Finite Group Databasebeta

← #30674288

Subgroups — #30674288

Order 16768.

688 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16862231C211yesmaximal, characteristic, complemented
2C2×#16862228C211yesmaximal, characteristic, complemented
3C2×#16862219C211yesmaximal, characteristic
4C2×C42×D131C211yesmaximal, characteristic, complemented
5#22553212C211yesmaximal, characteristic, complemented
6#22553212C211yesmaximal, characteristic, complemented
7#22553209C211yesmaximal, characteristic, complemented
8#22553207C211yesmaximal, characteristic, complemented
9#22553204C211yesmaximal, characteristic, complemented
10#22553203C211yesmaximal, characteristic, complemented
11#22553200C211yesmaximal, characteristic, complemented
12#22553190C211yesmaximal, characteristic, complemented
13#22553188C211yesmaximal, characteristic, complemented
14#22553171C211yesmaximal, characteristic, complemented
15#388×C131C211yesmaximal, characteristic, complemented
16C22×D524C2211yescharacteristic
17C22×C22⋊D131C2211yescharacteristic
18C22×C22⋊D131C2211yescharacteristic
19C22×C4×D131C2211yescharacteristic
20C22×C4×D131C2211yescharacteristic
21C22×C4×D131C2211yescharacteristic
22#16862231—24no—
23#16862228—24no—
24#16862219—24no—
25C2×C4⋊Dic131C2211yescharacteristic
26C2×Dic131⋊C4C2211yescharacteristic
27C2×Dic131⋊C4C2211yescharacteristic
28C2×Dic131⋊C4C2211yescharacteristic
29C2×Dic131⋊C4C2211yescharacteristic
30C2×Dic131⋊C4C2211yescharacteristic
31C2×C22⋊Dic131C2211yescharacteristic
32C2×C22⋊Dic131C2211yescharacteristic
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×(C2×D131)⋊C4C2211yescharacteristic
40C2×C4×Dic131C2211yescharacteristic
41C2×C4×Dic131C2211yescharacteristic
42C2×C4×Dic131C2211yescharacteristic
43C42×D131—24no—
44C131⋊#95C2211yescharacteristic
45C131⋊#95C2211yescharacteristic
46C131⋊#95C2211yescharacteristic
47C131⋊#95C2211yescharacteristic
48C2×C4⋊C4×C131C2211yescharacteristic
49C2×C22⋊C4×C131C2211yescharacteristic
50C2×C22⋊C4×C131C2211yescharacteristic
51C2×C22⋊C4×C131C2211yescharacteristic
52C2×C42×C131C2211yescharacteristic
53#95×C131C2211yescharacteristic
54#95×C131C2211yescharacteristic
55C23×D131C2311yescharacteristic
56C23×D131C2311yescharacteristic
57C2×D524—22no—
58C2×D524—22no—
59C2×D524—22no—
60C2×C22⋊D131—22no—
61C2×C22⋊D131—22no—
62C2×C22⋊D131—22no—
63C2×C22⋊D131—22no—
64C2×C22⋊D131—22no—
65C2×C22⋊D131—22no—
66C22×Dic131C2311yescharacteristic
67C22×Dic131C2311yescharacteristic
68C22×Dic131C2311yescharacteristic
69C22×Dic131C2311yescharacteristic
70C22×Dic131C2311yescharacteristic
71C22×Dic131C2311yescharacteristic
72C2×C4×D131—22no—
73C2×C4×D131—22no—
74C2×C4×D131—22no—
75C2×C4×D131—22no—
76C2×C4×D131—22no—
77C2×C4×D131—22no—
78C2×C4×D131—22no—
79C2×C4×D131D414yes—
80C2×C4×D131D414yescomplemented
81C4⋊Dic131—22no—
82Dic131⋊C4—22no—
83Dic131⋊C4—22no—
84Dic131⋊C4—22no—
85Dic131⋊C4—22no—
86Dic131⋊C4—22no—
87C22⋊Dic131—22no—
88C22⋊Dic131—22no—
89(C2×D131)⋊C4—22no—
90(C2×D131)⋊C4—22no—
91(C2×D131)⋊C4—22no—
92(C2×D131)⋊C4—22no—
93(C2×D131)⋊C4—22no—
94(C2×D131)⋊C4—22no—
95(C2×D131)⋊C4—22no—
96C4×Dic131—22no—
97C4×Dic131—22no—
98C4×Dic131—22no—
99C24×C131C2311yescharacteristic
100C22×C4×C131C2311yescharacteristic
101C22×C4×C131C2311yescharacteristic
102C22×C4×C131C2311yescharacteristic
103C22×C4×C131C2311yescharacteristic
104C22×C4×C131C2311yescharacteristic
105C22×C4×C131C2311yescharacteristic
106C4⋊C4×C131—22no—
107C22⋊C4×C131—22no—
108C22⋊C4×C131—22no—
109C22⋊C4×C131—22no—
110C42×C131—22no—
111C22×D131—21no—
112C22×D131—21no—
113C22×D131—21no—
114C22×D131—21no—
