Finite Group Databasebeta

← #30670308

Subgroups — #30670308

Order 16768.

560 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16862236C211yesmaximal, characteristic, complemented
2C2×#16862219C211yesmaximal, characteristic
3#22553241C211yesmaximal, characteristic, complemented
4#22553230C211yesmaximal, characteristic, complemented
5#22553228C211yesmaximal, characteristic, complemented
6#22553213C211yesmaximal, characteristic, complemented
7#22553212C211yesmaximal, characteristic, complemented
8#22553212C211yesmaximal, characteristic, complemented
9#22553212C211yesmaximal, characteristic, complemented
10#22553206C211yesmaximal, characteristic, complemented
11#22553204C211yesmaximal, characteristic, complemented
12#22553171C211yesmaximal, characteristic, complemented
13#22553163C211yesmaximal, characteristic, complemented
14#22553159C211yesmaximal, characteristic, complemented
15#397×C131C211yesmaximal, characteristic, complemented
16C22×D524C2211yescharacteristic
17C22×C22⋊D131C2211yescharacteristic
18C22×C22⋊D131C2211yescharacteristic
19C22×C4×D131C2211yescharacteristic
20#16862236—24no—
21#16862219—24no—
22C2×C4⋊Dic131C2211yescharacteristic
23C2×Dic131⋊C4C2211yescharacteristic
24C2×Dic131⋊C4C2211yescharacteristic
25C2×Dic131⋊C4C2211yescharacteristic
26C2×Dic131⋊C4C2211yescharacteristic
27C2×Dic131⋊C4C2211yescharacteristic
28C2×C22⋊Dic131C2211yescharacteristic
29C2×C22⋊Dic131C2211yescharacteristic
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×(C2×D131)⋊C4C2211yescharacteristic
39C2×C4×Dic131C2211yescharacteristic
40C2×C4×Dic131C2211yescharacteristic
41C2×C4×Dic131C2211yescharacteristic
42C2×C4×Dic131C2211yescharacteristic
43C131⋊#95C2211yescharacteristic
44C131⋊#95C2211yescharacteristic
45C131⋊#95C2211yescharacteristic
46C2×C4⋊C4×C131C2211yescharacteristic
47C2×C22⋊C4×C131C2211yescharacteristic
48C2×C22⋊C4×C131C2211yescharacteristic
49C2×C22⋊C4×C131C2211yescharacteristic
50#95×C131C2211yescharacteristic
51#95×C131C2211yescharacteristic
52#95×C131C2211yescharacteristic
53C23×D131C2311yescharacteristic
54C23×D131C2311yescharacteristic
55C2×D524—22no—
56C2×D524—22no—
57C2×D524—41no—
58C2×C22⋊D131—22no—
59C2×C22⋊D131—22no—
60C2×C22⋊D131—22no—
61C2×C22⋊D131—22no—
62C2×C22⋊D131—22no—
63C2×C22⋊D131—41no—
64C22×Dic131C2311yescharacteristic
65C22×Dic131C2311yescharacteristic
66C22×Dic131C2311yescharacteristic
67C22×Dic131C2311yescharacteristic
68C22×Dic131C2311yescharacteristic
69C22×Dic131C2311yescharacteristic
70C2×C4×D131—22no—
71C2×C4×D131—22no—
72C2×C4×D131—22no—
73C4⋊Dic131—22no—
74Dic131⋊C4—22no—
75Dic131⋊C4—22no—
76Dic131⋊C4—22no—
77Dic131⋊C4—22no—
78Dic131⋊C4D414yes—
79C22⋊Dic131—22no—
80C22⋊Dic131—41no—
81(C2×D131)⋊C4—22no—
82(C2×D131)⋊C4—22no—
83(C2×D131)⋊C4—22no—
84(C2×D131)⋊C4—22no—
85(C2×D131)⋊C4—41no—
86(C2×D131)⋊C4—41no—
87(C2×D131)⋊C4—41no—
88(C2×D131)⋊C4—41no—
89(C2×D131)⋊C4—41no—
90C4×Dic131—22no—
91C4×Dic131—22no—
92C4×Dic131—41no—
93C4×Dic131—41no—
94C24×C131C2311yescharacteristic
95C22×C4×C131C2311yescharacteristic
96C22×C4×C131C2311yescharacteristic
97C22×C4×C131C2311yescharacteristic
98C22×C4×C131C2311yescharacteristic
99C22×C4×C131C2311yescharacteristic
100C22×C4×C131C2311yescharacteristic
101C4⋊C4×C131—22no—
102C22⋊C4×C131—22no—
103C22⋊C4×C131—41no—
104C22⋊C4×C131—41no—
105C22×D131—21no—
106C22×D131—21no—
107C22×D131—21no—
108C22×D131—21no—
109C22×D131—21no—
110C22×D131—21no—
111C22×D131—21no—
112C22×D131—21no—
