Finite Group Databasebeta

← #29734283

Subgroups — #29734283

Order 15616.

1094 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic
2C2×#16840114C211yesmaximal, characteristic, complemented
3#21454166C211yesmaximal, characteristic
4#21447017C214yesmaximal
5#21446781C211yesmaximal, characteristic, complemented
6#21446621C211yesmaximal, characteristic, complemented
7#21446084C214yesmaximal, complemented
8C4×#14519689C211yesmaximal, characteristic
9C2×C61×#498C211yesmaximal, characteristic, complemented
10#16840317—24no—
11#16840317—24no—
12C2×#14519777C2211yescharacteristic
13C2×#14519768—21no—
14C2×#14519757C2212yes—
15C2×#14519753—21no—
16C2×#14519753—21no—
17C2×Q8×Dic61—21no—
18C2×#14519742C2212yes—
19C2×#14519735—21no—
20C2×C42×D61C2211yescharacteristic
21#16840114—24no—
22C2×#14519702C2214yes—
23C2×#14519700C2214yes—
24C2×#14519689C2212yes—
25C2×C4×#92418C2211yescharacteristic
26#16839669—21no—
27#16839658—21no—
28#16839658C2212yes—
29#16839632C2211yescharacteristic
30#16839619—21no—
31#16839619C2212yes—
32#16839524C2214yes—
33#16839516C2212yes—
34C2×C4⋊Q8×C61C2211yescharacteristic
35C2×C4⋊Q8×C61C2211yescharacteristic
36C61×#498—24no—
37C2×#106×C61C2214yes—
38C2×C4×C8×C61C2211yescharacteristic
39C22×Dic122C2312yes—
40C22×C4×D61D411yescharacteristic
41C22×C4×D61D411yescharacteristic
42C22×C4×D61C2311yescharacteristic
43C22×#92418C2312yes—
44#14519777—22no—
45#14519777D414yes—
46#14519768—28no—
47#14519757—24no—
48#14519757—24no—
49#14519753—28no—
50#14519753—44no—
51Q8×Dic61—28no—
52#14519742—24no—
53#14519742—24no—
54#14519735—28no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—22no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61C2312yes—
61C2×#11758196—22no—
62C2×#11758196—22no—
63C2×#11758198—22no—
64C2×#11758198—22no—
65C2×C4×Dic61—22no—
66C2×C4×Dic61D411yescharacteristic
67C2×C4×Dic61D411yescharacteristic
68C2×C4×Dic61C2311yescharacteristic
69C42×D61—22no—
70C42×D61—22no—
71#14519702—28no—
72#14519700—28no—
73#14519689—24no—
74C4×#92418—22no—
75#14519684—22no—
76#14519684—22no—
77C22×Q8×C61—21no—
78C22×Q8×C61—21no—
79C22×C8×C61C2312yes—
80C4⋊Q8×C61—22no—
81C4⋊Q8×C61—22no—
82C4⋊Q8×C61—22no—
83C4⋊Q8×C61D414yes—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61C2312yes—
87C2×C4⋊C4×C61C2312yes—
88C2×C42×C61C2311yescharacteristic
89#106×C61—28no—
90C4×C8×C61—22no—
91C23×D61C2×D411yescharacteristic
92C2×Q8×C61—24no—
93C2×Q8×C61—42no—
94C2×Q8×C61—42no—
95C2×Q8×C61—42no—
96C2×Q8×C61—42no—
97C2×Q8×C61—42no—
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2411yescharacteristic
103C22×Dic61—22no—
104C22×Dic61—22no—
105C22×Dic61C2×D411yescharacteristic
106C22×Dic61C2×D411yescharacteristic
107C22×Dic61C2×D411yescharacteristic
108C2×C4×D61—22no—
109C2×C4×D61—22no—
110C2×C4×D61—22no—
111C2×C4×D61—22no—
112C2×C4×D61—22no—
113C2×C4×D61—21no—
114C2×C4×D61—21no—
115C2×C4×D61—22no—
116C2×C4×D61—22no—
117C2×C4×D61SD1614yes—
118C2×Dic122—24no—
119C2×Dic122—24no—
120C2×Dic122—22no—
121C2×Dic122C2×D414yes—
122C2×C8×C61—22no—
123C2×C8×C61—22no—
124C2×C8×C61#4114yes—
125C4⋊C4×C61—24no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—42no—
130C4⋊C4×C61C2×D414yes—
131C42×C61C2×D412yes—
132C42×C61C2×D412yes—
133#11758198—28no—
134#11758198—44no—
135C4⋊Dic61—28no—
136C4⋊Dic61—24no—
137C4⋊Dic61—24no—
138C4⋊Dic61—24no—
139C4⋊Dic61—42no—
140C4⋊Dic61—42no—
141#11758196—28no—
142#11758196—44no—
143C4×Dic61—28no—
144C4×Dic61—22no—
145C4×Dic61—22no—
146C4×Dic61—21no—
147C4×Dic61—21no—
148C2×#92418—22no—