115C22×D131—21no—
116C22×D131—21no—
117C22×D131—21no—
118C22×D131—21no—
119C22×D131—21no—
120C22×D131—21no—
121C22×D131—21no—
122C22×D131C2×D412yes—
123C22×D131C2×D412yes—
124C22×D131#4112yes—
125D524—44no—
126C22⋊D131—44no—
127C22⋊D131—44no—
128C2×Dic131—21no—
129C2×Dic131—21no—
130C2×Dic131—21no—
131C2×Dic131—21no—
132C2×Dic131—21no—
133C2×Dic131—21no—
134C2×Dic131—21no—
135C2×Dic131—21no—
136C2×Dic131—21no—
137C2×Dic131—21no—
138C2×Dic131—21no—
139C2×Dic131—21no—
140C2×Dic131—21no—
141C2×Dic131C2×D412yes—
142C2×Dic131C2×D412yes—
143C2×Dic131#4112yes—
144C2×Dic131#4112yes—
145C2×Dic131#4112yes—
146C4×D131—28no—
147C4×D131—28no—
148C4×D131—28no—
149C23×C131—21no—
150C23×C131—21no—
151C23×C131—21no—
152C23×C131—21no—
153C23×C131—21no—
154C23×C131—21no—
155C23×C131—21no—
156C23×C131C2411yescharacteristic
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×C131—21no—
167C2×C4×C131—21no—
168C2×C4×C131—21no—
169C2×C4×C131—21no—
170C2×C4×C131—21no—
171C2×C4×C131—21no—
172C2×C4×C131C2×D412yes—
173C2×C4×C131C2×D412yes—
174C2×C4×C131#4112yes—
175C22×C131—41no—
176C22×C131—41no—
177C22×C131—41no—
178C22×C131—41no—
179C22×C131—41no—
180C22×C131—41no—
181C22×C131—41no—
182C22×C131C22×D411yescharacteristic
183C22×C131C22×D411yescharacteristic
184C22×C131C2×#4111yescharacteristic
185C22×C131C2×#4111yescharacteristic
186C22×C131C2×#4111yescharacteristic
187C22×C131C2×#4111yescharacteristic
188C22×C131C2×#4111yescharacteristic
189C2×D131—22no—
190C2×D131—22no—
191C2×D131—22no—
192C2×D131—22no—
193C2×D131—22no—
194C2×D131—22no—
195C2×D131—41no—
196C2×D131—41no—
197C2×D131—41no—
198C2×D131—41no—
199C2×D131—41no—
200C2×D131—41no—
201C2×D131—41no—
202C2×D131#12114yes—
203C4×C131—22no—
204C4×C131—22no—
205C4×C131—22no—
206C4×C131—41no—
207C4×C131—41no—
208C4×C131—41no—
209Dic131—22no—
210Dic131—22no—
211Dic131—41no—
212Dic131—41no—
213Dic131—41no—
214Dic131#12314yescomplemented
215C2×C131—81no—
216C2×C131C2×#12111yescharacteristic
217C2×C131C2×#12311yescharacteristic
218C2×C131#52511yescharacteristic
219C2×C131#52911yescharacteristic
220C2×C131#52911yescharacteristic
221C2×C131#53211yescharacteristic
222C2×C131#53211yescharacteristic
223D131—24no—
224D131—81no—
225C131#285411yescharacteristic, complemented, Sylow
226#2854—1311nomaximal, Sylow
227C2×#123—1311no—
228C2×#123—1311no—
229C2×#121—1311no—
230C22×C42—1311no—
231#390—1311no—
232#390—1311no—
233#389—1311no—
234#389—1311no—
235#388—1311no—
236#388—1311no—
237#388—1311no—
238#388D13111yescharacteristic, complemented
239#386—1311no—
240#386—1311no—
241#380—1311no—
242C22×D4—1311no—
243C22×D4—1311no—
244C22×D4—1311no—
245C23×C4—1311no—
246C23×C4—1311no—
247C23×C4—1311no—
248#123—2624no—
249#123—2624no—
250#121—2624no—
251C2×C4⋊C4—1311no—
252C2×C4⋊C4—1311no—
253C2×C4⋊C4—1311no—
254C2×C4⋊C4—1311no—
255C2×C4⋊C4—1311no—
256C2×C4⋊C4—1311no—
257C2×C4⋊C4C2×D13111yescharacteristic
258C2×C22⋊C4—1311no—
259C2×C22⋊C4—1311no—
260C2×C22⋊C4—1311no—
261C2×C22⋊C4—1311no—
262C2×C22⋊C4—1311no—
263C2×C22⋊C4—1311no—
264C2×C22⋊C4—1311no—
265C2×C22⋊C4—1311no—
266C2×C22⋊C4—1311no—
267C2×C22⋊C4C2×D13111yescharacteristic
268C2×C22⋊C4C2×D13111yescharacteristic
269C2×C22⋊C4C2×D13111yescharacteristic
270C2×C42—2624no—
271C2×C42—1311no—
272C2×C42—1311no—
273C2×C42—1311no—
274C2×C42C2×D13111yescharacteristic
275#95—1311no—
276#95—1311no—
277#95—1311no—
278#95—1311no—
279#95C2×D13111yescharacteristic
280#95C2×D13111yescharacteristic
281C24—1311no—
282C24—1311no—
283C24C22×D13111yescharacteristic
284C2×D4—2622no—
285C2×D4—2622no—
286C2×D4—2622no—
287C2×D4—2622no—
