Finite Group Databasebeta

← #30675431

Subgroups — #30675431

Order 16768.

484 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16862278C211yesmaximal, characteristic, complemented
2C2×#16862275C211yesmaximal, characteristic, complemented
3#22553861C211yesmaximal, characteristic
4C4×Q8×D131C211yesmaximal, characteristic, complemented
5#22553535C211yesmaximal, characteristic, complemented
6#22553534C211yesmaximal, characteristic, complemented
7#22553531C211yesmaximal, characteristic, complemented
8#22553521C211yesmaximal, characteristic, complemented
9#22553520C211yesmaximal, characteristic, complemented
10#22553511C211yesmaximal, characteristic, complemented
11#22553510C211yesmaximal, characteristic, complemented
12#22553334C211yesmaximal, characteristic, complemented
13#22553312C211yesmaximal, characteristic, complemented
14#22553310C211yesmaximal, characteristic, complemented
15#464×C131C211yesmaximal, characteristic, complemented
16C2×#14523514C2211yescharacteristic
17C2×#14523512C2211yescharacteristic
18C2×Q8×D131C2211yescharacteristic
19#16862278—24no—
20#16862275—24no—
21C2×Dic524C2211yescharacteristic
22C2×#14523503C2211yescharacteristic
23C2×#14523503C2211yescharacteristic
24C2×#14523502C2211yescharacteristic
25C2×#14523501C2211yescharacteristic
26C2×#14523500C2211yescharacteristic
27C2×#14523496C2211yescharacteristic
28C2×#14523496C2211yescharacteristic
29#16862244C2211yescharacteristic
30#16862242C2211yescharacteristic
31#16862234C2211yescharacteristic
32#16862219—21no—
33#16862219—21no—
34Q8×Dic131C2211yescharacteristic
35C4×Dic262—21no—
36C4×Dic262C2211yescharacteristic
37#16862211—21no—
38#16862211C2211yescharacteristic
39(C22⋊D131)⋊C4—21no—
40(C22⋊D131)⋊C4—21no—
41C4⋊C4×D131—21no—
42C4⋊C4×D131C2211yescharacteristic
43C42×D131—21no—
44C42×D131C2211yescharacteristic
45C4⋊(C131⋊C8)C2211yescharacteristic
46Dic131⋊C8C2211yescharacteristic
47Dic131⋊C8C2211yescharacteristic
48#16862171C2211yescharacteristic
49#16862170C2211yescharacteristic
50#16862169C2211yescharacteristic
51#16862168C2211yescharacteristic
52#16862167C2211yescharacteristic
53#16862166C2211yescharacteristic
54C2×Q16×C131C2211yescharacteristic
55C2×SD16×C131C2211yescharacteristic
56#124×C131C2211yescharacteristic
57C4×Q8×C131C2211yescharacteristic
58C4⋊C8×C131C2211yescharacteristic
59#103×C131C2211yescharacteristic
60#102×C131C2211yescharacteristic
61#14523514—22no—
62#14523514—22no—
63#14523512—22no—
64#14523512—22no—
65C2×Dic262—21no—
66C2×Dic262C2311yescharacteristic
67C2×Dic262C2311yescharacteristic
68Q8×D131—22no—
69Q8×D131D414yescomplemented
70C2×D524—21no—
71C2×D524C2311yescharacteristic
72C2×C22⋊D131—21no—
73C2×C22⋊D131—21no—
74C2×C22⋊D131—21no—
75C22×Dic131—21no—
76C2×C4×D131—21no—
77C2×C4×D131—21no—
78C2×C4×D131—21no—
79C2×C4×D131D411yescharacteristic
80C2×C4×D131D411yescharacteristic
81C2×C4×D131C2311yescharacteristic
82Dic524—22no—
83#14523503—22no—
84#14523503—22no—
85#14523502—22no—
86#14523501—22no—
87#14523500—22no—
88#14523496—22no—
89#14523496—22no—
90C2×C131⋊C8C2311yescharacteristic
91C2×C131⋊C8C2311yescharacteristic
92C4⋊Dic131—21no—
93C4⋊Dic131C2311yescharacteristic
94Dic131⋊C4—21no—
95Dic131⋊C4—21no—
96Dic131⋊C4—21no—
97Dic131⋊C4—21no—
98Dic131⋊C4—21no—
99(C2×D131)⋊C4—21no—
100(C2×D131)⋊C4—21no—
