Finite Group Databasebeta

← #30670257

Subgroups — #30670257

Order 16768.

540 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16862208C211yesmaximal, characteristic
2C2×#16862203C211yesmaximal, characteristic, complemented
3#22553241C211yesmaximal, characteristic, complemented
4#22553239C211yesmaximal, characteristic, complemented
5#22553230C211yesmaximal, characteristic
6#22553229C211yesmaximal, characteristic
7#22553209C211yesmaximal, characteristic, complemented
8#22553207C211yesmaximal, characteristic, complemented
9#22553204C211yesmaximal, characteristic, complemented
10#22553203C211yesmaximal, characteristic, complemented
11#22553193C211yesmaximal, characteristic, complemented
12#22553171C211yesmaximal, characteristic, complemented
13#22553171C211yesmaximal, characteristic, complemented
14#22553171C211yesmaximal, characteristic, complemented
15#387×C131C211yesmaximal, characteristic, complemented
16C22×D524C2211yescharacteristic
17C22×C4×D131C2211yescharacteristic
18C22×C4×D131C2211yescharacteristic
19#16862208—24no—
20#16862203—24no—
21C2×C4⋊Dic131C2211yescharacteristic
22C2×Dic131⋊C4C2211yescharacteristic
23C2×Dic131⋊C4C2211yescharacteristic
24C2×Dic131⋊C4C2211yescharacteristic
25C2×Dic131⋊C4C2211yescharacteristic
26C2×Dic131⋊C4C2211yescharacteristic
27C2×Dic131⋊C4C2211yescharacteristic
28C2×(C2×D131)⋊C4C2211yescharacteristic
29C2×(C2×D131)⋊C4C2211yescharacteristic
30C2×(C2×D131)⋊C4C2211yescharacteristic
31C2×(C2×D131)⋊C4C2211yescharacteristic
32C2×(C2×D131)⋊C4C2211yescharacteristic
33C2×(C2×D131)⋊C4C2211yescharacteristic
34C2×(C2×D131)⋊C4C2211yescharacteristic
35C2×(C2×D131)⋊C4C2211yescharacteristic
36C2×(C2×D131)⋊C4C2211yescharacteristic
37C2×(C2×D131)⋊C4C2211yescharacteristic
38C2×C4×Dic131C2211yescharacteristic
39C2×C4×Dic131C2211yescharacteristic
40C131⋊#95C2211yescharacteristic
41C131⋊#95C2211yescharacteristic
42C131⋊#95C2211yescharacteristic
43C131⋊#95C2211yescharacteristic
44C131⋊#95C2211yescharacteristic
45C131⋊#95C2211yescharacteristic
46C2×C4⋊C4×C131C2211yescharacteristic
47C2×C4⋊C4×C131C2211yescharacteristic
48C2×C42×C131C2211yescharacteristic
49#95×C131C2211yescharacteristic
50#95×C131C2211yescharacteristic
51#95×C131C2211yescharacteristic
52#95×C131C2211yescharacteristic
53C23×D131C2311yescharacteristic
54C23×D131C2311yescharacteristic
55C2×D524—22no—
56C2×D524—22no—
57C2×D524—22no—
58C22×Dic131C2311yescharacteristic
59C22×Dic131C2311yescharacteristic
60C22×Dic131C2311yescharacteristic
61C22×Dic131C2311yescharacteristic
62C22×Dic131C2311yescharacteristic
63C22×Dic131C2311yescharacteristic
64C2×C4×D131—22no—
65C2×C4×D131—22no—
66C2×C4×D131—22no—
67C2×C4×D131—22no—
68C2×C4×D131—22no—
69C2×C4×D131D414yes—
70C4⋊Dic131—41no—
71Dic131⋊C4—22no—
72Dic131⋊C4—22no—
73Dic131⋊C4—22no—
74Dic131⋊C4—22no—
75Dic131⋊C4—22no—
76Dic131⋊C4—41no—
77(C2×D131)⋊C4—22no—
78(C2×D131)⋊C4—22no—
79(C2×D131)⋊C4—22no—
80(C2×D131)⋊C4—22no—
81(C2×D131)⋊C4—22no—
82(C2×D131)⋊C4—22no—
83(C2×D131)⋊C4—41no—
84(C2×D131)⋊C4—41no—
85(C2×D131)⋊C4—41no—
86(C2×D131)⋊C4—41no—
87C4×Dic131—22no—
88C4×Dic131—22no—
89C22×C4×C131C2311yescharacteristic
