Finite Group Databasebeta

← #30671071

Subgroups — #30671071

Order 16768.

510 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16862236C211yesmaximal, characteristic, complemented
2#22553251C211yesmaximal, characteristic, complemented
3#22553235C211yesmaximal, characteristic, complemented
4#22553235C211yesmaximal, characteristic, complemented
5#22553226C211yesmaximal, characteristic
6#22553215C211yesmaximal, characteristic
7#22553207C211yesmaximal, characteristic, complemented
8#22553206C211yesmaximal, characteristic, complemented
9#22553206C211yesmaximal, characteristic, complemented
10#22553206C211yesmaximal, characteristic, complemented
11#22553200C211yesmaximal, characteristic, complemented
12#22553185C211yesmaximal, characteristic, complemented
13C4⋊C4×Dic131C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic131C211yesmaximal, characteristic, complemented
15C2×#123×C131C211yesmaximal, characteristic
16#16862236—24no—
17C2×C4⋊Dic131C2211yescharacteristic
18C2×C4⋊Dic131C2211yescharacteristic
19C2×Dic131⋊C4C2211yescharacteristic
20C2×Dic131⋊C4C2211yescharacteristic
21C2×Dic131⋊C4C2211yescharacteristic
22C2×Dic131⋊C4C2211yescharacteristic
23C2×Dic131⋊C4C2211yescharacteristic
24C2×Dic131⋊C4C2211yescharacteristic
25C2×C22⋊Dic131C2211yescharacteristic
26C2×C22⋊Dic131C2211yescharacteristic
27C2×C22⋊Dic131C2211yescharacteristic
28C2×C22⋊Dic131C2211yescharacteristic
29C2×C22⋊Dic131C2211yescharacteristic
30C2×C22⋊Dic131C2211yescharacteristic
31C2×C22⋊Dic131C2211yescharacteristic
32C2×C22⋊Dic131C2211yescharacteristic
33C2×C4×Dic131C2211yescharacteristic
34C2×C4×Dic131C2211yescharacteristic
35C2×C4×Dic131C2211yescharacteristic
36C2×C4×Dic131C2211yescharacteristic
37C131⋊#95C2211yescharacteristic
38C131⋊#95C2211yescharacteristic
39C131⋊#95C2211yescharacteristic
40C131⋊#95C2211yescharacteristic
41C131⋊#95C2211yescharacteristic
42C131⋊#95C2211yescharacteristic
43C131⋊#95C2211yescharacteristic
44C131⋊#95C2211yescharacteristic
45C22×D4×C131C2211yescharacteristic
46C23×C4×C131C2211yescharacteristic
47#123×C131—24no—
48C2×C4⋊C4×C131C2211yescharacteristic
49C2×C4⋊C4×C131C2211yescharacteristic
50C2×C22⋊C4×C131C2211yescharacteristic
51C2×C22⋊C4×C131C2211yescharacteristic
52C2×C22⋊C4×C131C2211yescharacteristic
53C22×Dic131C2311yescharacteristic
54C22×Dic131C2311yescharacteristic
55C22×Dic131C2311yescharacteristic
56C22×Dic131C2311yescharacteristic
57C22×Dic131C2311yescharacteristic
58C22×Dic131C2311yescharacteristic
59C22×Dic131C2311yescharacteristic
60C22×Dic131C2311yescharacteristic
61C4⋊Dic131—22no—
62C4⋊Dic131—22no—
63Dic131⋊C4—22no—
64Dic131⋊C4—22no—
65Dic131⋊C4—22no—
66Dic131⋊C4—22no—
67Dic131⋊C4—22no—
68Dic131⋊C4—41no—
69C22⋊Dic131—22no—
70C22⋊Dic131—22no—
71C22⋊Dic131—22no—
72C22⋊Dic131—22no—
73C22⋊Dic131—22no—
74C22⋊Dic131—22no—
75C22⋊Dic131—22no—
76C22⋊Dic131—41no—
77C4×Dic131—22no—
78C4×Dic131—22no—
79C4×Dic131—41no—
80C4×Dic131—41no—
81C24×C131C2311yescharacteristic
82C24×C131C2311yescharacteristic
83C2×D4×C131—22no—
84C2×D4×C131—22no—
85C2×D4×C131—41no—
86C22×C4×C131—22no—
87C22×C4×C131—22no—
88C22×C4×C131—41no—
89C22×C4×C131C2311yescharacteristic