149C2×#92418—22no—
150C2×#92418—22no—
151C23×C61C22×D411yescharacteristic
152C22×D61—21no—
153C22×D61—21no—
154C22×D61—21no—
155C22×D61—21no—
156C22×D61—21no—
157C22×D61—21no—
158C22×D61C2×SD1612yes—
159Q8×C61—48no—
160Q8×C61—84no—
161C2×C4×C61—24no—
162C2×C4×C61—42no—
163C2×C4×C61—42no—
164C2×C4×C61—42no—
165C2×C4×C61—42no—
166C2×C4×C61—42no—
167C2×C4×C61#12012yes—
168C2×C4×C61#12012yes—
169C2×C4×C61C2×SD1612yes—
170C2×C4×C61#13512yes—
171C2×C4×C61#13712yes—
172C2×C4×C61#13712yes—
173C2×C4×C61C22×D411yescharacteristic
174C2×C4×C61C22×D411yescharacteristic
175C2×C4×C61C2×#4112yes—
176C2×C4×C612−1+412yes—
177C2×Dic61—21no—
178C2×Dic61—21no—
179C2×Dic61—21no—
180C2×Dic61—21no—
181C2×Dic61—24no—
182C2×Dic61—21no—
183C2×Dic61—21no—
184C2×Dic61—21no—
185C2×Dic61—21no—
186C2×Dic61—42no—
187C2×Dic61—42no—
188C2×Dic61—42no—
189C2×Dic61—42no—
190C2×Dic61—42no—
191C2×Dic61C2×SD1612yes—
192C4×D61—28no—
193C4×D61—42no—
194C4×D61—42no—
195C4×D61—44no—
196Dic122—28no—
197Dic122—44no—
198C8×C61—24no—
199#92418—42no—
200#25672—611nomaximal, Sylow
201C22×C61C2×#12011yescharacteristic
202C22×C61#53211yescharacteristic
203C22×C61#53611yescharacteristic
204C22×C61C22×SD1611yescharacteristic
205C22×C61C2×#13511yescharacteristic
206C22×C61C2×#13711yescharacteristic
207C22×C61C2×#13711yescharacteristic
208C2×D61—22no—
209C2×D61—22no—
210C2×D61—22no—
211C2×D61—41no—
212C2×D61—41no—
213C2×D61—41no—
214C2×D61—41no—
215C4×C61—44no—
216C4×C61—82no—
217C4×C61#44814yes—
218C4×C61#45214yes—
219C4×C61#54712yes—
220C4×C61#54812yes—
221Dic61—22no—
222Dic61—44no—
223Dic61—41no—
224Dic61—41no—
225Dic61—82no—
226C2×#533—611no—
227C2×#498D6111yescharacteristic, complemented
228C2×#487—611no—
229#2857—611no—
230#2035—614no—
231#1990—611no—
232#1956—611no—
233#1865—614no—
234C4×C22⋊C8—611no—
235C2×C61#243211yescharacteristic
236C2×C61C2×#44811yescharacteristic
237C2×C61C2×#45211yescharacteristic
238C2×C61#316711yescharacteristic
239C2×C61#318511yescharacteristic
240C2×C61#322911yescharacteristic
241C2×C61#323111yescharacteristic
242D61—42no—
243#533—1224no—
244#533—1224no—
245C2×C4⋊Q8—611no—
246C2×C4⋊Q8C2×D6111yescharacteristic
247C2×C4⋊Q8C2×D6111yescharacteristic
248C2×#125—1221no—
249C2×C22⋊Q8—612no—
250C2×C22⋊Q8—1221no—
251C2×C22⋊Q8—1221no—
252C2×C4×Q8—612no—
253C2×C4×Q8—1221no—
254C2×#117—1221no—
255C22×C42—611no—
256#498—24no—
257#487—1224no—
258C2×#106C2×D6114yes—
259C2×#103—614no—
260C2×#103—614no—
261C2×C22⋊C8—612no—
262C2×C4×C8—611no—
263C2×C4×C8C2×D6111yescharacteristic
264#391—1221no—
265#389—612no—
266#389—1221no—
267#384—611no—
268#380—612no—
269#380—1221no—
270#340—614no—
271#336—612no—
272C61#2567211yescharacteristic, complemented, Sylow
273C22×Q8—21no—
274C22×Q8—21no—
275C22×Q8—612no—
276C23×C4—611no—
277C23×C4—611no—
278C23×C4—611no—
279C22×C8—612no—
280C22×C8C22×D6112yes—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—614no—
285C4⋊Q8—1222no—
286C4⋊Q8C22⋊D6114yes—
287#125—1228no—
288C22⋊Q8—1224no—
289C22⋊Q8—1228no—
290C22⋊Q8—1224no—
291C22⋊Q8—2444no—
292C4×Q8—1224no—
293C4×Q8—1228no—
294C4×Q8—1224no—
295#117—1228no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—612no—
299C2×C4⋊C4—1222no—
300C2×C4⋊C4—1222no—
301C2×C4⋊C4—1221no—
302C2×C4⋊C4—1221no—
303C2×C4⋊C4—1222no—
304C2×C4⋊C4—1221no—
305C2×C4⋊C4—1221no—
306C2×C4⋊C4C22×D6112yes—
307C2×C4⋊C4C22×D6112yes—
308C2×C22⋊C4—1222no—
309C2×C22⋊C4—1222no—
310C2×C42—611no—
311C2×C42—611no—
312C2×C42—611no—
313C2×C42—1222no—
314C2×C42—1222no—
315C2×C42—1222no—