101C4×Dic131—21no—
102C4×Dic131—21no—
103C4×Dic131—21no—
104C4×Dic131D411yescharacteristic
105C4×Dic131D411yescharacteristic
106C4×Dic131C2311yescharacteristic
107C2×Q8×C131C2311yescharacteristic
108C2×Q8×C131C2311yescharacteristic
109C2×D4×C131C2311yescharacteristic
110Q16×C131—22no—
111SD16×C131—22no—
112C2×C8×C131C2311yescharacteristic
113C2×C8×C131C2311yescharacteristic
114C4⋊C4×C131—21no—
115C4⋊C4×C131C2311yescharacteristic
116C22⋊C4×C131—21no—
117C22⋊C4×C131—21no—
118C42×C131—21no—
119C42×C131C2311yescharacteristic
120C22×D131—21no—
121C22×D131C2×D411yescharacteristic
122Dic262—21no—
123Dic262—24no—
124Dic262—21no—
125Dic262—21no—
126Dic262C2×D412yes—
127D524—21no—
128D524—21no—
129D524—42no—
130C22⋊D131—42no—
131C22⋊D131—42no—
132C22⋊D131—42no—
133C2×Dic131—21no—
134C2×Dic131—21no—
135C2×Dic131—21no—
136C2×Dic131—42no—
137C2×Dic131C2×D411yescharacteristic
138C2×Dic131C2×D411yescharacteristic
139C2×Dic131C2×D411yescharacteristic
140C4×D131—24no—
141C4×D131—22no—
142C4×D131—41no—
143C4×D131—42no—
144C4×D131—42no—
145C4×D131C2×D412yes—
146C4×D131#4112yes—
147C131⋊C8—21no—
148C131⋊C8—21no—
149C23×C131—21no—
150Q8×C131—21no—
151Q8×C131—21no—
152Q8×C131—21no—
153Q8×C131C2×D412yes—
154D4×C131—21no—
155D4×C131—21no—
156C2×C4×C131—21no—
157C2×C4×C131—21no—
158C2×C4×C131—21no—
159C2×C4×C131C2×D411yescharacteristic
160C2×C4×C131C2×D411yescharacteristic
161C2×C4×C131C2411yescharacteristic
162C8×C131—21no—
163C8×C131—21no—
164C22×C131—41no—
165C22×C131—41no—
166C22×C131—41no—
167C22×C131C22×D411yescharacteristic
168C2×D131—21no—
169C2×D131—21no—
170C2×D131—41no—
171C2×D131—41no—
172C2×D131—41no—
173C2×D131#12112yes—
174C4×C131—22no—
175C4×C131—21no—
176C4×C131—21no—
177C4×C131—41no—
178C4×C131—41no—
179C4×C131C22×D411yescharacteristic
180C4×C131C2×#4111yescharacteristic
181Dic131—22no—
182Dic131—21no—
183Dic131—41no—
184Dic131—41no—
185Dic131#12112yes—
186Dic131#13712yescomplemented
187C2×C131—81no—
188C2×C131C2×#12111yescharacteristic
189C2×C131C2×#13711yescharacteristic
190C2×C131#57511yescharacteristic
191D131—22no—
192D131—81no—
193C131#303711yescharacteristic, complemented, Sylow
194#3037—1311nomaximal, Sylow
195C2×#137—1311no—
196C2×#137—1311no—
197#532—1311no—
198C2×C4×Q8—1311no—
199#464—1311no—
200#464—1311no—
201#464—1311no—
202#464D13111yescharacteristic, complemented
203#462—1311no—
204#462—1311no—
205#460—1311no—
206#460—1311no—
207#423—1311no—
208#417—1311no—
209#417—1311no—
210C2×#41—1311no—
211C2×#41—1311no—
212C22×Q8—1311no—
213#137—2624no—
214#137—2624no—
215C2×Q16—1311no—
216C2×Q16—1311no—
217C2×Q16—1311no—
218C2×Q16C2×D13111yescharacteristic
219C2×SD16—1311no—
220C2×SD16—1311no—
221C2×SD16—1311no—
222C2×SD16C2×D13111yescharacteristic
223C2×M16—1311no—
224C2×M16—1311no—
225C4⋊Q8—1311no—
226#127—1311no—
227#124—1311no—
228#124C2×D13111yescharacteristic
229#121—2621no—
230#121—2621no—
231C4×Q8—2621no—
232C4×Q8—2621no—
233C4×Q8—1311no—
234C4×Q8—1311no—
235C4×Q8—1311no—
236C4×Q8C2×D13111yescharacteristic
237C4×D4—2621no—
238C4×D4—2621no—
239C2×C4⋊C4—2621no—
240C2×C4⋊C4—1311no—
241C2×C42—2621no—
242C2×C42—1311no—
243C4⋊C8—1311no—
244C4⋊C8—1311no—
245C4⋊C8—1311no—
246C4⋊C8C2×D13111yescharacteristic
247#103—1311no—
248#103—1311no—
249#103—1311no—
250#103C2×D13111yescharacteristic
251#102—1311no—
