Finite Group Databasebeta

← #30677487

Subgroups — #30677487

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×#16862271C211yesmaximal, characteristic, complemented
2C2×#16862266C211yesmaximal, characteristic, complemented
3#22553958C211yesmaximal, characteristic, complemented
4#22553704C211yesmaximal, characteristic, complemented
5#22553568C211yesmaximal, characteristic, complemented
6#22553543C211yesmaximal, characteristic, complemented
7#22553527C211yesmaximal, characteristic, complemented
8#22553514C211yesmaximal, characteristic, complemented
9#22553474C211yesmaximal, characteristic, complemented
10#22553472C211yesmaximal, characteristic, complemented
11#22553462C211yesmaximal, characteristic, complemented
12#22553457C211yesmaximal, characteristic, complemented
13#22553422C211yesmaximal, characteristic, complemented
14#22553384C211yesmaximal, characteristic, complemented
15#547×C131C211yesmaximal, characteristic, complemented
16C2×#14523513C2211yescharacteristic
17C22×D524C2211yescharacteristic
18#16862271—24no—
19#16862266—24no—
20C2×#14523503C2211yescharacteristic, complemented
21C2×#14523503C2211yescharacteristic, complemented
22C2×#14523501C2211yescharacteristic, complemented
23C2×#14523500C2211yescharacteristic
24C2×#14523500C2211yescharacteristic
25C2×#14523500C2211yescharacteristic
26C2×#14523500C2211yescharacteristic
27C2×#14523498C2211yescharacteristic
28C2×#14523498C2211yescharacteristic
29C2×#14523497C2211yescharacteristic
30C22×C131⋊C8C2211yescharacteristic, complemented
31#16862234C2211yescharacteristic
32#16862231—21no—
33#16862225C2211yescharacteristic
34#16862220C2211yescharacteristic
35#16862219—21no—
36#16862214—21no—
37C4×D524C2211yescharacteristic
38C22⋊C4×D131—21no—
39#16862177C2211yescharacteristic, complemented
40C4⋊(C131⋊C8)C2211yescharacteristic, complemented
41#16862170C2211yescharacteristic, complemented
42#16862170C2211yescharacteristic, complemented
43#16862169C2211yescharacteristic, complemented
44#16862167C2211yescharacteristic, complemented
45#16862166C2211yescharacteristic
46#16862166C2211yescharacteristic
47C22⋊(C131⋊C8)C2211yescharacteristic
48C22⋊(C131⋊C8)C2211yescharacteristic, complemented
49#16862157C2211yescharacteristic, complemented
50C2×#41×C131C2211yescharacteristic
51C2×#41×C131C2211yescharacteristic, complemented
52C4⋊Q8×C131C2211yescharacteristic, complemented
53#123×C131—21no—
54#123×C131—21no—
55C22⋊Q8×C131C2211yescharacteristic
56C22⋊Q8×C131C2211yescharacteristic, complemented
57#121×C131—21no—
58C4×D4×C131C2211yescharacteristic
59C4×D4×C131C2211yescharacteristic, complemented
60C2×C4⋊C4×C131—21no—
61C23×D131—21no—
62#14523513—22no—
63#14523513—41no—
64C2×Dic262C2311yescharacteristic
65C2×D524—22no—
66C2×D524—21no—
67C2×D524—41no—
68C2×D524C2311yescharacteristic
69C2×D524C2311yescharacteristic
70C2×C22⋊D131—21no—
71C2×C22⋊D131—21no—
72C2×C4×D131—21no—
73C2×C4×D131—21no—
74#14523503—22no—
75#14523503—41no—
76#14523501—22no—
77#14523500—22no—
78#14523500—22no—
79#14523500—22no—
80#14523500—41no—
81#14523498—22no—
82#14523498D414yescomplemented
83#14523497—22no—
84C2×C131⋊C8—22no—
85C2×C131⋊C8C2311yescharacteristic
86C2×C131⋊C8C2311yescharacteristic
87C2×C131⋊C8C2311yescharacteristic
88C2×C131⋊C8C2311yescharacteristic
89C4⋊Dic131C2311yescharacteristic
90Dic131⋊C4—21no—
91C22⋊Dic131—21no—
92(C2×D131)⋊C4—21no—
93(C2×D131)⋊C4—21no—
94(C2×D131)⋊C4—21no—
95(C2×D131)⋊C4—21no—