113C22×D131—21no—
114C22×D131—21no—
115C22×D131—21no—
116C22×D131—21no—
117C22×D131—21no—
118C22×D131#4112yes—
119D524—44no—
120C22⋊D131—44no—
121C22⋊D131—44no—
122C2×Dic131—21no—
123C2×Dic131—21no—
124C2×Dic131—21no—
125C2×Dic131—21no—
126C2×Dic131—21no—
127C2×Dic131—21no—
128C2×Dic131—21no—
129C2×Dic131—21no—
130C2×Dic131—21no—
131C2×Dic131—21no—
132C2×Dic131—21no—
133C2×Dic131—21no—
134C2×Dic131—21no—
135C2×Dic131C2×D412yes—
136C2×Dic131C2×D412yes—
137C2×Dic131#4112yes—
138C2×Dic131#4112yes—
139C2×Dic131#4112yes—
140C4×D131—44no—
141C23×C131—21no—
142C23×C131—21no—
143C23×C131—21no—
144C23×C131—21no—
145C23×C131—21no—
146C23×C131—21no—
147C23×C131—21no—
148C23×C131C2411yescharacteristic
149C2×C4×C131—21no—
150C2×C4×C131—21no—
151C2×C4×C131—21no—
152C2×C4×C131—21no—
153C2×C4×C131—21no—
154C2×C4×C131—21no—
155C2×C4×C131—21no—
156C2×C4×C131—21no—
157C2×C4×C131—21no—
158C2×C4×C131—21no—
159C2×C4×C131—21no—
160C2×C4×C131—21no—
161C2×C4×C131—21no—
162C2×C4×C131—21no—
163C2×C4×C131—21no—
164C2×C4×C131—21no—
165C2×C4×C131—21no—
166C2×C4×C131C2×D412yes—
167C22×C131—41no—
168C22×C131—41no—
169C22×C131—41no—
170C22×C131—41no—
171C22×C131—41no—
172C22×C131—41no—
173C22×C131—41no—
174C22×C131C22×D411yescharacteristic
175C22×C131C2×#4111yescharacteristic
176C22×C131C2×#4111yescharacteristic
177C22×C131C2×#4111yescharacteristic
178C22×C131C2×#4111yescharacteristic
179C22×C131H(5,𝔽2)11yescharacteristic
180C22×C131H(5,𝔽2)11yescharacteristic
181C2×D131—22no—
182C2×D131—22no—
183C2×D131—22no—
184C2×D131—41no—
185C2×D131—41no—
186C2×D131—41no—
187C2×D131—41no—
188C2×D131—41no—
189C2×D131—41no—
190C2×D131—41no—
191C2×D131—41no—
192C2×D131—41no—
193C2×D131—41no—
194C2×D131—41no—
195C4×C131—22no—
196C4×C131—41no—
197C4×C131—41no—
198C4×C131—41no—
199C4×C131—41no—
200C4×C131—41no—
201Dic131—22no—
202Dic131—22no—
203Dic131—41no—
204Dic131—41no—
205Dic131—41no—
206Dic131#12414yes—
207C2×C131—81no—
208C2×C131C2×#12411yescharacteristic
209C2×C131#52911yescharacteristic
210C2×C131#54011yescharacteristic
211C2×C131#54611yescharacteristic
212C2×C131#55011yescharacteristic
213C2×C131#55311yescharacteristic
214C2×C131#55311yescharacteristic
215D131—42no—
216D131—81no—
217C131#249011yescharacteristic, complemented, Sylow
218#2490—1311nomaximal, Sylow
219C2×#125—1311no—
220C2×#121—1311no—
221#397—1311no—
222#397D13111yescharacteristic, complemented
223#394—1311no—
224#394—1311no—
225#390—1311no—
226#390—1311no—
227#390—1311no—
228#390—1311no—
229#388—1311no—
230#388—1311no—
231#380—1311no—
232C4×C4⋊C4—1311no—
233C4×C22⋊C4—1311no—
234C22×D4—1311no—
235C22×D4—1311no—
236C22×D4—1311no—
237C23×C4—1311no—
238#125—2624no—
239#121—2624no—
240C2×C4⋊C4—1311no—
241C2×C4⋊C4—1311no—
242C2×C4⋊C4—1311no—
243C2×C4⋊C4—1311no—
244C2×C4⋊C4—1311no—
245C2×C4⋊C4—1311no—
246C2×C4⋊C4C2×D13111yescharacteristic
247C2×C22⋊C4—1311no—
248C2×C22⋊C4—1311no—
249C2×C22⋊C4—1311no—
250C2×C22⋊C4—1311no—
251C2×C22⋊C4—1311no—
252C2×C22⋊C4—1311no—
253C2×C22⋊C4—1311no—
254C2×C22⋊C4—1311no—
255C2×C22⋊C4—1311no—
256C2×C22⋊C4—1311no—
257C2×C22⋊C4—1311no—
258C2×C22⋊C4C2×D13111yescharacteristic
259C2×C22⋊C4C2×D13111yescharacteristic
260C2×C22⋊C4C2×D13111yescharacteristic
261C2×C42—1311no—
262C2×C42—1311no—
263C2×C42—1311no—
264C2×C42—1311no—
265#95—1311no—
266#95—1311no—
267#95—1311no—
268#95C2×D13111yescharacteristic
269#95C2×D13111yescharacteristic
270#95C2×D13111yescharacteristic
271C24—1311no—
272C24—1311no—