252#102—1311no—
253#102—1311no—
254#102C2×D13111yescharacteristic
255#41—2622no—
256#41—2622no—
257#41—2622no—
258#41—2622no—
259C2×Q8—2621no—
260C2×Q8—2622no—
261C2×Q8—1314no—
262C2×Q8—1311no—
263C2×Q8—1311no—
264C2×Q8C22×D13111yescharacteristic
265C2×Q8C22×D13111yescharacteristic
266C2×D4—2621no—
267C2×D4—2621no—
268C2×D4—2621no—
269C2×D4—2621no—
270C2×D4—1311no—
271C2×D4C22×D13111yescharacteristic
272C22×C4—2621no—
273C22×C4—2621no—
274C22×C4—2621no—
275C22×C4—2621no—
276C22×C4—1311no—
277C22×C4—1311no—
278C22×C4—1311no—
279Q16—22no—
280Q16—2622no—
281Q16—2622no—
282Q16—2622no—
283SD16—22no—
284SD16—2622no—
285SD16—2622no—
286SD16—2622no—
287M16—2622no—
288M16—2622no—
289C2×C8—1311no—
290C2×C8—1311no—
291C2×C8C22×D13111yescharacteristic
292C2×C8C22×D13111yescharacteristic
293C4⋊C4—21no—
294C4⋊C4—2621no—
295C4⋊C4—2621no—
296C4⋊C4—2621no—
297C4⋊C4—2621no—
298C4⋊C4—2621no—
299C4⋊C4—2621no—
300C4⋊C4—1311no—
301C4⋊C4C22×D13111yescharacteristic
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—2621no—
305C22⋊C4—2621no—
306C42—21no—
307C42—2621no—
308C42—2621no—
309C42—2621no—
310C42—1311no—
311C42—1311no—
312C42—1311no—
313C42C22×D13111yescharacteristic
314C23—21no—
315C23—2621no—
316C23—1311no—
317Q8—21no—
318Q8—21no—
319Q8—21no—
320Q8—2624no—
321Q8—2621no—
322Q8—2621no—
323Q8—2621no—
324Q8—1312no—
325Q8D4×D13112yes—
326D4—21no—
327D4—21no—
328D4—2621no—
329D4—2621no—
330D4—5242no—
331D4—5242no—
332D4—5242no—
333D4—5242no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—2624no—
338C2×C4—2622no—
339C2×C4—2621no—
340C2×C4—2621no—
341C2×C4—2621no—
342C2×C4—5241no—
343C2×C4—5242no—
344C2×C4—5242no—
345C2×C4—5242no—
346C2×C4—1311no—
347C2×C4—1311no—
348C2×C4—1311no—
349C2×C4—1312no—
350C2×C4—1312no—
351C2×C4D4×D13111yescharacteristic
352C2×C4D4×D13111yescharacteristic
353C2×C4C23×D13111yescharacteristic
354C8—21no—
355C8—21no—
356C8—2621no—
357C8—2621no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—2621no—
362C22—2621no—
363C22—5241no—
364C22—5241no—
365C22—5241no—
366C22—1312no—
367C22C2×D4×D13111yescharacteristic
368C4—22no—
369C4—21no—
370C4—21no—
371C4—41no—
372C4—41no—
373C4—2622no—
374C4—2621no—
375C4—5241no—
376C4—5241no—
377C4—1312no—
378C4—1312no—
379C4C2×D4×D13111yescharacteristic
380C4#41×D13111yescharacteristic
381C2—2622no—
382C2—81no—
383C2—10481no—
384C2#121×D13111yescharacteristic
385C2#137×D13111yescharacteristic
386C2#2255410911yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#464×C131C2
socleC22×C131C22×D4
prime radical#22553861C2
hypercenter#464D131
hypocenterC131#3037
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#16862278C2semidirect
2. C2×#16862275C2semidirect
4. C4×Q8×D131C2semidirect
5. #22553535C2semidirect
6. #22553534C2semidirect
7. #22553531C2semidirect
8. #22553521C2semidirect
9. #22553520C2semidirect
10. #22553511C2semidirect
11. #22553510C2semidirect
12. #22553334C2semidirect
13. #22553312C2semidirect
14. #22553310C2semidirect
15. #464×C131C2semidirect
69. Q8×D131D4semidirect
186. Dic131#137semidirect
193. C131#3037semidirect
202. #464D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622782
C2×#168622752
#225538612
C4×Q8×D1312
#225535352
#225535342
#225535312
#225535212
#225535202
#225535112
#225535102
#225533342
#225533122
#225533102
#464×C1312
#3037131