Finite Group Databasebeta

← #30675645

Subgroups — #30675645

Order 16768.

472 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16862275C211yesmaximal, characteristic, complemented
2C2×#16862252C211yesmaximal, characteristic, complemented
3#22553891C211yesmaximal, characteristic
4#22553885C211yesmaximal, characteristic, complemented
5#22553571C211yesmaximal, characteristic, complemented
6#22553568C211yesmaximal, characteristic, complemented
7#22553556C211yesmaximal, characteristic, complemented
8#22553555C211yesmaximal, characteristic, complemented
9#22553553C211yesmaximal, characteristic, complemented
10#22553552C211yesmaximal, characteristic, complemented
11#22553540C211yesmaximal, characteristic, complemented
12#22553351C211yesmaximal, characteristic, complemented
13#22553305C211yesmaximal, characteristic, complemented
14#103×D131C211yesmaximal, characteristic, complemented
15#465×C131C211yesmaximal, characteristic, complemented
16C2×#14523513C2211yescharacteristic
17C2×#14523512C2211yescharacteristic
18C2×Q8×D131C2211yescharacteristic
19#16862275—24no—
20C2×Dic524C2211yescharacteristic
21C2×#14523503C2211yescharacteristic
22C2×#14523501C2211yescharacteristic
23#16862252—24no—
24C2×#14523497C2211yescharacteristic
25C2×#14523496C2211yescharacteristic
26C2×#14523496C2211yescharacteristic
27C2×C8×D131C2211yescharacteristic
28C2×C8×D131C2211yescharacteristic
29#16862245—21no—
30#16862232—21no—
31#16862231—21no—
32#16862228—21no—
33#16862226C2211yescharacteristic
34#16862226C2211yescharacteristic
35#16862225C2211yescharacteristic
36#16862225C2211yescharacteristic
37#16862223—21no—
38#16862222C2211yescharacteristic
39#16862219—21no—
40#16862218—21no—
41#16862217—21no—
42#16862217C2211yescharacteristic
43#16862204C2211yescharacteristic
44C4⋊C4×D131C2211yescharacteristic
45#16862180C2211yescharacteristic
46#16862179C2211yescharacteristic
47#16862177C2211yescharacteristic
48#16862171C2211yescharacteristic
49#16862171C2211yescharacteristic
50#16862170C2211yescharacteristic
51#16862169C2211yescharacteristic
52#16862169C2211yescharacteristic
53#16862167C2211yescharacteristic
54C2×SD16×C131C2211yescharacteristic
55C22×C8×C131C2211yescharacteristic
56C22⋊Q8×C131C2211yescharacteristic
57#121×C131C2211yescharacteristic
58#106×C131C2211yescharacteristic
59#103×C131C2211yescharacteristic
60#102×C131C2211yescharacteristic
61#14523513—22no—
62#14523513—22no—
63#14523512—22no—
64#14523512—22no—
65C2×Dic262—21no—
66C2×Dic262D411yescharacteristic
67C2×Dic262C2311yescharacteristic
68C2×Dic262C2311yescharacteristic
69Q8×D131—22no—
70Q8×D131—22no—
71C2×D524D411yescharacteristic
72C2×C22⋊D131—21no—
73C2×C22⋊D131—21no—
74C2×C22⋊D131D411yescharacteristic
75C2×C22⋊D131D411yescharacteristic
76C22×Dic131—21no—
77C2×C4×D131—21no—
78C2×C4×D131—21no—
79C2×C4×D131C2311yescharacteristic
80C2×C4×D131C2311yescharacteristic
81Dic524—22no—
82#14523503—22no—
83#14523501—22no—
84#14523497—22no—
85#14523496—22no—
86#14523496D414yescomplemented
87C2×C131⋊C8C2311yescharacteristic
88C2×C131⋊C8C2311yescharacteristic
89C8×D131—22no—
90C8×D131—22no—
91C4⋊Dic131—21no—
92C4⋊Dic131—21no—
93C4⋊Dic131C2311yescharacteristic
94C4⋊Dic131C2311yescharacteristic
95Dic131⋊C4—21no—
96Dic131⋊C4—21no—
97Dic131⋊C4—21no—
98Dic131⋊C4—21no—
99C22⋊Dic131—21no—
