Finite Group Databasebeta

← #30674267

Subgroups — #30674267

Order 16768.

556 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16862236C211yesmaximal, characteristic, complemented
2C22×C4×Dic131C211yesmaximal, characteristic
3#22553209C211yesmaximal, characteristic, complemented
4#22553207C211yesmaximal, characteristic, complemented
5#22553198C211yesmaximal, characteristic, complemented
6#22553197C211yesmaximal, characteristic, complemented
7#22553194C211yesmaximal, characteristic, complemented
8#22553193C211yesmaximal, characteristic, complemented
9#22553187C211yesmaximal, characteristic
10#22553185C211yesmaximal, characteristic
11#22553184C211yesmaximal, characteristic
12#22553176C211yesmaximal, characteristic, complemented
13#22553170C211yesmaximal, characteristic
14#22553170C211yesmaximal, characteristic
15#385×C131C211yesmaximal, characteristic
16C23×Dic131C2211yescharacteristic
17C23×Dic131C2211yescharacteristic
18#16862236—24no—
19C2×C4⋊Dic131C2211yescharacteristic
20C2×C4⋊Dic131C2211yescharacteristic
21C2×Dic131⋊C4C2211yescharacteristic
22C2×Dic131⋊C4C2211yescharacteristic
23C2×Dic131⋊C4C2211yescharacteristic
24C2×Dic131⋊C4C2211yescharacteristic
25C2×Dic131⋊C4C2211yescharacteristic
26C2×Dic131⋊C4C2211yescharacteristic
27C2×Dic131⋊C4C2211yescharacteristic
28C2×Dic131⋊C4C2211yescharacteristic
29C2×C22⋊Dic131C2211yescharacteristic
30C2×C22⋊Dic131C2211yescharacteristic
31C2×C4×Dic131—24no—
32C2×C4×Dic131C2211yescharacteristic
33C2×C4×Dic131C2211yescharacteristic
34C2×C4×Dic131C2211yescharacteristic
35C2×C4×Dic131C2211yescharacteristic
36C131⋊#95C2211yescharacteristic
37C131⋊#95C2211yescharacteristic
38C131⋊#95C2211yescharacteristic
39C131⋊#95C2211yescharacteristic
40C131⋊#95C2211yescharacteristic
41C131⋊#95C2211yescharacteristic
42C131⋊#95C2211yescharacteristic
43C131⋊#95C2211yescharacteristic
44C131⋊#95C2211yescharacteristic
45C131⋊#95C2211yescharacteristic
46C23×C4×C131C2211yescharacteristic
47C2×C4⋊C4×C131C2211yescharacteristic
48C2×C4⋊C4×C131C2211yescharacteristic
49C2×C22⋊C4×C131C2211yescharacteristic
50C2×C22⋊C4×C131C2211yescharacteristic
51#95×C131C2211yescharacteristic
52#95×C131C2211yescharacteristic
53C22×Dic131—22no—
54C22×Dic131—22no—
55C22×Dic131—22no—
56C22×Dic131—22no—
57C22×Dic131—22no—
58C22×Dic131Q814yes—
59C22×Dic131C2311yescharacteristic
60C22×Dic131C2311yescharacteristic
61C22×Dic131C2311yescharacteristic
62C22×Dic131C2311yescharacteristic
63C22×Dic131C2311yescharacteristic
64C22×Dic131C2311yescharacteristic
65C22×Dic131C2311yescharacteristic
66C22×Dic131C2311yescharacteristic
67C4⋊Dic131—22no—
68C4⋊Dic131—22no—
69Dic131⋊C4—22no—
70Dic131⋊C4—22no—
71Dic131⋊C4—22no—
72Dic131⋊C4—22no—
73Dic131⋊C4—22no—
74Dic131⋊C4—22no—
75Dic131⋊C4—22no—
76Dic131⋊C4—22no—
77C22⋊Dic131—22no—
78C22⋊Dic131—22no—
79C4×Dic131—22no—
80C4×Dic131—22no—
81C4×Dic131—22no—
82C4×Dic131D414yes—
83C24×C131C2311yescharacteristic
84C22×C4×C131—22no—
85C22×C4×C131—22no—
86C22×C4×C131—22no—
87C22×C4×C131C2311yescharacteristic
88C22×C4×C131C2311yescharacteristic
89C22×C4×C131C2311yescharacteristic
90C22×C4×C131C2311yescharacteristic
91C22×C4×C131C2311yescharacteristic
92C22×C4×C131C2311yescharacteristic
