Finite Group Databasebeta

← #30677486

Subgroups — #30677486

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×#16862270C211yesmaximal, characteristic, complemented
2#22553961C211yesmaximal, characteristic, complemented
3#22553950C211yesmaximal, characteristic, complemented
4#22553705C211yesmaximal, characteristic, complemented
5#22553567C211yesmaximal, characteristic, complemented
6#22553541C211yesmaximal, characteristic, complemented
7#22553529C211yesmaximal, characteristic, complemented
8#22553515C211yesmaximal, characteristic, complemented
9#22553474C211yesmaximal, characteristic, complemented
10#22553471C211yesmaximal, characteristic, complemented
11#22553461C211yesmaximal, characteristic, complemented
12#22553457C211yesmaximal, characteristic, complemented
13#22553424C211yesmaximal, characteristic, complemented
14#22553383C211yesmaximal, characteristic, complemented
15C2×#135×C131C211yesmaximal, characteristic, complemented
16C2×#14523512C2211yescharacteristic
17C2×D4×D131C2211yescharacteristic
18#16862270—24no—
19C2×#14523503C2211yescharacteristic, complemented
20C2×#14523502C2211yescharacteristic
21C2×#14523501C2211yescharacteristic
22C2×#14523500C2211yescharacteristic
23C2×#14523498C2211yescharacteristic
24C2×#14523496C2211yescharacteristic
25#16862244C2211yescharacteristic, complemented
26#16862235C2211yescharacteristic
27#16862229—21no—
28#16862229—21no—
29#16862227—21no—
30#16862225C2211yescharacteristic, complemented
31#16862222C2211yescharacteristic
32#16862222C2211yescharacteristic
33#16862220C2211yescharacteristic
34#16862218—21no—
35#16862218—21no—
36#16862214—21no—
37D4×Dic131C2211yescharacteristic
38D4×Dic131C2211yescharacteristic, complemented
39C4×C22⋊D131C2211yescharacteristic
40C2×Dic131⋊C4—21no—
41C2×(C2×D131)⋊C4—21no—
42#16862178C2211yescharacteristic, complemented
43Dic131⋊C8C2211yescharacteristic, complemented
44#16862171C2211yescharacteristic, complemented
45#16862170C2211yescharacteristic, complemented
46#16862170C2211yescharacteristic, complemented
47#16862168C2211yescharacteristic
48#16862167C2211yescharacteristic
49#16862167C2211yescharacteristic, complemented
50C22⋊(C131⋊C8)C2211yescharacteristic, complemented
51(C2×D131)⋊C8C2211yescharacteristic
52#16862156C2211yescharacteristic, complemented
53C2×#41×C131C2211yescharacteristic
54C2×#41×C131C2211yescharacteristic, complemented
55#135×C131—24no—
56C2×Q16×C131C2211yescharacteristic, complemented
57C2×SD16×C131C2211yescharacteristic
58C2×SD16×C131C2211yescharacteristic, complemented
59C2×D8×C131C2211yescharacteristic
60C22×C8×C131C2211yescharacteristic, complemented
61C23×D131—21no—
62#14523512—22no—
63#14523512—41no—
64C2×Dic262C2311yescharacteristic
65C2×D524C2311yescharacteristic
66C2×C22⋊D131—21no—
67C2×C22⋊D131—21no—
68C2×C22⋊D131—21no—
69C2×C22⋊D131D411yescharacteristic
70D4×D131—22no—
71D4×D131—41no—
72C22×Dic131—21no—
73C22×Dic131—21no—
74C22×Dic131—21no—
75C2×C4×D131C2311yescharacteristic
76#14523503—41no—
77#14523502—22no—
78#14523501—22no—
79#14523500—41no—
80#14523498—22no—
81#14523496—22no—
82C2×C131⋊C8C2311yescharacteristic
83C2×C131⋊C8C2311yescharacteristic
84C4⋊Dic131C2311yescharacteristic
85C4⋊Dic131C2311yescharacteristic
86Dic131⋊C4—21no—
87Dic131⋊C4—21no—
88Dic131⋊C4—21no—
89Dic131⋊C4—21no—
90Dic131⋊C4—21no—
91Dic131⋊C4D411yescharacteristic
92C22⋊Dic131—21no—
93C22⋊Dic131—21no—
94C22⋊Dic131—21no—
95C22⋊Dic131D411yescharacteristic