90C22×C4×C131C2311yescharacteristic
91C22×C4×C131C2311yescharacteristic
92C22×C4×C131C2311yescharacteristic
93C22×C4×C131C2311yescharacteristic
94C4⋊C4×C131—22no—
95C4⋊C4×C131D414yes—
96C22⋊C4×C131—22no—
97C22⋊C4×C131—22no—
98C22⋊C4×C131—22no—
99C2×Dic131—21no—
100C2×Dic131—21no—
101C2×Dic131—21no—
102C2×Dic131—21no—
103C2×Dic131—21no—
104C2×Dic131—21no—
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×Dic131—21no—
118C2×Dic131—21no—
119C2×Dic131—21no—
120C2×Dic131#4112yes—
121C2×Dic131#4112yes—
122C2×Dic131#4112yes—
123C23×C131—21no—
124C23×C131—21no—
125C23×C131—21no—
126C23×C131—21no—
127C23×C131—21no—
128C23×C131—21no—
129C23×C131—21no—
130C23×C131—21no—
131C23×C131—21no—
132C23×C131—21no—
133C23×C131—21no—
134C23×C131—21no—
135C23×C131—21no—
136C23×C131#4112yes—
137C23×C131C2411yescharacteristic
138D4×C131—44no—
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×C131—21no—
149C2×C4×C131—21no—
150C2×C4×C131—21no—
151C2×C4×C131—44no—
152C2×C4×C131C2×D412yes—
153C2×C4×C131C2×D412yes—
154C2×C4×C131C2×D412yes—
155C22×C131—22no—
156C22×C131—22no—
157C22×C131—22no—
158C22×C131—41no—
159C22×C131—41no—
160C22×C131—41no—
161C22×C131—41no—
162C22×C131—41no—
163C22×C131—41no—
164C22×C131—41no—
165C22×C131—41no—
166C22×C131—41no—
167C22×C131—41no—
168C22×C131—41no—
169C22×C131C22×D411yescharacteristic
170C22×C131C2×#4111yescharacteristic
171C22×C131C2×#4111yescharacteristic
172C22×C131C2×#4111yescharacteristic
173C22×C131C2×#4111yescharacteristic
174C22×C131H(5,𝔽2)11yescharacteristic
175C22×C1312−1+411yescharacteristic
176C4×C131—22no—
177C4×C131—22no—
178C4×C131—22no—
179C4×C131—41no—
180C4×C131—41no—
181Dic131—22no—
182Dic131—22no—
183Dic131—22no—
184Dic131—41no—
185Dic131—41no—
186Dic131—41no—
187Dic131—41no—
188Dic131—41no—
189C2×C131—42no—
190C2×C131—81no—
191C2×C131#52911yescharacteristic
192C2×C131#53211yescharacteristic
193C2×C131#53911yescharacteristic
194C2×C131#54611yescharacteristic
195C2×C131#54811yescharacteristic
196C2×C131#55211yescharacteristic
197C2×C131#55311yescharacteristic
198C131#255811yescharacteristic, complemented, Sylow
199#2558—1311nomaximal, Sylow
200C2×#125—1311no—
201C2×#123D13111yescharacteristic
202#400—1311no—
203#396—1311no—
204#396—1311no—
205#394—1311no—
206#390—1311no—
207#389—1311no—
208#388—1311no—
209#388—1311no—
210#388—1311no—
211#388—1311no—
212#385—1311no—
213C4×C4⋊C4—1311no—
214C4×C22⋊C4—1311no—
215C22×D4C2×D13111yescharacteristic
216C23×C4C2×D13111yescharacteristic
217#125—2624no—
218#123—24no—
219C2×C4⋊C4—1311no—
220C2×C4⋊C4—1311no—
221C2×C4⋊C4—1311no—
222C2×C4⋊C4—1311no—
223C2×C4⋊C4—1311no—
224C2×C4⋊C4—1311no—
225C2×C4⋊C4—1311no—
226C2×C4⋊C4—1311no—
227C2×C4⋊C4C2×D13111yescharacteristic
228C2×C4⋊C4C2×D13111yescharacteristic
229C2×C22⋊C4—1311no—
230C2×C22⋊C4—1311no—
231C2×C22⋊C4—1311no—
232C2×C22⋊C4—1311no—
233C2×C22⋊C4—1311no—
234C2×C22⋊C4—1311no—
235C2×C22⋊C4—1311no—
236C2×C22⋊C4—1311no—
237C2×C22⋊C4C2×D13111yescharacteristic
238C2×C22⋊C4C2×D13111yescharacteristic
239C2×C22⋊C4C2×D13111yescharacteristic
240C2×C42—1311no—
241C2×C42—1311no—
242C2×C42—1311no—
243C2×C42—1311no—
244#95—1311no—
245#95—1311no—
246#95—1311no—
247#95—1311no—