316C2×C42C22×D6111yescharacteristic
317#106—28no—
318#103—1228no—
319#103—1228no—
320C22⋊C8—1224no—
321C4×C8—22no—
322C4×C8—1222no—
323#95—1222no—
324#95—1222no—
325C24—611no—
326C2×Q8—24no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—614no—
333C2×Q8—1222no—
334C2×Q8—1224no—
335C2×Q8—1224no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—611no—
339C22×C4—614no—
340C22×C4—611no—
341C22×C4—611no—
342C22×C4—1222no—
343C22×C4—1222no—
344C22×C4—1222no—
345C22×C4—1221no—
346C22×C4—1221no—
347C22×C4—1222no—
348C22×C4—1222no—
349C22×C4—1222no—
350C22×C4—1222no—
351C22×C4—1222no—
352C22×C4—1222no—
353C22×C4D4×D6111yescharacteristic
354C22×C4D4×D6111yescharacteristic
355C22×C4C23×D6111yescharacteristic
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—1222no—
359C2×C8—1222no—
360C2×C8—1222no—
361C2×C8#1175822614yes—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—42no—
367C4⋊C4—1228no—
368C4⋊C4—1228no—
369C4⋊C4—1224no—
370C4⋊C4—1224no—
371C4⋊C4—1224no—
372C4⋊C4—2444no—
373C4⋊C4—2442no—
374C4⋊C4—2442no—
375C4⋊C4C2×C22⋊D6114yes—
376C22⋊C4—1228no—
377C22⋊C4—2444no—
378C42—1228no—
379C42—1221no—
380C42—1221no—
381C42—1222no—
382C42—1222no—
383C42D4×D6112yes—
384C42C2×C22⋊D6112yes—
385C23—612no—
386C23—1221no—
387C23—1221no—
388C23—1221no—
389C23—1221no—
390C23—1221no—
391C23—1221no—
392C23C2×D4×D6111yescharacteristic
393Q8—48no—
394Q8—84no—
395Q8—1228no—
396Q8—2444no—
397C2×C4—24no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—612no—
404C2×C4—1221no—
405C2×C4—1221no—
406C2×C4—1228no—
407C2×C4—1221no—
408C2×C4—1221no—
409C2×C4—1224no—
410C2×C4—1221no—
411C2×C4—1221no—
412C2×C4—1221no—
413C2×C4—1221no—
414C2×C4—2442no—
415C2×C4—2442no—
416C2×C4—2442no—
417C2×C4—2442no—
418C2×C4—2442no—
419C2×C4—2442no—
420C2×C4—2442no—
421C2×C4—2444no—
422C2×C4#1451974512yes—
423C2×C4#1451974512yes—
424C2×C4SD16×D6112yes—
425C2×C4#1451979812yes—
426C2×C4#1451980612yes—
427C2×C4#1451980612yes—
428C2×C4C2×D4×D6111yescharacteristic
429C2×C4C22×C22⋊D6111yescharacteristic
430C2×C4C2×#1175822612yes—
431C2×C4#1451982512yes—
432C8—24no—
433C8—2442no—
434C22—1222no—
435C22—1222no—
436C22—1222no—
437C22—2441no—
438C22—2441no—
439C22—2441no—
440C22—2441no—
441C22C2×#1451974511yescharacteristic
442C22#1684031311yescharacteristic
443C22#1684033711yescharacteristic
444C22C2×SD16×D6111yescharacteristic
445C22C2×#1451979811yescharacteristic
446C22C2×#1451980611yescharacteristic
447C22C2×#1451980611yescharacteristic
448C4—44no—
449C4—82no—
450C4—1222no—
451C4—4882no—
452C4—2441no—
453C4—2441no—
454C4—2444no—
455C4#1683989914yes—
456C4#1683992114yes—
457C4#1684041512yes—
458C4#1684042312yes—
459C2—2442no—
460C2#2144944411yescharacteristic
461C2C2×#1683989911yescharacteristic
462C2C2×#1683992111yescharacteristic
463C2#2145638011yescharacteristic
464C2#2145656511yescharacteristic
465C2#2145696411yescharacteristic
466C2#2145698111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#498C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenterC2×#498D61
hypocenterC61#25672
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#498

Complements

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

NH ≅ G/Ntype
2. C2×#16840114C2semidirect
5. #21446781C2semidirect
6. #21446621C2semidirect
7. #21446084C2semidirect
9. C2×C61×#498C2semidirect
227. C2×#498D61semidirect
272. C61#25672semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168401142
#214541662
#214470172
#214467812
#214466212
#214460842
C4×#145196892
C2×C61×#4982
#2567261