96(C2×D131)⋊C4—21no—
97(C2×D131)⋊C4—21no—
98(C2×D131)⋊C4—21no—
99(C2×D131)⋊C4—21no—
100(C2×D131)⋊C4—21no—
101(C2×D131)⋊C4D411yescharacteristic
102C4×Dic131C2311yescharacteristic
103#41×C131—22no—
104#41×C131—22no—
105#41×C131—41no—
106#41×C131—41no—
107C2×Q8×C131C2311yescharacteristic
108C2×Q8×C131C2311yescharacteristic
109C2×D4×C131—21no—
110C2×D4×C131—21no—
111C2×D4×C131C2311yescharacteristic
112C2×D4×C131C2311yescharacteristic
113C22×C4×C131—21no—
114C22×C4×C131—21no—
115C22×C4×C131C2311yescharacteristic
116Q16×C131—22no—
117SD16×C131—22no—
118SD16×C131—22no—
119D8×C131D414yescomplemented
120C2×C8×C131—22no—
121C2×C8×C131C2311yescharacteristic
122C2×C8×C131C2311yescharacteristic
123C22×D131—21no—
124C22×D131—42no—
125C22×D131—42no—
126C22×D131—81no—
127C22×D131C2×D411yescharacteristic
128Dic262—21no—
129Dic262—21no—
130D524—21no—
131D524—21no—
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—42no—
141C2×Dic131—81no—
142C2×Dic131C2×D411yescharacteristic
143C2×Dic131C2×D411yescharacteristic
144C2×Dic131C2×D411yescharacteristic
145C4×D131—21no—
146C4×D131—21no—
147C131⋊C8—21no—
148C131⋊C8—21no—
149C23×C131—21no—
150C23×C131—21no—
151C23×C131C2×D411yescharacteristic
152Q8×C131—21no—
153Q8×C131—21no—
154Q8×C131—21no—
155Q8×C131—21no—
156D4×C131—21no—
157D4×C131—21no—
158D4×C131—42no—
159D4×C131—42no—
160D4×C131C2×D412yes—
161D4×C131C2×D412yes—
162C2×C4×C131—21no—
163C2×C4×C131—21no—
164C2×C4×C131—21no—
165C2×C4×C131—21no—
166C2×C4×C131—42no—
167C2×C4×C131—42no—
168C2×C4×C131C2×D411yescharacteristic
169C2×C4×C131C2411yescharacteristic
170C8×C131—21no—
171C8×C131C2×D412yes—
172C22×C131—21no—
173C22×C131—22no—
174C22×C131—22no—
175C22×C131—21no—
176C22×C131—21no—
177C22×C131—41no—
178C22×C131—41no—
179C22×C131—41no—
180C22×C131—41no—
181C22×C131C22×D411yescharacteristic
182C2×D131—21no—
183C2×D131—21no—
184C2×D131—21no—
185C2×D131—41no—
186C2×D131—41no—
187C2×D131—41no—
188C2×D131—84no—
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—42no—
201C2×C131—41no—
202C2×C131—42no—
203C2×C131D4211yescharacteristic
204C2×C131#57511yescharacteristic
205C2×C131#57711yescharacteristic
206D131—41no—
207D131—81no—
208C131#326711yescharacteristic, complemented, Sylow
209#3267—1311nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1311no—
211C2×#135D13111yescharacteristic, complemented
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#450—1311no—
222#449—1311no—
223#442—1311no—
224#435—1311no—
225C2×#41—1311no—
226C2×#41C2×D13111yescharacteristic
227C2×#41C2×D13111yescharacteristic, complemented
228C22×D4—1311no—
229AGL(1,ℤ/8)—2624no—
230#135—24no—
231C2×Q16—1311no—
232C2×Q16C2×D13111yescharacteristic, complemented
233C2×SD16—1311no—
234C2×SD16—1311no—
235C2×SD16—1311no—
236C2×SD16C2×D13111yescharacteristic
237C2×SD16C2×D13111yescharacteristic, complemented
238C2×D8—1311no—
239C2×D8C2×D13111yescharacteristic
240C2×M16—1311no—
241C22×C8C2×D13111yescharacteristic, complemented
242C4⋊Q8—1311no—
243#124—1311no—
244#123—2621no—
245#123—2621no—
246#123—2621no—
247C22⋊Q8—1311no—
248C22⋊Q8—1311no—
249C22⋊Q8—1311no—
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#102—1311no—
266#102—1311no—
267C22⋊C8—1311no—
268C22⋊C8—1311no—
269#97—1311no—
270C24—2621no—
271#41—22no—
272#41—22no—
273#41—41no—
274#41—41no—
275#41—2622no—
276#41—5241no—
277C2×Q8—1311no—
278C2×Q8C22×D13111yescharacteristic
279C2×Q8C22×D13111yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—2622no—