248#95—1311no—
249#95—1311no—
250#95—1311no—
251#95—1311no—
252C24C22×D13111yescharacteristic
253C24C22×D13111yescharacteristic
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—41no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—41no—
260C22×C4—1311no—
261C22×C4—1311no—
262C22×C4—1311no—
263C22×C4—1311no—
264C22×C4—1311no—
265C22×C4—1311no—
266C22×C4—1311no—
267C22×C4—1311no—
268C22×C4C22×D13111yescharacteristic
269C22×C4C22×D13111yescharacteristic
270C22×C4C22×D13111yescharacteristic
271C22×C4C22×D13111yescharacteristic
272C22×C4C22×D13111yescharacteristic
273C4⋊C4—22no—
274C4⋊C4—2622no—
275C4⋊C4—2622no—
276C4⋊C4—2622no—
277C4⋊C4—2622no—
278C4⋊C4—2622no—
279C4⋊C4—2622no—
280C4⋊C4—2622no—
281C4⋊C4—5241no—
282C4⋊C4C22⋊D13114yes—
283C22⋊C4—22no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—2622no—
287C22⋊C4—2622no—
288C22⋊C4—2622no—
289C22⋊C4—2622no—
290C22⋊C4—2622no—
291C22⋊C4—2622no—
292C22⋊C4—2622no—
293C22⋊C4—5241no—
294C42—2622no—
295C42—2622no—
296C42—5241no—
297C42—5241no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23#1452351212yes—
312C23C23×D13111yescharacteristic
313D4—44no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—44no—
327C2×C4—2621no—
328C2×C4—2621no—
329C2×C4—2621no—
330C2×C4—2621no—
331C2×C4—2621no—
332C2×C4—2621no—
333C2×C4—2621no—
334C2×C4—2621no—
335C2×C4—2621no—
336C2×C4—2621no—
337C2×C4—2621no—
338C2×C4—2621no—
339C2×C4—2621no—
340C2×C4—2621no—
341C2×C4—2621no—
342C2×C4—2621no—
343C2×C4—2621no—
344C2×C4—2621no—
345C2×C4—2621no—
346C2×C4—2621no—
347C2×C4—2621no—
348C2×C4—1312no—
349C2×C4—1312no—
350C2×C4—1312no—
351C2×C4C2×C22⋊D13112yes—
352C2×C4C2×C22⋊D13112yes—
353C2×C4C2×C22⋊D13112yes—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22C22×C22⋊D13111yescharacteristic
369C22#41×D13111yescharacteristic
370C22#41×D13111yescharacteristic
371C22#41×D13111yescharacteristic
372C22C2×#1452351211yescharacteristic
373C22#1686229111yescharacteristic
374C22#1686229311yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—2622no—
381C4—2622no—
382C4—2622no—
383C4—5241no—
384C4—5241no—
385C4—5241no—
386C4—5241no—
387C4—5241no—
388C2—42no—
389C2—81no—
390C2#2255383911yescharacteristic
391C2#2255385811yescharacteristic
392C2#2255390511yescharacteristic
393C2#2255395611yescharacteristic
394C2#2255396911yescharacteristic
395C2#2255399911yescharacteristic
396C2#2255400711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
FittingC2×#123×C131C2
prime radicalC22×D4×C131C22
hypercenterC2×#123D131
hypocenterC131#2558
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#16862236C2semidirect
2. #22553251C2semidirect
3. #22553235C2semidirect
4. #22553235C2semidirect
7. #22553207C2semidirect
8. #22553206C2semidirect
9. #22553206C2semidirect
10. #22553206C2semidirect
11. #22553200C2semidirect
12. #22553185C2semidirect
13. C4⋊C4×Dic131C2semidirect
14. C4×C22⋊Dic131C2semidirect
198. C131#2558semidirect

Maximal subgroups

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

subgroupindex
C2×#168622362
#225532512
#225532352
#225532262
#225532152
#225532072
#225532062
#225532002
#225531852
C4⋊C4×Dic1312
C4×C22⋊Dic1312
C2×#123×C1312
#2558131