96#41×C131—22no—
97#41×C131—22no—
98#41×C131—41no—
99#41×C131—41no—
100C2×Q8×C131C2311yescharacteristic
101C2×Q8×C131C2311yescharacteristic
102C2×D4×C131—21no—
103C2×D4×C131—21no—
104C2×D4×C131D411yescharacteristic
105C2×D4×C131C2311yescharacteristic
106C22×C4×C131—21no—
107C22×C4×C131—21no—
108C22×C4×C131—21no—
109C22×C4×C131C2311yescharacteristic
110C22×C4×C131C2311yescharacteristic
111C4⋊C4×C131—21no—
112C4⋊C4×C131—21no—
113C4⋊C4×C131—21no—
114C4⋊C4×C131—21no—
115C4⋊C4×C131D411yescharacteristic
116C4⋊C4×C131C2311yescharacteristic
117C22⋊C4×C131—21no—
118C22⋊C4×C131—21no—
119C22⋊C4×C131—21no—
120C22⋊C4×C131D411yescharacteristic
121C22⋊C4×C131D411yescharacteristic
122C42×C131C2311yescharacteristic
123C22×D131—21no—
124C22×D131—21no—
125C22×D131—42no—
126C22×D131—42no—
127C22×D131—81no—
128Dic262—21no—
129Dic262—21no—
130D524—21no—
131D524—21no—
132D524—21no—
133D524—42no—
134D524C2×D412yes—
135C22⋊D131—42no—
136C22⋊D131—81no—
137C2×Dic131—21no—
138C2×Dic131—21no—
139C4×D131—42no—
140C4×D131—81no—
141C131⋊C8—21no—
142C131⋊C8—21no—
143C131⋊C8—21no—
144C131⋊C8C2×D412yes—
145C23×C131—21no—
146C23×C131C2×D411yescharacteristic
147C23×C131C2×D411yescharacteristic
148Q8×C131—21no—
149Q8×C131—21no—
150Q8×C131—21no—
151Q8×C131—21no—
152D4×C131—21no—
153D4×C131—41no—
154D4×C131—42no—
155D4×C131—42no—
156D4×C131C2×D412yes—
157C2×C4×C131—21no—
158C2×C4×C131—21no—
159C2×C4×C131—21no—
160C2×C4×C131—21no—
161C2×C4×C131—21no—
162C2×C4×C131—21no—
163C2×C4×C131—21no—
164C2×C4×C131—42no—
165C2×C4×C131—42no—
166C2×C4×C131—42no—
167C2×C4×C131C2×D411yescharacteristic
168C2×C4×C131C2×D411yescharacteristic
169C2×C4×C131C2×D411yescharacteristic
170C2×C4×C131C2×D411yescharacteristic
171C2×C4×C131C2411yescharacteristic
172C22×C131—21no—
173C22×C131—21no—
174C22×C131—22no—
175C22×C131—21no—
176C22×C131—21no—
177C22×C131—21no—
178C22×C131—21no—
179C22×C131—41no—
180C22×C131—41no—
181C22×C131C22×D411yescharacteristic
182C2×D131—22no—
183C2×D131—41no—
184C2×D131—41no—
185C2×D131—41no—
186C2×D131—41no—
187C2×D131—41no—
188C2×D131—84no—
189C4×C131—21no—
190C4×C131—21no—
191C4×C131—21no—
192C4×C131—21no—
193C4×C131—41no—
194C4×C131—41no—
195C4×C131—41no—
196C4×C131C22×D411yescharacteristic
197C4×C131H(5,𝔽2)11yescharacteristic
198Dic131—41no—
199Dic131—41no—
200C2×C131—41no—
201C2×C131—41no—
202C2×C131—42no—
203C2×C131D4211yescharacteristic
204C2×C131#57511yescharacteristic
205C2×C131#57711yescharacteristic
206D131—42no—
207D131—81no—
208C131#326711yescharacteristic, complemented, Sylow
209#3267—1311nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1311no—
211C2×#135—1311no—
212#547D13111yescharacteristic, complemented
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—2624no—
231C2×Q16—1311no—
232C2×Q16—1311no—
233C2×SD16—1311no—
234C2×SD16—1311no—
235C2×SD16—1311no—
236C2×SD16—1311no—
237C2×SD16—1311no—
238C2×D8—1311no—
239C2×D8—1311no—
240C2×M16—1311no—
241C22×C8—1311no—
242C4⋊Q8C2×D13111yescharacteristic, complemented
243#124—1311no—
244#123—21no—
245#123—21no—
246#123—2621no—
247C22⋊Q8—1311no—
248C22⋊Q8C2×D13111yescharacteristic
249C22⋊Q8C2×D13111yescharacteristic, complemented
250#121—21no—
251#121—2621no—
252#121—1311no—
253#120—2621no—
254C4×D4—1311no—
255C4×D4C2×D13111yescharacteristic
256C4×D4C2×D13111yescharacteristic, complemented
257C2×C4⋊C4—21no—
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—21no—
294C22×C4—2621no—