283C2×D4—2621no—
284C2×D4—2621no—
285C2×D4—2621no—
286C2×D4—5241no—
287C2×D4—1311no—
288C2×D4—1311no—
289C2×D4C22×D13111yescharacteristic
290C2×D4C22×D13111yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—2621no—
294C22×C4—2621no—
295C22×C4—2621no—
296C22×C4—1311no—
297C22×C4C22×D13111yescharacteristic
298Q16—22no—
299Q16—5241no—
300SD16—22no—
301SD16—22no—
302SD16—2622no—
303SD16—2622no—
304SD16—5241no—
305D8—2622no—
306D8C22⋊D13114yescomplemented
307M16—2622no—
308C2×C8—22no—
309C2×C8—1311no—
310C2×C8—1311no—
311C2×C8C22×D13111yescharacteristic
312C2×C8C22×D13111yescharacteristic
313C4⋊C4—2621no—
314C4⋊C4—2621no—
315C4⋊C4—2621no—
316C4⋊C4—2621no—
317C4⋊C4—2621no—
318C4⋊C4—1311no—
319C4⋊C4—1311no—
320C4⋊C4—1311no—
321C22⋊C4—2621no—
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—2621no—
335C23—5242no—
336C23—5242no—
337C23—10481no—
338C23—1311no—
339C23D4×D13111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—2621no—
345Q8—2621no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—42no—
350D4—2621no—
351D4—2621no—
352D4—5241no—
353D4—5242no—
354D4—5242no—
355D4—10481no—
356D4C2×C22⋊D13112yes—
357D4C2×C22⋊D13112yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—2621no—
365C2×C4—2621no—
366C2×C4—2621no—
367C2×C4—2621no—
368C2×C4—2621no—
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—2621no—
379C8—2621no—
380C8C2×C22⋊D13112yes—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—22no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—2621no—
391C22—2621no—
392C22—2621no—
393C22—5241no—
394C22—5241no—
395C22—5241no—
396C22—10484no—
397C22C2×D4×D13111yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—2621no—
402C4—2621no—
403C4—2621no—
404C4—5241no—
405C4—5241no—
406C4—5241no—
407C4C22×C22⋊D13111yescharacteristic
408C4#1686229111yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—5241no—
413C2—10481no—
414C2D4×C22⋊D13111yescharacteristic
415C2#2255410911yescharacteristic
416C2#2255411911yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC2×#135×C131C2
socleC22×C131C22×D4
hypercenterC2×#135D131
hypocenterC131#3267
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#16862270C2semidirect
2. #22553961C2semidirect
3. #22553950C2semidirect
4. #22553705C2semidirect
5. #22553567C2semidirect
6. #22553541C2semidirect
7. #22553529C2semidirect
8. #22553515C2semidirect
9. #22553474C2semidirect
10. #22553471C2semidirect
11. #22553461C2semidirect
12. #22553457C2semidirect
13. #22553424C2semidirect
14. #22553383C2semidirect
15. C2×#135×C131C2semidirect
19. C2×#14523503C22semidirect
25. #16862244C22semidirect
30. #16862225C22semidirect
38. D4×Dic131C22semidirect
42. #16862178C22semidirect
43. Dic131⋊C8C22semidirect
44. #16862171C22semidirect
45. #16862170C22semidirect
46. #16862170C22semidirect
49. #16862167C22semidirect
50. C22⋊(C131⋊C8)C22semidirect
52. #16862156C22semidirect
54. C2×#41×C131C22semidirect
56. C2×Q16×C131C22semidirect
58. C2×SD16×C131C22semidirect
60. C22×C8×C131C22semidirect
119. D8×C131D4semidirect
208. C131#3267semidirect
211. C2×#135D131semidirect
227. C2×#41C2×D131semidirect
232. C2×Q16C2×D131semidirect
237. C2×SD16C2×D131semidirect
241. C22×C8C2×D131semidirect
306. D8C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622702
#225539612
#225539502
#225537052
#225535672
#225535412
#225535292
#225535152
#225534742
#225534712
#225534612
#225534572
#225534242
#225533832
C2×#135×C1312
#3267131