90C22×C4×C131C2311yescharacteristic
91C22×C4×C131C2311yescharacteristic
92C22×C4×C131C2311yescharacteristic
93C22×C4×C131C2311yescharacteristic
94C22×C4×C131C2311yescharacteristic
95C22×C4×C131C2311yescharacteristic
96C4⋊C4×C131—22no—
97C4⋊C4×C131—22no—
98C42×C131—22no—
99C22×D131—21no—
100C22×D131—21no—
101C22×D131—21no—
102C22×D131—21no—
103C22×D131—21no—
104C22×D131—21no—
105C22×D131—21no—
106C22×D131—21no—
107C22×D131—21no—
108C22×D131—21no—
109C22×D131—21no—
110C22×D131—21no—
111C22×D131C2×D412yes—
112C22×D131#4112yes—
113D524—28no—
114C2×Dic131—21no—
115C2×Dic131—21no—
116C2×Dic131—21no—
117C2×Dic131—21no—
118C2×Dic131—21no—
119C2×Dic131—21no—
120C2×Dic131—21no—
121C2×Dic131—21no—
122C2×Dic131—21no—
123C2×Dic131—21no—
124C2×Dic131—21no—
125C2×Dic131—21no—
126C2×Dic131—21no—
127C2×Dic131—21no—
128C2×Dic131—21no—
129C2×Dic131C2×D412yes—
130C2×Dic131#4112yes—
131C2×Dic131#4112yes—
132C4×D131—28no—
133C4×D131—44no—
134C23×C131C2411yescharacteristic
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×C131—21no—
149C2×C4×C131—21no—
150C2×C4×C131—21no—
151C2×C4×C131—21no—
152C2×C4×C131—21no—
153C2×C4×C131—21no—
154C2×C4×C131C2×D412yes—
155C2×C4×C131#4112yes—
156C22×C131C22×D411yescharacteristic
157C22×C131C2×#4111yescharacteristic
158C22×C131C2×#4111yescharacteristic
159C22×C131C2×#4111yescharacteristic
160C22×C131C2×#4111yescharacteristic
161C22×C131H(5,𝔽2)11yescharacteristic
162C22×C1312−1+411yescharacteristic
163C2×D131—22no—
164C2×D131—22no—
165C2×D131—22no—
166C2×D131—22no—
167C2×D131—22no—
168C2×D131—22no—
169C2×D131—41no—
170C2×D131—41no—
171C2×D131—41no—
172C2×D131—41no—
173C2×D131—41no—
174C2×D131—41no—
175C2×D131—41no—
176C2×D131—41no—
177C4×C131—22no—
178C4×C131—22no—
179C4×C131—41no—
180C4×C131—41no—
181C4×C131—41no—
182C4×C131—41no—
183C4×C131—41no—
184Dic131—41no—
185Dic131—41no—
186Dic131—41no—
187Dic131—41no—
188Dic131—41no—
189Dic131#12314yes—
190C2×C131C2×#12311yescharacteristic
191C2×C131#52511yescharacteristic
192C2×C131#53711yescharacteristic
193C2×C131#55211yescharacteristic
194C2×C131#55311yescharacteristic
195C2×C131#55511yescharacteristic
196C2×C131#55611yescharacteristic
197D131—42no—
198D131—42no—
199C131#248511yescharacteristic, complemented, Sylow
200#2485—1311nomaximal, Sylow
201C2×C4×D4—1311no—
202C2×#117—1311no—
203#397—1311no—
204#397—1311no—
205#394—1311no—
206#394—1311no—
207#389—1311no—
208#389—1311no—
209#388—1311no—
210#388—1311no—
211#387—1311no—
212#387D13111yescharacteristic, complemented
213#380—1311no—
214#380—1311no—
215#380—1311no—
216C22×D4—1311no—
217C23×C4—1311no—
218C23×C4—1311no—
219C4×D4—2624no—
220#117—2624no—
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⋊C4—1311no—
228C2×C4⋊C4C2×D13111yescharacteristic
229C2×C4⋊C4C2×D13111yescharacteristic
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⋊C4—1311no—
238C2×C22⋊C4—1311no—
239C2×C22⋊C4—1311no—
240C2×C42—1311no—
241C2×C42—1311no—
242C2×C42C2×D13111yescharacteristic
243#95—1311no—
244#95—1311no—
245#95—1311no—
246#95—1311no—
247#95—1311no—
248#95—1311no—
249#95C2×D13111yescharacteristic
250#95C2×D13111yescharacteristic
251#95C2×D13111yescharacteristic
