Finite Group Databasebeta

← #29733671

Subgroups — #29733671

Order 15616.

1094 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840374C211yesmaximal, characteristic, complemented
2C2×#16840157C211yesmaximal, characteristic, complemented
3C2×#16840116C211yesmaximal, characteristic, complemented
4#21446827C212yesmaximal, complemented
5#21446822C212yesmaximal
6#21446787C211yesmaximal, characteristic, complemented
7#21446620C211yesmaximal, characteristic, complemented
8#21446177C212yesmaximal, complemented
9#21446163C212yesmaximal, complemented
10#21445585C211yesmaximal, characteristic, complemented
11C61×#2620C211yesmaximal, characteristic
12C22×#11758202C2212yes—
13C22×#11758200C2212yes—
14#16840374—24no—
15#16840374—24no—
16C2×#14519777C2211yescharacteristic
17C2×#14519770—22no—
18C2×#14519769C2211yescharacteristic
19C2×#14519767—22no—
20C2×#14519756C2212yes—
21C2×C4×Dic122C2212yes—
22#16840157—24no—
23#16840116—24no—
24C2×#14519701C2212yes—
25C2×#14519700C2212yes—
26C2×#14519700C2212yes—
27C2×#14519698C2212yes—
28C2×C22⋊#92418C2212yes—
29C2×#14519688C2211yescharacteristic
30C2×C4×#92418C2211yescharacteristic
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34C2×#124×C61C2211yescharacteristic
35C2×C22⋊Q8×C61C2211yescharacteristic
36C2×#121×C61C2211yescharacteristic
37C2×#117×C61C2211yescharacteristic
38C61×#396—22no—
39C61×#394—22no—
40C61×#391C2211yescharacteristic
41C61×#390C2211yescharacteristic
42C61×#384C2211yescharacteristic
43C22×Dic122C2312yes—
44C2×#11758202—24no—
45C2×#11758202—24no—
46C2×#11758202—42no—
47C2×#11758200—24no—
48C2×#11758200—24no—
49C2×#11758200—42no—
50C22×#92418C2312yes—
51C22×#92418C2312yes—
52#14519777—22no—
53#14519777D414yescomplemented
54#14519770—216no—
55#14519769—22no—
56#14519769D414yescomplemented
57#14519767—216no—
58#14519756—24no—
59#14519756—24no—
60C4×Dic122—24no—
61C4×Dic122—24no—
62C2×C4⋊Dic61—22no—
63C2×C4⋊Dic61C2312yes—
64C2×#11758196—22no—
65C2×#11758196—22no—
66C2×#11758196—22no—
67C2×C22⋊Dic61—22no—
68C2×C4×Dic61—22no—
69#14519701—24no—
70#14519700—42no—
71#14519700—42no—
72#14519698—24no—
73C22⋊#92418—24no—
74#14519688—22no—
75C4×#92418—22no—
76C22×Q8×C61C2311yescharacteristic
77C22×D4×C61—21no—
78C22×D4×C61C2311yescharacteristic
79C23×C4×C61C2311yescharacteristic
80#124×C61—22no—
81#124×C61—22no—
82C22⋊Q8×C61—42no—
83#121×C61—42no—
84#117×C61—22no—
85#117×C61—22no—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—22no—
88C2×C4⋊C4×C61D412yes—
89C2×C4⋊C4×C61C2311yescharacteristic
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C22⋊C4×C61—21no—
92C2×C22⋊C4×C61—22no—
93C2×C22⋊C4×C61—21no—
94C2×C22⋊C4×C61—22no—
95C2×C22⋊C4×C61D412yes—
96C2×C42×C61C2311yescharacteristic
97C2×C42×C61C2311yescharacteristic
98#95×C61—22no—
99C24×C61—21no—
100C24×C61C2×D411yescharacteristic
101C2×Q8×C61—21no—
102C2×Q8×C61—21no—
103C2×Q8×C61—22no—
104C2×Q8×C61—22no—
105C2×D4×C61—21no—
106C2×D4×C61—21no—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61—42no—
110C2×D4×C61—82no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61C2×D412yes—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2411yescharacteristic
122C22×Dic61—22no—
123C22×Dic61—22no—
124C2×Dic122—22no—
125C2×Dic122—22no—
126C2×Dic122—22no—
127C2×Dic122—22no—
128C2×Dic122—22no—
129C2×Dic122C2×D414yes—
130C4⋊C4×C61—24no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—42no—
134C4⋊C4×C61—82no—
135C22⋊C4×C61—24no—
