Finite Group Databasebeta

← #30677484

Subgroups — #30677484

Order 16768.

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×#16862272C211yesmaximal, characteristic, complemented
2C2×#16862267C211yesmaximal, characteristic, complemented
3#22553964C211yesmaximal, characteristic, complemented
4#22553956C211yesmaximal, characteristic, complemented
5#22553703C211yesmaximal, characteristic, complemented
6#22553568C211yesmaximal, characteristic, complemented
7#22553540C211yesmaximal, characteristic, complemented
8#22553526C211yesmaximal, characteristic, complemented
9#22553515C211yesmaximal, characteristic, complemented
10#22553476C211yesmaximal, characteristic, complemented
11#22553475C211yesmaximal, characteristic, complemented
12#22553456C211yesmaximal, characteristic, complemented
13#22553421C211yesmaximal, characteristic, complemented
14#22553386C211yesmaximal, characteristic, complemented
15#450×C131C211yesmaximal, characteristic, complemented
16C2×#14523514C2211yescharacteristic, complemented
17C2×#14523513C2211yescharacteristic
18C2×#14523512C2211yescharacteristic
19#16862272—24no—
20#16862267—24no—
21C2×Dic524C2211yescharacteristic, complemented
22C2×#14523503C2211yescharacteristic, complemented
23C2×#14523502C2211yescharacteristic, complemented
24C2×#14523501C2211yescharacteristic
25C2×#14523501C2211yescharacteristic
26C2×#14523498C2211yescharacteristic
27C2×#14523498C2211yescharacteristic
28C2×#14523497C2211yescharacteristic
29C2×C8×D131C2211yescharacteristic, complemented
30#16862244C2211yescharacteristic, complemented
31#16862233C2211yescharacteristic
32#16862231—21no—
33#16862228—21no—
34#16862227—21no—
35#16862226C2211yescharacteristic, complemented
36#16862225C2211yescharacteristic
37#16862219—21no—
38#16862218—21no—
39#16862217C2211yescharacteristic
40#16862215—21no—
41D4×Dic131C2211yescharacteristic
42#16862208C2211yescharacteristic, complemented
43C4×C22⋊D131C2211yescharacteristic
44C4⋊C4×D131—21no—
45C2×C22⋊Dic131—21no—
46#16862177C2211yescharacteristic, complemented
47Dic131⋊C8C2211yescharacteristic, complemented
48#16862171C2211yescharacteristic, complemented
49#16862170C2211yescharacteristic, complemented
50#16862169C2211yescharacteristic, complemented
51#16862168C2211yescharacteristic, complemented
52(C2×D131)⋊C8C2211yescharacteristic, complemented
53#16862156C2211yescharacteristic, complemented
54C22×D4×C131C2211yescharacteristic
55C2×SD16×C131C2211yescharacteristic
56C2×SD16×C131C2211yescharacteristic
57C22⋊Q8×C131C2211yescharacteristic
58#102×C131C2211yescharacteristic
59#102×C131C2211yescharacteristic
60C22⋊C8×C131C2211yescharacteristic
61#14523514—22no—
62#14523514—41no—
63#14523513—22no—
64#14523513—41no—
65#14523512—22no—
66#14523512—41no—
67C2×Dic262C2311yescharacteristic
68C2×Dic262C2311yescharacteristic
69C2×D524—21no—
70C2×D524C2311yescharacteristic
71C2×C22⋊D131—21no—
72C2×C22⋊D131—21no—
73C2×C22⋊D131—21no—
74C2×C22⋊D131D411yescharacteristic
75C22×Dic131—21no—
76C22×Dic131—21no—
77C2×C4×D131—21no—
78C2×C4×D131—21no—
79C2×C4×D131—21no—
80C2×C4×D131C2311yescharacteristic
81Dic524—41no—
82#14523503—22no—
83#14523502—22no—
84#14523501—22no—
85#14523501—22no—
86#14523498—22no—
87#14523498D414yescomplemented
88#14523497—22no—
89C2×C131⋊C8C2311yescharacteristic
90C2×C131⋊C8C2311yescharacteristic
91C8×D131—22no—
92C4⋊Dic131—21no—
93C4⋊Dic131C2311yescharacteristic
94Dic131⋊C4—21no—
95Dic131⋊C4—21no—
96Dic131⋊C4—21no—
97Dic131⋊C4D411yescharacteristic
98C22⋊Dic131—21no—
99C22⋊Dic131—21no—
100C22⋊Dic131—21no—
101C22⋊Dic131—21no—
102C22⋊Dic131—21no—
103C22⋊Dic131D411yescharacteristic
104(C2×D131)⋊C4—21no—
105(C2×D131)⋊C4—21no—