288C2×D4—2622no—
289C2×D4—2622no—
290C2×D4—2622no—
291C2×D4—2622no—
292C2×D4—2622no—
293C22×C4—2622no—
294C22×C4—2622no—
295C22×C4—2622no—
296C22×C4—2622no—
297C22×C4—2622no—
298C22×C4—2622no—
299C22×C4—2622no—
300C22×C4—1311no—
301C22×C4—1311no—
302C22×C4—1311no—
303C22×C4—1311no—
304C22×C4—1311no—
305C22×C4—1314no—
306C22×C4—1314no—
307C22×C4—1311no—
308C22×C4C22×D13111yescharacteristic
309C22×C4C22×D13111yescharacteristic
310C22×C4C22×D13111yescharacteristic
311C22×C4C22×D13111yescharacteristic
312C22×C4C22×D13111yescharacteristic
313C22×C4C22×D13111yescharacteristic
314C4⋊C4—22no—
315C4⋊C4—2622no—
316C4⋊C4—2622no—
317C4⋊C4—2622no—
318C4⋊C4—2622no—
319C4⋊C4—2622no—
320C4⋊C4—2622no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—2622no—
325C22⋊C4—2622no—
326C22⋊C4—2622no—
327C22⋊C4—2622no—
328C22⋊C4—2622no—
329C22⋊C4—2622no—
330C22⋊C4—2622no—
331C22⋊C4—2622no—
332C22⋊C4—2622no—
333C42—22no—
334C42—2622no—
335C42—2622no—
336C42—2622no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—2621no—
345C23—2621no—
346C23—2621no—
347C23—2621no—
348C23—2621no—
349C23—2621no—
350C23—2621no—
351C23—2621no—
352C23—2621no—
353C23—2621no—
354C23—2621no—
355C23—1312no—
356C23—1312no—
357C23—1312no—
358C23C23×D13111yescharacteristic
359D4—5244no—
360D4—5244no—
361D4—5244no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—2628no—
378C2×C4—2628no—
379C2×C4—2628no—
380C2×C4—2621no—
381C2×C4—2621no—
382C2×C4—2621no—
383C2×C4—2621no—
384C2×C4—2621no—
385C2×C4—2621no—
386C2×C4—2621no—
387C2×C4—2621no—
388C2×C4—2621no—
389C2×C4—2621no—
390C2×C4—2621no—
391C2×C4—2621no—
392C2×C4—2621no—
393C2×C4—1312no—
394C2×C4—1312no—
395C2×C4—1312no—
396C2×C4—1312no—
397C2×C4—1312no—
398C2×C4D4×D13112yes—
399C2×C4D4×D13112yes—
400C2×C4#1452351312yes—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—2622no—
409C22—2622no—
410C22—2622no—
411C22—2622no—
412C22—2622no—
413C22—2622no—
414C22—5241no—
415C22—5241no—
416C22—5241no—
417C22—5241no—
418C22—5241no—
419C22—5241no—
420C22—5241no—
421C22—1314no—
422C22C2×D4×D13111yescharacteristic
423C22C2×D4×D13111yescharacteristic
424C22#41×D13111yescharacteristic
425C22#41×D13111yescharacteristic
426C22#41×D13111yescharacteristic
427C22#41×D13111yescharacteristic
428C22C2×#1452351311yescharacteristic
429C4—22no—
430C4—22no—
431C4—22no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—2622no—
436C4—2622no—
437C4—5241no—
438C4—5241no—
439C4—5241no—
440C4—1314no—
441C2—2624no—
442C2—81no—
443C2—10481no—
444C2#121×D13111yescharacteristic
445C2#123×D13111yescharacteristic
446C2#2255382111yescharacteristic
447C2#2255384011yescharacteristic
448C2#2255384511yescharacteristic
449C2#2255385911yescharacteristic
450C2#2255386111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#388×C131C2
prime radicalC2×#16862219C2
hypercenter#388D131
hypocenterC131#2854
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#16862231C2semidirect
2. C2×#16862228C2semidirect
4. C2×C42×D131C2semidirect
5. #22553212C2semidirect
6. #22553212C2semidirect
7. #22553209C2semidirect
8. #22553207C2semidirect
9. #22553204C2semidirect
10. #22553203C2semidirect
11. #22553200C2semidirect
12. #22553190C2semidirect
13. #22553188C2semidirect
14. #22553171C2semidirect
15. #388×C131C2semidirect
80. C2×C4×D131D4semidirect
214. Dic131#123semidirect
225. C131#2854semidirect
238. #388D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622312
C2×#168622282
C2×#168622192
C2×C42×D1312
#225532122
#225532092
#225532072
#225532042
#225532032
#225532002
#225531902
#225531882
#225531712
#388×C1312
#2854131