, characteristic, complemented
14#20295098C211yesmaximal, characteristic, complemented
15#547×C109C211yesmaximal, characteristic, complemented
16C2×#13182029C2211yescharacteristic
17C22×D436C2211yescharacteristic
18#16812724—24no—
19#16812719—24no—
20C2×#13182019C2211yescharacteristic, complemented
21C2×#13182019C2211yescharacteristic, complemented
22C2×#13182017C2211yescharacteristic, complemented
23C2×#13182016C2211yescharacteristic
24C2×#13182016C2211yescharacteristic
25C2×#13182016C2211yescharacteristic
26C2×#13182016C2211yescharacteristic
27C2×#13182014C2211yescharacteristic
28C2×#13182014C2211yescharacteristic
29C2×#13182013C2211yescharacteristic
30C22×#11724784C2211yescharacteristic, complemented
31#16812687C2211yescharacteristic
32#16812684—21no—
33#16812678C2211yescharacteristic
34#16812673C2211yescharacteristic
35#16812672—21no—
36#16812667—21no—
37C4×D436C2211yescharacteristic
38C22⋊C4×D109—21no—
39#16812630C2211yescharacteristic, complemented
40C4⋊#11724784C2211yescharacteristic, complemented
41#16812623C2211yescharacteristic, complemented
42#16812623C2211yescharacteristic, complemented
43#16812622C2211yescharacteristic, complemented
44#16812620C2211yescharacteristic, complemented
45#16812619C2211yescharacteristic
46#16812619C2211yescharacteristic
47C22⋊#11724784C2211yescharacteristic
48C22⋊#11724784C2211yescharacteristic, complemented
49#16812610C2211yescharacteristic, complemented
50C2×#41×C109C2211yescharacteristic
51C2×#41×C109C2211yescharacteristic, complemented
52C4⋊Q8×C109C2211yescharacteristic, complemented
53#123×C109—21no—
54#123×C109—21no—
55C22⋊Q8×C109C2211yescharacteristic
56C22⋊Q8×C109C2211yescharacteristic, complemented
57#121×C109—21no—
58C4×D4×C109C2211yescharacteristic
59C4×D4×C109C2211yescharacteristic, complemented
60C2×C4⋊C4×C109—21no—
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—41no—
76#13182017—22no—
77#13182016—22no—
78#13182016—22no—
79#13182016—22no—
80#13182016—41no—
81#13182014—22no—
82#13182014D414yescomplemented
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—
96#41×C109—22no—
97#41×C109—22no—
98#41×C109—41no—
99#41×C109—41no—
100C2×Q8×C109C2311yescharacteristic
101C2×Q8×C109C2311yescharacteristic
102C2×D4×C109—21no—
103C2×D4×C109—21no—
104C2×D4×C109D411yescharacteristic
105C2×D4×C109C2311yescharacteristic
106C22×C4×C109—21no—
107C22×C4×C109—21no—
108C22×C4×C109—21no—
109C22×C4×C109C2311yescharacteristic
110C22×C4×C109C2311yescharacteristic
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×C109—21no—
120C22⋊C4×C109D411yescharacteristic
121C22⋊C4×C109D411yescharacteristic
122C42×C109C2311yescharacteristic
123C23×C109—21no—
124C23×C109C2×D411yescharacteristic
125C23×C109C2×D411yescharacteristic
126C22×D109—21no—
127C22×D109—21no—
128C22×D109—42no—
129C22×D109—42no—
130C22×D109—81no—
131Q8×C109—21no—
132Q8×C109—21no—
133Q8×C109—21no—
134Q8×C109—21no—
135D4×C109—21no—
136D4×C109—41no—
137D4×C109—42no—
138D4×C109—42no—
139D4×C109C2×D412yes—
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×C109—42no—
150C2×C4×C109C2×D411yescharacteristic
151C2×C4×C109C2×D411yescharacteristic
152C2×C4×C109C2×D411yescharacteristic
153C2×C4×C109C2×D411yescharacteristic
154C2×C4×C109C2411yescharacteristic
155C22⋊D109—42no—
156C22⋊D109—81no—
157C2×Dic109—21no—
158C2×Dic109—21no—
159D436—21no—
160D436—21no—
161D436—21no—
162D436—42no—
163D436C2×D412yes—
164C4×D109—42no—
165C4×D109—81no—
166Dic218—21no—
167Dic218—21no—
168#11724784—21no—
169#11724784—21no—
170#11724784—21no—
171#11724784C2×D412yes—
172C22×C109—21no—
173C22×C109—21no—
174C22×C109—22no—
175C22×C109—21no—
176C22×C109—21no—
177C22×C109—21no—
178C22×C109—21no—
179C22×C109—41no—
180C22×C109—41no—
181C22×C109C22×D411yescharacteristic
182C2×D109—22no—
183C2×D109—41no—
184C2×D109—41no—
185C2×D109—41no—
186C2×D109—41no—
187C2×D109—41no—
188C2×D109—84no—
189C4×C109—21no—
190C4×C109—21no—
191C4×C109—21no—
192C4×C109—21no—
193C4×C109—41no—
194C4×C109—41no—
195C4×C109—41no—
196C4×C109C22×D411yescharacteristic
197C4×C109H(5,𝔽2)11yescharacteristic
198Dic109—41no—
199Dic109—41no—
200C2×C109—41no—
201C2×C109—41no—
202C2×C109—42no—
203C2×C109D4211yescharacteristic
204C2×C109#57511yescharacteristic
205C2×C109#57711yescharacteristic
206D109—42no—
207D109—81no—
208#3267—1091nomaximal, Sylow
209C109#326711yescharacteristic, complemented, Sylow
210C2×AGL(1,ℤ/8)—1091no—
211C2×#135—1091no—
212#547D10911yescharacteristic, complemented
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×#41C2×D10911yescharacteristic