106(C2×D131)⋊C4D411yescharacteristic
107C4×Dic131C2311yescharacteristic
108C24×C131—21no—
109C2×Q8×C131C2311yescharacteristic
110C2×D4×C131—22no—
111C2×D4×C131—21no—
112C2×D4×C131—41no—
113C2×D4×C131C2311yescharacteristic
114C2×D4×C131C2311yescharacteristic
115C22×C4×C131C2311yescharacteristic
116SD16×C131—22no—
117SD16×C131—41no—
118C2×C8×C131C2311yescharacteristic
119C2×C8×C131C2311yescharacteristic
120C4⋊C4×C131—21no—
121C4⋊C4×C131C2311yescharacteristic
122C22⋊C4×C131—21no—
123C22×D131—21no—
124C22×D131C2×D411yescharacteristic
125Dic262—21no—
126Dic262—21no—
127Dic262—21no—
128Dic262—21no—
129D524—21no—
130D524—42no—
131D524C2×D412yes—
132C22⋊D131—41no—
133C22⋊D131—42no—
134C22⋊D131—42no—
135C22⋊D131—81no—
136C2×Dic131—21no—
137C2×Dic131—21no—
138C2×Dic131—21no—
139C2×Dic131—42no—
140C2×Dic131—81no—
141C2×Dic131C2×D411yescharacteristic
142C2×Dic131C2×D411yescharacteristic
143C2×Dic131C2×D411yescharacteristic
144C4×D131—21no—
145C4×D131—21no—
146C4×D131—42no—
147C4×D131—42no—
148C4×D131—42no—
149C131⋊C8—21no—
150C131⋊C8C2×D412yes—
151C23×C131—21no—
152C23×C131—21no—
153C23×C131—42no—
154C23×C131—42no—
155C23×C131—81no—
156C23×C131C2×D411yescharacteristic
157Q8×C131—21no—
158Q8×C131—21no—
159D4×C131—21no—
160D4×C131—21no—
161D4×C131—21no—
162D4×C131—42no—
163D4×C131C2×D412yes—
164C2×C4×C131—21no—
165C2×C4×C131—21no—
166C2×C4×C131—21no—
167C2×C4×C131—21no—
168C2×C4×C131C2×D411yescharacteristic
169C2×C4×C131C2411yescharacteristic
170C8×C131—21no—
171C8×C131—21no—
172C22×C131—21no—
173C22×C131—22no—
174C22×C131—21no—
175C22×C131—21no—
176C22×C131—41no—
177C22×C131—41no—
178C22×C131—41no—
179C22×C131—41no—
180C22×C131—41no—
181C22×C131—84no—
182C22×C131C22×D411yescharacteristic
183C2×D131—21no—
184C2×D131—22no—
185C2×D131—21no—
186C2×D131—21no—
187C2×D131—41no—
188C2×D131—41no—
189C4×C131—21no—
190C4×C131—41no—
191C4×C131—41no—
192C4×C131C22×D411yescharacteristic
193C4×C131H(5,𝔽2)11yescharacteristic
194Dic131—21no—
195Dic131—21no—
196Dic131—21no—
197Dic131—41no—
198Dic131—41no—
199Dic131—41no—
200C2×C131—41no—
201C2×C131—42no—
202C2×C131—81no—
203C2×C131D4211yescharacteristic
204C2×C131#57511yescharacteristic
205C2×C131#57711yescharacteristic
206D131—41no—
207D131—42no—
208C131#326711yescharacteristic, complemented, Sylow
209#3267—1311nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1311no—
211C2×#135—1311no—
212#547—1311no—
213#546—1311no—
214#497—1311no—
215#470—1311no—
216#465—1311no—
217#463—1311no—
218#461—1311no—
219#452—1311no—
220#452—1311no—
221#450D13111yescharacteristic, complemented
222#449—1311no—
223#442—1311no—
224#435—1311no—
225C2×#41—1311no—
226C2×#41—1311no—
227C2×#41—1311no—
228C22×D4C2×D13111yescharacteristic
229AGL(1,ℤ/8)—2624no—
230#135—2624no—
231C2×Q16—1311no—
232C2×Q16—1311no—
233C2×SD16—1311no—
234C2×SD16—1311no—
235C2×SD16—1311no—
236C2×SD16C2×D13111yescharacteristic
237C2×SD16C2×D13111yescharacteristic
238C2×D8—1311no—
239C2×D8—1311no—
240C2×M16—1311no—
241C22×C8—1311no—
242C4⋊Q8—1311no—
243#124—1311no—
244#123—2621no—
245#123—2621no—
246#123—2621no—
247C22⋊Q8—1311no—
248C22⋊Q8—1311no—
249C22⋊Q8C2×D13111yescharacteristic
250#121—2621no—
251#121—2621no—
252#121—1311no—
253#120—2621no—
254C4×D4—1311no—
255C4×D4—1311no—
256C4×D4—1311no—
257C2×C4⋊C4—2621no—
258C2×C22⋊C4—2621no—
259#106—1311no—
260C4⋊C8—1311no—
261#103—1311no—
262#103—1311no—
263#103—1311no—
264#102—1311no—
265#102C2×D13111yescharacteristic
266#102C2×D13111yescharacteristic
267C22⋊C8—1311no—
268C22⋊C8C2×D13111yescharacteristic
269#97—1311no—
270C24—21no—
271#41—2622no—
272#41—2622no—
273#41—2622no—
274#41—5241no—
275#41—5241no—
276#41—5241no—
277C2×Q8—1311no—
278C2×Q8—1311no—