273C24C22×D13111yescharacteristic
274C2×D4—2622no—
275C2×D4—2622no—
276C2×D4—2622no—
277C2×D4—2622no—
278C2×D4—2622no—
279C2×D4—2622no—
280C2×D4—2622no—
281C2×D4—5241no—
282C2×D4—5241no—
283C22×C4—2622no—
284C22×C4—2622no—
285C22×C4—2622no—
286C22×C4—1311no—
287C22×C4—1311no—
288C22×C4—1311no—
289C22×C4—1311no—
290C22×C4—1311no—
291C22×C4—1311no—
292C22×C4C22×D13111yescharacteristic
293C22×C4C22×D13111yescharacteristic
294C22×C4C22×D13111yescharacteristic
295C22×C4C22×D13111yescharacteristic
296C22×C4C22×D13111yescharacteristic
297C22×C4C22×D13111yescharacteristic
298C4⋊C4—22no—
299C4⋊C4—2622no—
300C4⋊C4—2622no—
301C4⋊C4—2622no—
302C4⋊C4—2622no—
303C4⋊C4—2622no—
304C4⋊C4—1314no—
305C22⋊C4—22no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—2622no—
309C22⋊C4—2622no—
310C22⋊C4—2622no—
311C22⋊C4—2622no—
312C22⋊C4—2622no—
313C22⋊C4—5241no—
314C22⋊C4—5241no—
315C22⋊C4—5241no—
316C22⋊C4—5241no—
317C22⋊C4—5241no—
318C22⋊C4—5241no—
319C42—2622no—
320C42—2622no—
321C42—5241no—
322C42—5241no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—2621no—
331C23—2621no—
332C23—2621no—
333C23—2621no—
334C23—2621no—
335C23—2621no—
336C23—2621no—
337C23—2621no—
338C23—2621no—
339C23—2621no—
340C23—2621no—
341C23—2621no—
342C23—2621no—
343C23—1312no—
344C23C23×D13111yescharacteristic
345D4—5244no—
346D4—5244no—
347D4—5244no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—2621no—
366C2×C4—2621no—
367C2×C4—2621no—
368C2×C4—2621no—
369C2×C4—2621no—
370C2×C4—2621no—
371C2×C4—2621no—
372C2×C4—2621no—
373C2×C4—2621no—
374C2×C4—2621no—
375C2×C4—2621no—
376C2×C4—2621no—
377C2×C4—2621no—
378C2×C4—5244no—
379C2×C4—1312no—
380C2×C4—1312no—
381C2×C4—1312no—
382C2×C4—1312no—
383C2×C4—1312no—
384C2×C4D4×D13112yes—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—2622no—
393C22—2622no—
394C22—2622no—
395C22—5241no—
396C22—5241no—
397C22—5241no—
398C22—5241no—
399C22—5241no—
400C22—5241no—
401C22—5241no—
402C22—5241no—
403C22—5241no—
404C22—5241no—
405C22—5241no—
406C22C2×D4×D13111yescharacteristic
407C22#41×D13111yescharacteristic
408C22#41×D13111yescharacteristic
409C22#41×D13111yescharacteristic
410C22#41×D13111yescharacteristic
411C22#1686229111yescharacteristic
412C22#1686229211yescharacteristic
413C4—22no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4—2622no—
420C4—2622no—
421C4—5241no—
422C4—5241no—
423C4—5241no—
424C4—1314no—
425C2—81no—
426C2—5242no—
427C2—10481no—
428C2#124×D13111yescharacteristic
429C2#2255384511yescharacteristic
430C2#2255391411yescharacteristic
431C2#2255395411yescharacteristic
432C2#2255398011yescharacteristic
433C2#2255400711yescharacteristic
434C2#2255401311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#397×C131C2
prime radicalC2×#16862219C2
hypercenter#397D131
hypocenterC131#2490
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#16862236C2semidirect
3. #22553241C2semidirect
4. #22553230C2semidirect
5. #22553228C2semidirect
6. #22553213C2semidirect
7. #22553212C2semidirect
8. #22553212C2semidirect
9. #22553212C2semidirect
10. #22553206C2semidirect
11. #22553204C2semidirect
12. #22553171C2semidirect
13. #22553163C2semidirect
14. #22553159C2semidirect
15. #397×C131C2semidirect
217. C131#2490semidirect
222. #397D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622362
C2×#168622192
#225532412
#225532302
#225532282
#225532132
#225532122
#225532062
#225532042
#225531712
#225531632
#225531592
#397×C1312
#2490131