100C22⋊Dic131—21no—
101C22⋊Dic131—21no—
102(C2×D131)⋊C4—21no—
103(C2×D131)⋊C4—21no—
104(C2×D131)⋊C4—21no—
105(C2×D131)⋊C4—21no—
106C4×Dic131—21no—
107C2×Q8×C131C2311yescharacteristic
108C2×D4×C131—21no—
109C2×D4×C131C2311yescharacteristic
110C22×C4×C131C2311yescharacteristic
111SD16×C131—22no—
112C2×C8×C131—22no—
113C2×C8×C131C2311yescharacteristic
114C2×C8×C131C2311yescharacteristic
115C4⋊C4×C131—21no—
116C4⋊C4×C131C2311yescharacteristic
117C4⋊C4×C131C2311yescharacteristic
118C22⋊C4×C131—21no—
119C22⋊C4×C131—21no—
120C22×D131C2×D411yescharacteristic
121C22×D131C2×D411yescharacteristic
122Dic262—22no—
123Dic262—21no—
124Dic262—21no—
125Dic262—21no—
126Dic262—21no—
127Dic262—42no—
128D524—22no—
129C22⋊D131—22no—
130C22⋊D131—22no—
131C22⋊D131—42no—
132C22⋊D131—81no—
133C2×Dic131—21no—
134C2×Dic131—21no—
135C2×Dic131—21no—
136C2×Dic131—21no—
137C2×Dic131—42no—
138C2×Dic131C2×D411yescharacteristic
139C2×Dic131C2×D411yescharacteristic
140C4×D131—21no—
141C4×D131—21no—
142C4×D131—42no—
143C4×D131—81no—
144C4×D131C2×D412yes—
145C4×D131#4112yes—
146C131⋊C8—21no—
147C131⋊C8C2×D412yes—
148C23×C131—21no—
149C23×C131C2×D411yescharacteristic
150Q8×C131—21no—
151Q8×C131—21no—
152D4×C131—21no—
153D4×C131—21no—
154D4×C131—81no—
155C2×C4×C131—21no—
156C2×C4×C131—21no—
157C2×C4×C131—21no—
158C2×C4×C131—21no—
159C2×C4×C131—21no—
160C2×C4×C131C2×D411yescharacteristic
161C2×C4×C131C2411yescharacteristic
162C8×C131—21no—
163C8×C131C2×D412yes—
164C22×C131—21no—
165C22×C131—21no—
166C22×C131—21no—
167C22×C131—41no—
168C22×C131—41no—
169C22×C131—41no—
170C22×C131C22×D411yescharacteristic
171C2×D131—21no—
172C2×D131—21no—
173C2×D131—21no—
174C2×D131—21no—
175C2×D131—21no—
176C2×D131#12112yes—
177C4×C131—21no—
178C4×C131—41no—
179C4×C131—41no—
180C4×C131—41no—
181C4×C131C22×D411yescharacteristic
182C4×C131C2×#4111yescharacteristic
183Dic131—21no—
184Dic131—41no—
185Dic131—41no—
186Dic131—41no—
187Dic131—41no—
188Dic131#12112yes—
189C2×C131—41no—
190C2×C131—81no—
191C2×C131C2×#12111yescharacteristic
192C2×C131#57711yescharacteristic
193C2×C131#57811yescharacteristic
194D131—22no—
195D131—41no—
196C131#306311yescharacteristic, complemented, Sylow
197#3063—1311nomaximal, Sylow
198C2×#137—1311no—
199C2×#131—1311no—
200#537—1311no—
201#536—1311no—
202#470—1311no—
203#470—1311no—
204#468—1311no—
205#468—1311no—
206#467—1311no—
207#467—1311no—
208#465—1311no—
209#465D13111yescharacteristic, complemented
210#428—1311no—
211#416—1311no—
212C2×#103—1311no—
213C2×#41—1311no—
214C2×#41—1311no—
215C22×Q8—1311no—
216#137—2624no—
217C2×Q16—1311no—
218C2×Q16—1311no—
219C2×SD16—1311no—
220C2×SD16C2×D13111yescharacteristic
221#131—2624no—
222C2×M16—1311no—
223C2×M16—1311no—
224C2×M16—1311no—
225C22×C8—1311no—
226C22×C8—1311no—
227C22×C8C2×D13111yescharacteristic
228C4⋊Q8—2621no—
229#124—2621no—
230#123—2621no—
231#123—2621no—
232C22⋊Q8—2621no—
233C22⋊Q8—1311no—
234C22⋊Q8—1311no—
235C22⋊Q8—1311no—
236C22⋊Q8—1311no—
237C22⋊Q8—1311no—
238C22⋊Q8C2×D13111yescharacteristic
239#121—2621no—
240#121—2621no—
241#121—2621no—
242#121—1311no—
243#121C2×D13111yescharacteristic
244#117—1311no—
245C2×C4⋊C4—1311no—
246#107—1311no—
247#107—1311no—
248#106—1311no—
249#106C2×D13111yescharacteristic
250#103—1311no—
251#103—1311no—
252#103—1311no—
253#103—1311no—
254#103—1311no—