93C4⋊C4×C131—22no—
94C4⋊C4×C131—22no—
95C22⋊C4×C131—22no—
96C22⋊C4×C131—22no—
97C2×Dic131—28no—
98C2×Dic131—21no—
99C2×Dic131—21no—
100C2×Dic131—21no—
101C2×Dic131—21no—
102C2×Dic131—21no—
103C2×Dic131—21no—
104C2×Dic131—28no—
105C2×Dic131—21no—
106C2×Dic131—21no—
107C2×Dic131—21no—
108C2×Dic131—21no—
109C2×Dic131—21no—
110C2×Dic131—21no—
111C2×Dic131—21no—
112C2×Dic131—21no—
113C2×Dic131—21no—
114C2×Dic131—21no—
115C2×Dic131—21no—
116C2×Dic131—21no—
117C2×Dic131C2×D412yes—
118C2×Dic131C2×D412yes—
119C2×Dic131C2×Q812yes—
120C2×Dic131C2×Q812yes—
121C2×Dic131#4112yes—
122C2×Dic131#4112yes—
123C23×C131—21no—
124C23×C131—21no—
125C23×C131—21no—
126C23×C131—21no—
127C23×C131C2×Q812yes—
128C23×C131#4112yes—
129C23×C131#4112yes—
130C23×C131C2411yescharacteristic
131C2×C4×C131—28no—
132C2×C4×C131—21no—
133C2×C4×C131—21no—
134C2×C4×C131—21no—
135C2×C4×C131—21no—
136C2×C4×C131—21no—
137C2×C4×C131—21no—
138C2×C4×C131—21no—
139C2×C4×C131—21no—
140C2×C4×C131—21no—
141C2×C4×C131—21no—
142C2×C4×C131—21no—
143C2×C4×C131—21no—
144C2×C4×C131—21no—
145C2×C4×C131—21no—
146C2×C4×C131—21no—
147C2×C4×C131—21no—
148C2×C4×C131C2×D412yes—
149C2×C4×C131#4112yes—
150C22×C131—22no—
151C22×C131—22no—
152C22×C131—22no—
153C22×C131—22no—
154C22×C131—22no—
155C22×C131—22no—
156C22×C131#12514yescomplemented
157C22×C131C22×D411yescharacteristic
158C22×C131C22×Q811yescharacteristic
159C22×C131C2×#4111yescharacteristic
160C22×C131C2×#4111yescharacteristic
161C22×C131C2×#4111yescharacteristic
162C22×C131C2×#4111yescharacteristic
163C22×C131C2×#4111yescharacteristic
164C4×C131—22no—
165C4×C131—22no—
166C4×C131—41no—
167C4×C131—41no—
168C4×C131—41no—
169C4×C131—41no—
170Dic131—22no—
171Dic131—22no—
172Dic131—22no—
173Dic131—41no—
174Dic131—41no—
175Dic131—41no—
176Dic131—41no—
177Dic131C22⋊Q814yes—
178C2×C131—24no—
179C2×C131C2×C22⋊Q811yescharacteristic
180C2×C131#52511yescharacteristic
181C2×C131#52511yescharacteristic
182C2×C131C2×#12511yescharacteristic
183C2×C131#52911yescharacteristic
184C2×C131#52911yescharacteristic
185C2×C131#53311yescharacteristic
186C131#285111yescharacteristic, complemented, Sylow
187#2851—1311nomaximal, Sylow
188C2×#125—1311no—
189C22×C42—1311no—
190#389—1311no—
191#389—1311no—
192#387—1311no—
193#387—1311no—
194#387—1311no—
195#387—1311no—
196#385—1311no—
197#385—1311no—
198#385—1311no—
199#385D13111yescharacteristic
200#382—1311no—
201#380—1311no—
202#380—1311no—
203C23×C4—1311no—
204C23×C4—1311no—
205C23×C4C2×D13111yescharacteristic
206#125—2624no—
207C2×C4⋊C4—1311no—
208C2×C4⋊C4—1311no—
209C2×C4⋊C4—1311no—
210C2×C4⋊C4—1311no—
211C2×C4⋊C4—1311no—
212C2×C4⋊C4—1311no—
213C2×C4⋊C4—1311no—
214C2×C4⋊C4—1311no—
215C2×C4⋊C4—1311no—
216C2×C4⋊C4—1311no—
217C2×C4⋊C4C2×D13111yescharacteristic
218C2×C4⋊C4C2×D13111yescharacteristic
219C2×C22⋊C4—1311no—
220C2×C22⋊C4—1311no—
221C2×C22⋊C4C2×D13111yescharacteristic
222C2×C22⋊C4C2×D13111yescharacteristic
223C2×C42—2624no—
224C2×C42—1311no—
225C2×C42—1311no—
226C2×C42—1311no—
227C2×C42—1311no—
228#95—1311no—
229#95—1311no—
230#95—1311no—
231#95—1311no—
232#95—1311no—
233#95—1311no—
234#95—1311no—
235#95—1311no—
236#95—1311no—
237#95—1311no—