227C2×#41C2×D10911yescharacteristic, complemented
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×D8—1091no—
240C2×M16—1091no—
241C22×C8—1091no—
242C4⋊Q8C2×D10911yescharacteristic, complemented
243#124—1091no—
244#123—21no—
245#123—21no—
246#123—2181no—
247C22⋊Q8—1091no—
248C22⋊Q8C2×D10911yescharacteristic
249C22⋊Q8C2×D10911yescharacteristic, complemented
250#121—21no—
251#121—1091no—
252#121—2181no—
253#120—2181no—
254C4×D4—1091no—
255C4×D4C2×D10911yescharacteristic
256C4×D4C2×D10911yescharacteristic, complemented
257C2×C4⋊C4—21no—
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—2181no—
271#41—22no—
272#41—22no—
273#41—41no—
274#41—41no—
275#41—4361no—
276#41—2182no—
277C2×Q8—1091no—
278C2×Q8C22×D10911yescharacteristic
279C2×Q8C22×D10911yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—1091no—
283C2×D4—1091no—
284C2×D4—4361no—
285C2×D4—2182no—
286C2×D4—2181no—
287C2×D4—2181no—
288C2×D4—2181no—
289C2×D4C22⋊D10911yescharacteristic
290C2×D4C22×D10911yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—2181no—
295C22×C4—2181no—
296C22×C4C22×D10911yescharacteristic
297C22×C4C22×D10911yescharacteristic
298Q16—4361no—
299Q16—2182no—
300SD16—4361no—
301SD16—2182no—
302SD16—2182no—
303SD16—2182no—
304SD16—2182no—
305D8—1094no—
306D8—2182no—
307M16—2182no—
308C2×C8—1091no—
309C2×C8—1091no—
310C2×C8—1091no—
311C2×C8—1091no—
312C2×C8—2182no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—1091no—
318C4⋊C4—2181no—
319C4⋊C4C22⋊D10911yescharacteristic
320C4⋊C4C22×D10911yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—2181no—
325C22⋊C4—2181no—
326C22⋊C4—2181no—
327C22⋊C4—2181no—
328C22⋊C4—2181no—
329C22⋊C4C22⋊D10911yescharacteristic
330C22⋊C4C22⋊D10911yescharacteristic
331C42C22×D10911yescharacteristic
332C23—21no—
333C23—8721no—
334C23—4362no—
335C23—4362no—
336C23—2181no—
337C23—2181no—
338C23C2×C22⋊D10911yescharacteristic
339C23C2×C22⋊D10911yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—2181no—
345Q8—2181no—
346D4—21no—
347D4—41no—
348D4—42no—
349D4—42no—
350D4—8721no—
351D4—1092no—
352D4—4362no—
353D4—4362no—
354D4—2181no—
355D4—2181no—
356D4—2181no—
357D4D4×D10912yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—8721no—
369C2×C4—4362no—
370C2×C4—2181no—
371C2×C4—2181no—
372C2×C4C2×C22⋊D10911yescharacteristic
373C2×C4C2×C22⋊D10911yescharacteristic
374C2×C4C2×C22⋊D10911yescharacteristic
375C2×C4C2×C22⋊D10911yescharacteristic
376C2×C4C23×D10911yescharacteristic
377C8—1092no—
378C8—2181no—
379C8—2181no—
380C8—2181no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—22no—
388C22—41no—
389C22—41no—
390C22—8724no—
391C22—4361no—
392C22—4361no—
393C22—4361no—
394C22—4361no—
395C22—4361no—
396C22—2182no—
397C22C22×C22⋊D10911yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—4361no—
406C4—4361no—
407C4C2×D4×D10911yescharacteristic
408C4#1681274411yescharacteristic
409C2—41no—
410C2—41no—
411C2—42no—
412C2—8721no—
413C2—4362no—
414C2D4×C22⋊D10911yescharacteristic
415C2#2029582411yescharacteristic
416C2#2029583411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#547×C109C2
socleC22×C109C22×D4
hypercenter#547D109
hypocenterC109#3267
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#16812724C2semidirect
2. C2×#16812719C2semidirect
3. #20295672C2semidirect
4. #20295418C2semidirect
5. #20295282C2semidirect
6. #20295257C2semidirect
7. #20295241C2semidirect
8. #20295228C2semidirect
9. #20295188C2semidirect
10. #20295186C2semidirect
11. #20295176C2semidirect
12. #20295171C2semidirect
13. #20295136C2semidirect
14. #20295098C2semidirect
15. #547×C109C2semidirect
20. C2×#13182019C22semidirect
21. C2×#13182019C22semidirect
22. C2×#13182017C22semidirect
30. C22×#11724784C22semidirect
39. #16812630C22semidirect
40. C4⋊#11724784C22semidirect
41. #16812623C22semidirect
42. #16812623C22semidirect
43. #16812622C22semidirect
44. #16812620C22semidirect
48. C22⋊#11724784C22semidirect
49. #16812610C22semidirect
51. C2×#41×C109C22semidirect
52. C4⋊Q8×C109C22semidirect
56. C22⋊Q8×C109C22semidirect
59. C4×D4×C109C22semidirect
82. #13182014D4semidirect
209. C109#3267semidirect
212. #547D109semidirect
227. C2×#41C2×D109semidirect
242. C4⋊Q8C2×D109semidirect
249. C22⋊Q8C2×D109semidirect
256. C4×D4C2×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127242
C2×#168127192
#202956722
#202954182
#202952822
#202952572
#202952412
#202952282
#202951882
#202951862
#202951762
#202951712
#202951362
#202950982
#547×C1092
#3267109