136C22⋊C4×C61—42no—
137C22⋊C4×C61—42no—
138C22⋊C4×C61—44no—
139C22⋊C4×C61—82no—
140C42×C61—21no—
141C42×C61—21no—
142C42×C61C2×D412yes—
143C42×C61C2×D412yes—
144C22⋊Dic61—28no—
145#11758202—48no—
146#11758200—48no—
147C4⋊Dic61—28no—
148C4⋊Dic61—22no—
149C4⋊Dic61C2×D414yes—
150#11758196—28no—
151#11758196—28no—
152#11758196—28no—
153C4×Dic61—28no—
154C2×#92418—22no—
155C2×#92418—22no—
156C2×#92418—22no—
157C2×#92418—22no—
158C2×#92418—22no—
159C2×#92418#4114yes—
160C23×C61—21no—
161C23×C61—21no—
162C23×C61—21no—
163C23×C61—22no—
164C23×C61—22no—
165C23×C61—41no—
166C23×C61—41no—
167C23×C61—41no—
168C23×C61—42no—
169C23×C61—42no—
170C23×C61C22×D411yescharacteristic
171Q8×C61—42no—
172Q8×C61—42no—
173D4×C61—42no—
174D4×C61—42no—
175D4×C61—84no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61#12014yes—
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C2×#4112yes—
197C2×C4×C61H(5,𝔽2)12yes—
198C2×Dic61—24no—
199C2×Dic61—24no—
200C2×Dic61—42no—
201C2×Dic61—42no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204Dic122—24no—
205Dic122—24no—
206Dic122—42no—
207Dic122—42no—
208#92418—24no—
209#92418—42no—
210#25612—611nomaximal, Sylow
211C22×C61—42no—
212C22×C61—41no—
213C22×C61—41no—
214C22×C61—41no—
215C22×C61—42no—
216C22×C61—82no—
217C22×C61—81no—
218C22×C61—81no—
219C22×C61—81no—
220C22×C61—82no—
221C22×C61C2×#12011yescharacteristic
222C22×C61#53211yescharacteristic
223C22×C61#53511yescharacteristic
224C22×C61#57712yes—
225C22×C61#57812yes—
226C4×C61—24no—
227C4×C61—41no—
228C4×C61—42no—
229C4×C61—81no—
230C4×C61—81no—
231C4×C61—81no—
232C4×C61#54612yes—
233C4×C61#55012yes—
234Dic61—44no—
235Dic61—44no—
236C2×#541—611no—
237C2×#497—611no—
238C2×#488—611no—
239#2620D6111yescharacteristic
240#2001—612no—
241#2000—612no—
242#1992—611no—
243#1956—611no—
244#1880—612no—
245#1878—612no—
246#1750—611no—
247C2×C61—81no—
248C2×C61—161no—
249C2×C61#243111yescharacteristic
250C2×C61#301312yes—
251C2×C61#317812yes—
252C2×C61#322211yescharacteristic
253C2×C61#322311yescharacteristic
254C22×Q16—612no—
255C22×SD16—612no—
256#541—1224no—
257#541—1224no—
258C2×C4⋊Q8—611no—
259C2×#126—1222no—
260C2×#125—611no—
261C2×#125—1222no—
262C2×#124C2×D6111yescharacteristic
263C2×C22⋊Q8—612no—
264C2×C22⋊Q8C2×D6111yescharacteristic
265C2×#121C2×D6111yescharacteristic
266C2×C4×Q8—612no—
267C2×#117C2×D6111yescharacteristic
268#497—1224no—
269#488—1224no—
270C2×#103—612no—
271C2×#103—612no—
272C2×#103—612no—
273C2×#102—612no—
274C2×C22⋊C8—612no—
275C2×#97—611no—
276C2×C4×C8—611no—
277#396—22no—
278#394—22no—
279#391C2×D6111yescharacteristic
280#390C2×D6111yescharacteristic
281#384C2×D6111yescharacteristic
282#340—612no—
283#340—612no—
284#336—612no—
285C61#2561211yescharacteristic, complemented, Sylow
286C22×Q8—612no—
287C22×Q8C22×D6111yescharacteristic
288C22×D4—21no—
289C22×D4C22×D6111yescharacteristic
290C23×C4C22×D6111yescharacteristic
291C2×Q16—1224no—
292C2×Q16—1224no—
293C2×Q16—2442no—
294C2×SD16—1224no—
295C2×SD16—1224no—
296C2×SD16—2442no—
297C22×C8—612no—
298C22×C8—612no—
299C4⋊Q8—614no—
300C4⋊Q8—1222no—
301#126—12216no—
302#125—614no—
303#125—12216no—
304#125—1222no—
305#124—22no—
306#124—22no—
307C22⋊Q8—42no—
308C22⋊Q8—1224no—
309C22⋊Q8—1224no—
310#121—42no—
311C4×Q8—1224no—
312C4×Q8—1224no—
313#117—22no—
314#117—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—612no—
318C2×C4⋊C4—1222no—
319C2×C4⋊C4—1222no—
320C2×C4⋊C4—1222no—