238#95C2×D13111yescharacteristic
239#95C2×D13111yescharacteristic
240C24C22×D13111yescharacteristic
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—2622no—
245C22×C4—2622no—
246C22×C4—2622no—
247C22×C4—2622no—
248C22×C4—2622no—
249C22×C4—1311no—
250C22×C4—1311no—
251C22×C4—1311no—
252C22×C4—1311no—
253C22×C4—1314no—
254C22×C4—1311no—
255C22×C4—1311no—
256C22×C4—1311no—
257C22×C4—1311no—
258C22×C4C22×D13111yescharacteristic
259C22×C4C22×D13111yescharacteristic
260C22×C4C22×D13111yescharacteristic
261C22×C4C22×D13111yescharacteristic
262C22×C4C22×D13111yescharacteristic
263C22×C4C22×D13111yescharacteristic
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—2622no—
267C4⋊C4—2622no—
268C4⋊C4—2622no—
269C4⋊C4—2622no—
270C4⋊C4—2622no—
271C4⋊C4—2622no—
272C4⋊C4—2622no—
273C4⋊C4—2622no—
274C4⋊C4—2622no—
275C4⋊C4—2622no—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—2622no—
279C22⋊C4—2622no—
280C42—2622no—
281C42—2622no—
282C42—2622no—
283C42—1314no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23Q8×D13112yes—
289C23#1452351212yes—
290C23#1452351312yes—
291C23C23×D13111yescharacteristic
292C2×C4—28no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—2628no—
310C2×C4—2621no—
311C2×C4—2621no—
312C2×C4—2621no—
313C2×C4—2621no—
314C2×C4—2621no—
315C2×C4—2621no—
316C2×C4—2628no—
317C2×C4—2621no—
318C2×C4—2621no—
319C2×C4—2621no—
320C2×C4—2621no—
321C2×C4—2621no—
322C2×C4—2621no—
323C2×C4—2621no—
324C2×C4—2621no—
325C2×C4—2621no—
326C2×C4—2621no—
327C2×C4—2621no—
328C2×C4—2621no—
329C2×C4—1312no—
330C2×C4—1312no—
331C2×C4—1312no—
332C2×C4—1312no—
333C2×C4—1312no—
334C2×C4—1312no—
335C2×C4D4×D13112yes—
336C2×C4#1452351212yes—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22#1686223614yescomplemented
344C22C2×D4×D13111yescharacteristic
345C22C2×Q8×D13111yescharacteristic
346C22#41×D13111yescharacteristic
347C22#41×D13111yescharacteristic
348C22C2×#1452351211yescharacteristic
349C22C2×#1452351211yescharacteristic
350C22C2×#1452351311yescharacteristic
351C4—22no—
352C4—22no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—2622no—
358C4—2622no—
359C4—2622no—
360C4—5241no—
361C4—5241no—
362C4—5241no—
363C4—5241no—
364C4—1314no—
365C2—24no—
366C2C22⋊Q8×D13111yescharacteristic
367C2#2255381811yescharacteristic
368C2#2255382111yescharacteristic
369C2C2×#1686223611yescharacteristic
370C2#2255383911yescharacteristic
371C2#2255384211yescharacteristic
372C2#2255386411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#385×C131C2
prime radicalC24×C131C23
hypercenter#385D131
hypocenterC131#2851
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#16862236C2semidirect
3. #22553209C2semidirect
4. #22553207C2semidirect
5. #22553198C2semidirect
6. #22553197C2semidirect
7. #22553194C2semidirect
8. #22553193C2semidirect
12. #22553176C2semidirect
156. C22×C131#125semidirect
186. C131#2851semidirect
343. C22#16862236semidirect

Maximal subgroups

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

subgroupindex
C2×#168622362
C22×C4×Dic1312
#225532092
#225532072
#225531982
#225531972
#225531942
#225531932
#225531872
#225531852
#225531842
#225531762
#225531702
#385×C1312
#2851131