295C22×C4—2621no—
296C22×C4C22×D13111yescharacteristic
297C22×C4C22×D13111yescharacteristic
298Q16—2622no—
299Q16—5241no—
300SD16—2622no—
301SD16—2622no—
302SD16—2622no—
303SD16—2622no—
304SD16—5241no—
305D8—2622no—
306D8—1314no—
307M16—2622no—
308C2×C8—2622no—
309C2×C8—1311no—
310C2×C8—1311no—
311C2×C8—1311no—
312C2×C8—1311no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—2621no—
318C4⋊C4—1311no—
319C4⋊C4C22⋊D13111yescharacteristic
320C4⋊C4C22×D13111yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—2621no—
325C22⋊C4—2621no—
326C22⋊C4—2621no—
327C22⋊C4—2621no—
328C22⋊C4—2621no—
329C22⋊C4C22⋊D13111yescharacteristic
330C22⋊C4C22⋊D13111yescharacteristic
331C42C22×D13111yescharacteristic
332C23—21no—
333C23—2621no—
334C23—2621no—
335C23—5242no—
336C23—5242no—
337C23—10481no—
338C23C2×C22⋊D13111yescharacteristic
339C23C2×C22⋊D13111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—2621no—
345Q8—2621no—
346D4—21no—
347D4—41no—
348D4—42no—
349D4—42no—
350D4—2621no—
351D4—2621no—
352D4—2621no—
353D4—5242no—
354D4—5242no—
355D4—10481no—
356D4—1312no—
357D4D4×D13112yes—
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—2621no—
369C2×C4—2621no—
370C2×C4—5242no—
371C2×C4—10481no—
372C2×C4C2×C22⋊D13111yescharacteristic
373C2×C4C2×C22⋊D13111yescharacteristic
374C2×C4C2×C22⋊D13111yescharacteristic
375C2×C4C2×C22⋊D13111yescharacteristic
376C2×C4C23×D13111yescharacteristic
377C8—2621no—
378C8—2621no—
379C8—2621no—
380C8—1312no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—22no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22—2622no—
391C22—5241no—
392C22—5241no—
393C22—5241no—
394C22—5241no—
395C22—5241no—
396C22—10484no—
397C22C22×C22⋊D13111yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—5241no—
406C4—5241no—
407C4C2×D4×D13111yescharacteristic
408C4#1686229111yescharacteristic
409C2—41no—
410C2—41no—
411C2—42no—
412C2—5242no—
413C2—10481no—
414C2D4×C22⋊D13111yescharacteristic
415C2#2255411011yescharacteristic
416C2#2255412011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#547×C131C2
socleC22×C131C22×D4
hypercenter#547D131
hypocenterC131#3267
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#16862271C2semidirect
2. C2×#16862266C2semidirect
3. #22553958C2semidirect
4. #22553704C2semidirect
5. #22553568C2semidirect
6. #22553543C2semidirect
7. #22553527C2semidirect
8. #22553514C2semidirect
9. #22553474C2semidirect
10. #22553472C2semidirect
11. #22553462C2semidirect
12. #22553457C2semidirect
13. #22553422C2semidirect
14. #22553384C2semidirect
15. #547×C131C2semidirect
20. C2×#14523503C22semidirect
21. C2×#14523503C22semidirect
22. C2×#14523501C22semidirect
30. C22×C131⋊C8C22semidirect
39. #16862177C22semidirect
40. C4⋊(C131⋊C8)C22semidirect
41. #16862170C22semidirect
42. #16862170C22semidirect
43. #16862169C22semidirect
44. #16862167C22semidirect
48. C22⋊(C131⋊C8)C22semidirect
49. #16862157C22semidirect
51. C2×#41×C131C22semidirect
52. C4⋊Q8×C131C22semidirect
56. C22⋊Q8×C131C22semidirect
59. C4×D4×C131C22semidirect
82. #14523498D4semidirect
208. C131#3267semidirect
212. #547D131semidirect
227. C2×#41C2×D131semidirect
242. C4⋊Q8C2×D131semidirect
249. C22⋊Q8C2×D131semidirect
256. C4×D4C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622712
C2×#168622662
#225539582
#225537042
#225535682
#225535432
#225535272
#225535142
#225534742
#225534722
#225534622
#225534572
#225534222
#225533842
#547×C1312
#3267131