255#103C2×D13111yescharacteristic
256#102—1311no—
257#102C2×D13111yescharacteristic
258#41—2622no—
259#41—2622no—
260#41—2622no—
261#41—2622no—
262C2×Q8—2621no—
263C2×Q8—2622no—
264C2×Q8—2622no—
265C2×Q8—1311no—
266C2×Q8—1311no—
267C2×Q8—1311no—
268C2×Q8C22×D13111yescharacteristic
269C2×D4—21no—
270C2×D4—2621no—
271C2×D4—2621no—
272C2×D4—1311no—
273C2×D4—1311no—
274C2×D4—1311no—
275C2×D4C22×D13111yescharacteristic
276C22×C4—2621no—
277C22×C4—2621no—
278C22×C4—2621no—
279C22×C4—1311no—
280C22×C4—1311no—
281C22×C4C22×D13111yescharacteristic
282Q16—2622no—
283Q16—2622no—
284SD16—22no—
285SD16—2622no—
286M16—2622no—
287M16—2622no—
288M16—1314no—
289C2×C8—22no—
290C2×C8—2622no—
291C2×C8—2622no—
292C2×C8—1311no—
293C2×C8—1311no—
294C2×C8C22×D13111yescharacteristic
295C2×C8C22×D13111yescharacteristic
296C4⋊C4—21no—
297C4⋊C4—2621no—
298C4⋊C4—2621no—
299C4⋊C4—2621no—
300C4⋊C4—2621no—
301C4⋊C4—2621no—
302C4⋊C4—2621no—
303C4⋊C4—1311no—
304C4⋊C4—1311no—
305C4⋊C4C22×D13111yescharacteristic
306C4⋊C4C22×D13111yescharacteristic
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—2621no—
310C22⋊C4—2621no—
311C22⋊C4—2621no—
312C22⋊C4—2621no—
313C22⋊C4—2621no—
314C22⋊C4—2621no—
315C22⋊C4—2621no—
316C42—2621no—
317C23—21no—
318C23—1311no—
319C23—1311no—
320C23D4×D13111yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—2622no—
324Q8—2621no—
325Q8—2621no—
326Q8—2621no—
327Q8—2621no—
328Q8—5242no—
329D4—21no—
330D4—21no—
331D4—2622no—
332D4—2622no—
333D4—2622no—
334D4—81no—
335D4—5242no—
336D4—10481no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—2621no—
343C2×C4—2621no—
344C2×C4—2621no—
345C2×C4—2621no—
346C2×C4—2621no—
347C2×C4—2621no—
348C2×C4—5242no—
349C2×C4—5242no—
350C2×C4—10481no—
351C2×C4—1311no—
352C2×C4—1311no—
353C2×C4—1312no—
354C2×C4—1312no—
355C2×C4D4×D13111yescharacteristic
356C2×C4C23×D13111yescharacteristic
357C8—21no—
358C8—2621no—
359C8—1312no—
360C8D4×D13112yes—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—2621no—
368C22—2621no—
369C22—2621no—
370C22—2621no—
371C22—2621no—
372C22—1312no—
373C22C2×D4×D13111yescharacteristic
374C4—21no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—2621no—
379C4—5241no—
380C4—5241no—
381C4—5241no—
382C4—5241no—
383C4—1312no—
384C4C2×D4×D13111yescharacteristic
385C4#41×D13111yescharacteristic
386C2—41no—
387C2—2622no—
388C2—81no—
389C2—5241no—
390C2#121×D13111yescharacteristic
391C2#2255411811yescharacteristic
392C2#2255412611yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#465×C131C2
socleC22×C131C22×D4
prime radical#22553891C2
hypercenter#465D131
hypocenterC131#3063
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#16862275C2semidirect
2. C2×#16862252C2semidirect
4. #22553885C2semidirect
5. #22553571C2semidirect
6. #22553568C2semidirect
7. #22553556C2semidirect
8. #22553555C2semidirect
9. #22553553C2semidirect
10. #22553552C2semidirect
11. #22553540C2semidirect
12. #22553351C2semidirect
13. #22553305C2semidirect
14. #103×D131C2semidirect
15. #465×C131C2semidirect
86. #14523496D4semidirect
196. C131#3063semidirect
209. #465D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622752
C2×#168622522
#225538912
#225538852
#225535712
#225535682
#225535562
#225535552
#225535532
#225535522
#225535402
#225533512
#225533052
#103×D1312
#465×C1312
#3063131