279C2×Q8C22×D13111yescharacteristic
280C2×D4—22no—
281C2×D4—21no—
282C2×D4—41no—
283C2×D4—2621no—
284C2×D4—2621no—
285C2×D4—2621no—
286C2×D4—2621no—
287C2×D4—1311no—
288C2×D4—1311no—
289C2×D4C22×D13111yescharacteristic
290C2×D4C22×D13111yescharacteristic
291C22×C4—2621no—
292C22×C4—2621no—
293C22×C4—2621no—
294C22×C4—2621no—
295C22×C4—2621no—
296C22×C4—1311no—
297C22×C4C22×D13111yescharacteristic
298Q16—2622no—
299Q16—5241no—
300SD16—22no—
301SD16—41no—
302SD16—2622no—
303SD16—2622no—
304SD16—2622no—
305D8—2622no—
306D8—1314no—
307M16—2622no—
308C2×C8—2622no—
309C2×C8—1311no—
310C2×C8—1311no—
311C2×C8C22×D13111yescharacteristic
312C2×C8C22×D13111yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—2621no—
315C4⋊C4—2621no—
316C4⋊C4—2621no—
317C4⋊C4—2621no—
318C4⋊C4—1311no—
319C4⋊C4—1311no—
320C4⋊C4C22×D13111yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—2621no—
323C22⋊C4—2621no—
324C22⋊C4—2621no—
325C22⋊C4—2621no—
326C22⋊C4—2621no—
327C22⋊C4—2621no—
328C22⋊C4—2621no—
329C22⋊C4—1311no—
330C22⋊C4—1311no—
331C42—1311no—
332C23—21no—
333C23—21no—
334C23—42no—
335C23—42no—
336C23—2621no—
337C23—81no—
338C23—1311no—
339C23D4×D13111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—2621no—
343Q8—2621no—
344Q8—2621no—
345Q8—2621no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—2621no—
351D4—5242no—
352D4—5241no—
353D4—5242no—
354D4—5242no—
355D4—10481no—
356D4—1312no—
357D4D4×D13112yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—2621no—
363C2×C4—2621no—
364C2×C4—2621no—
365C2×C4—2621no—
366C2×C4—2621no—
367C2×C4—5242no—
368C2×C4—5242no—
369C2×C4—5242no—
370C2×C4—5242no—
371C2×C4—10481no—
372C2×C4—1311no—
373C2×C4—1311no—
374C2×C4—1311no—
375C2×C4D4×D13111yescharacteristic
376C2×C4C23×D13111yescharacteristic
377C8—21no—
378C8—21no—
379C8—2621no—
380C8—1312no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—2621no—
391C22—2621no—
392C22—2622no—
393C22—2621no—
394C22—84no—
395C22—5241no—
396C22—5241no—
397C22C2×D4×D13111yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—2621no—
402C4—2621no—
403C4—2621no—
404C4—5241no—
405C4—5241no—
406C4—5241no—
407C4C2×D4×D13111yescharacteristic
408C4#1686229111yescharacteristic
409C2—41no—
410C2—42no—
411C2—81no—
412C2—5241no—
413C2—5242no—
414C2#2255394411yescharacteristic
415C2#2255410811yescharacteristic
416C2#2255411811yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#450×C131C2
socleC22×C131C22×D4
hypercenter#450D131
hypocenterC131#3267
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#16862272C2semidirect
2. C2×#16862267C2semidirect
3. #22553964C2semidirect
4. #22553956C2semidirect
5. #22553703C2semidirect
6. #22553568C2semidirect
7. #22553540C2semidirect
8. #22553526C2semidirect
9. #22553515C2semidirect
10. #22553476C2semidirect
11. #22553475C2semidirect
12. #22553456C2semidirect
13. #22553421C2semidirect
14. #22553386C2semidirect
15. #450×C131C2semidirect
16. C2×#14523514C22semidirect
21. C2×Dic524C22semidirect
22. C2×#14523503C22semidirect
23. C2×#14523502C22semidirect
29. C2×C8×D131C22semidirect
30. #16862244C22semidirect
35. #16862226C22semidirect
42. #16862208C22semidirect
46. #16862177C22semidirect
47. Dic131⋊C8C22semidirect
48. #16862171C22semidirect
49. #16862170C22semidirect
50. #16862169C22semidirect
51. #16862168C22semidirect
52. (C2×D131)⋊C8C22semidirect
53. #16862156C22semidirect
87. #14523498D4semidirect
208. C131#3267semidirect
221. #450D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622722
C2×#168622672
#225539642
#225539562
#225537032
#225535682
#225535402
#225535262
#225535152
#225534762
#225534752
#225534562
#225534212
#225533862
#450×C1312
#3267131