321C2×C4⋊C4—1222no—
322C2×C4⋊C4C22⋊D6112yes—
323C2×C4⋊C4C22×D6111yescharacteristic
324C2×C4⋊C4C22×D6111yescharacteristic
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—1222no—
330C2×C22⋊C4C22⋊D6112yes—
331C2×C42—1222no—
332C2×C42C22×D6111yescharacteristic
333C2×C42C22×D6111yescharacteristic
334#103—1224no—
335#103—2442no—
336#103—2442no—
337#102—1224no—
338C22⋊C8—1224no—
339#97—1222no—
340C4×C8—1222no—
341#95—22no—
342C24—21no—
343C24C2×C22⋊D6111yescharacteristic
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—614no—
349C2×Q8—1222no—
350C2×Q8—1222no—
351C2×Q8—1222no—
352C2×Q8—1222no—
353C2×Q8—1222no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—42no—
359C2×D4—82no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—1222no—
368C22×C4—1222no—
369C22×C4C2×C22⋊D6112yes—
370C22×C4C2×C22⋊D6111yescharacteristic
371C22×C4C2×C22⋊D6112yes—
372C22×C4C23×D6111yescharacteristic
373Q16—2448no—
374SD16—2448no—
375C2×C8—614no—
376C2×C8—1222no—
377C2×C8—1222no—
378C2×C8—1222no—
379C2×C8—1222no—
380C2×C8—1222no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—42no—
385C4⋊C4—82no—
386C4⋊C4—614no—
387C4⋊C4—1228no—
388C4⋊C4—1228no—
389C4⋊C4—1222no—
390C4⋊C4—1228no—
391C4⋊C4—1228no—
392C22⋊C4—24no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—44no—
396C22⋊C4—82no—
397C22⋊C4—1228no—
398C42—21no—
399C42—21no—
400C42—1228no—
401C42D4×D6112yes—
402C42D4×D6112yes—
403C23—21no—
404C23—21no—
405C23—22no—
406C23—22no—
407C23—21no—
408C23—41no—
409C23—41no—
410C23—42no—
411C23—42no—
412C23—41no—
413C23C22×C22⋊D6111yescharacteristic
414Q8—42no—
415Q8—42no—
416Q8—1224no—
417Q8—1224no—
418Q8—2442no—
419Q8—2442no—
420D4—42no—
421D4—42no—
422D4—84no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—1224no—
441C2×C4—1224no—
442C2×C4—2442no—
443C2×C4—2442no—
444C2×C4—2442no—
445C2×C4—2442no—
446C2×C4#1451974514yes—
447C2×C4C2×D4×D6111yescharacteristic
448C2×C4C2×D4×D6111yescharacteristic
449C2×C4#41×D6112yes—
450C2×C4#1451982212yes—
451C8—1224no—
452C8—2442no—
453C22—41no—
454C22—42no—
455C22—42no—
456C22—41no—
457C22—41no—
458C22—81no—
459C22—82no—
460C22—82no—
461C22—81no—
462C22—81no—
463C22C2×#1451974511yescharacteristic
464C22#1684030811yescharacteristic
465C22#1684032911yescharacteristic
466C22#1684057312yes—
467C22#1684057712yes—
468C4—24no—
469C4—42no—
470C4—41no—
471C4—81no—
472C4—81no—
473C4—81no—
474C4—2444no—
475C4—2444no—
476C4#1684040912yes—
477C4#1684043012yes—
478C2—81no—
479C2—161no—
480C2#2144943911yescharacteristic
481C2#2145513112yes—
482C2#2145648112yes—
483C2#2145689111yescharacteristic
484C2#2145690211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2620C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
hypercenter#2620D61
hypocenterC61#25612
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2620

Complements

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

NH ≅ G/Ntype
1. C2×#16840374C2semidirect
2. C2×#16840157C2semidirect
3. C2×#16840116C2semidirect
4. #21446827C2semidirect
6. #21446787C2semidirect
7. #21446620C2semidirect
8. #21446177C2semidirect
9. #21446163C2semidirect
10. #21445585C2semidirect
53. #14519777D4semidirect
56. #14519769D4semidirect
285. C61#25612semidirect

Maximal subgroups

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

subgroupindex
C2×#168403742
C2×#168401572
C2×#168401162
#214468272
#214468222
#214467872
#214466202
#214461772
#214461632
#214455852
C61×#26202
#2561261