252#95C2×D13111yescharacteristic
253C24—1311no—
254C24—1311no—
255C2×D4—2622no—
256C2×D4—2622no—
257C2×D4—2622no—
258C22×C4—2622no—
259C22×C4—2622no—
260C22×C4—2622no—
261C22×C4—2622no—
262C22×C4—2622no—
263C22×C4—1311no—
264C22×C4—1311no—
265C22×C4—1311no—
266C22×C4—1311no—
267C22×C4—1311no—
268C22×C4—1314no—
269C22×C4—1311no—
270C22×C4C22×D13111yescharacteristic
271C22×C4C22×D13111yescharacteristic
272C22×C4C22×D13111yescharacteristic
273C22×C4C22×D13111yescharacteristic
274C22×C4C22×D13111yescharacteristic
275C22×C4C22×D13111yescharacteristic
276C22×C4C22×D13111yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—2622no—
280C4⋊C4—2622no—
281C4⋊C4—2622no—
282C4⋊C4—2622no—
283C4⋊C4—2622no—
284C4⋊C4—5241no—
285C4⋊C4—5241no—
286C22⋊C4—2622no—
287C22⋊C4—2622no—
288C22⋊C4—2622no—
289C22⋊C4—2622no—
290C22⋊C4—2622no—
291C22⋊C4—2622no—
292C22⋊C4—5241no—
293C22⋊C4—5241no—
294C22⋊C4—5241no—
295C22⋊C4—5241no—
296C42—22no—
297C42—2622no—
298C42—2622no—
299C23—2621no—
300C23—2621no—
301C23—2621no—
302C23—2621no—
303C23—2621no—
304C23—2621no—
305C23—2621no—
306C23—2621no—
307C23—2621no—
308C23—2621no—
309C23—2621no—
310C23—2621no—
311C23—1312no—
312C23—1312no—
313C23C23×D13111yescharacteristic
314D4—2628no—
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—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—2628no—
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—2621no—
349C2×C4—2621no—
350C2×C4—5244no—
351C2×C4—1312no—
352C2×C4—1312no—
353C2×C4—1312no—
354C2×C4D4×D13112yes—
355C2×C4#1452351312yes—
356C22—2622no—
357C22—2622no—
358C22—2622no—
359C22—2622no—
360C22—2622no—
361C22—2622no—
362C22—5241no—
363C22—5241no—
364C22—5241no—
365C22—5241no—
366C22—5241no—
367C22—5241no—
368C22—5241no—
369C22—5241no—
370C22C2×D4×D13111yescharacteristic
371C22#41×D13111yescharacteristic
372C22#41×D13111yescharacteristic
373C22#41×D13111yescharacteristic
374C22C2×#1452351311yescharacteristic
375C22#1686229211yescharacteristic
376C22#1686229411yescharacteristic
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—5241no—
385C4—5241no—
386C4—5241no—
387C4—5241no—
388C4—5241no—
389C4—1314no—
390C2—5242no—
391C2—5242no—
392C2#123×D13111yescharacteristic
393C2#2255382111yescharacteristic
394C2#2255389211yescharacteristic
395C2#2255399411yescharacteristic
396C2#2255401311yescharacteristic
397C2#2255402511yescharacteristic
398C2#2255403411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#387×C131C2
prime radicalC22×D524C22
hypercenter#387D131
hypocenterC131#2485
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. C2×#16862203C2semidirect
3. #22553241C2semidirect
4. #22553239C2semidirect
7. #22553209C2semidirect
8. #22553207C2semidirect
9. #22553204C2semidirect
10. #22553203C2semidirect
11. #22553193C2semidirect
12. #22553171C2semidirect
13. #22553171C2semidirect
14. #22553171C2semidirect
15. #387×C131C2semidirect
199. C131#2485semidirect
212. #387D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622082
C2×#168622032
#225532412
#225532392
#225532302
#225532292
#225532092
#225532072
#225532042
#225532032
#225531932
#225531712
#387×C1312
#2485131