Finite Group Databasebeta

← #29663403

Subgroups — #29663403

Order 15616.

628 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457146C212yesmaximal, complemented
2#21456398C211yesmaximal, characteristic, complemented
3#21454794C211yesmaximal, characteristic, complemented
4#21445360C212yesmaximal, complemented
5#21445353C212yesmaximal, complemented
6#21445224C211yesmaximal, characteristic, complemented
7#21445219C211yesmaximal, characteristic, complemented
8#21445180C211yesmaximal, characteristic, complemented
9#21445175C211yesmaximal, characteristic, complemented
10#21444512C212yesmaximal, complemented
11C2×C61×#335C211yesmaximal, characteristic, complemented
12#16840312C2212yes—
13#16840296C2211yescharacteristic
14#16840167C2211yescharacteristic
15#16840139C2211yescharacteristic
16#16840109C2212yes—
17#16840063—22no—
18#16840058C2212yes—
19#16840053C2212yes—
20#16839998—22no—
21#16839998—21no—
22#16839998—21no—
23#16839991—22no—
24#16839978—22no—
25#16839963C2212yes—
26#16839958C2212yes—
27#16839892C2211yescharacteristic
28C8×Dic122C2211yescharacteristic
29C4×D488—21no—
30C4×D488—21no—
31#16839841C2211yescharacteristic
32#16839833—21no—
33C8×D244C2211yescharacteristic
34C8×C22⋊D61—21no—
35#16839821—21no—
36#16839819C2211yescharacteristic
37#16839716—21no—
38#16839510C2212yes—
39#16839510C2212yes—
40#16839505C2212yes—
41#16839505C2212yes—
42C2×C4⋊C8×C61C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C61×#335C2211yescharacteristic
45C61×#335C2211yescharacteristic
46C61×#335C2212yes—
47C2×#11758222—22no—
48C2×#11758190—22no—
49C2×D488—21no—
50C2×D488—21no—
51C2×D488—22no—
52C2×#11758189—21no—
53C2×C8×D61—21no—
54#14519777C2312yes—
55#14519774C2312yes—
56#14519771—21no—
57#14519769D411yescharacteristic
58#14519765—22no—
59#14519765D411yescharacteristic
60#14519760—21no—
61#14519756D411yescharacteristic
62#14519751—22no—
63#14519751—22no—
64#14519751D411yescharacteristic
65C4×Dic122C2311yescharacteristic
66C4×D244—22no—
67C4×D244—22no—
68C4×D244C2311yescharacteristic
69C4×C22⋊D61—21no—
70#14519734—21no—
71#14519711—21no—
72#14519711—21no—
73#14519711—22no—
74#14519709—22no—
75C4⋊#92418C2311yescharacteristic
76C4⋊#92418C2311yescharacteristic
77#14519706—21no—
78#14519706—21no—
79#14519702—22no—
80#14519699—22no—
81#14519699—22no—
82#14519699—22no—
83#14519699—22no—
84#14519699—22no—
85C22⋊#92418—21no—
86#14519689—21no—
87#14519689—21no—
88#14519688—21no—
89#14519687—21no—
90C8×Dic61—21no—
91C22×C8×C61—22no—
92C22×C8×C61D411yescharacteristic
93C22×C8×C61D411yescharacteristic
94C2×C42×C61C2311yescharacteristic
95C4⋊C8×C61—22no—
96C4⋊C8×C61C2312yes—
97C4⋊C8×C61C2312yes—
98C4×C8×C61D411yescharacteristic
99C4×C8×C61D411yescharacteristic
100C4×C8×C61C2311yescharacteristic
101C4×C8×C61C2311yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61#4111yescharacteristic
105C2×C22⋊D61—21no—
106C2×C22⋊D61—42no—
107#11758222—84no—
108C2×D244—22no—
109C2×D244—22no—
110C2×D244—22no—
111C2×D244C2×D411yescharacteristic
112C2×C4×D61—21no—
113C2×C4×D61—42no—
114C2×Dic122—22no—
115C2×Dic122C2×D411yescharacteristic
116C2×C8×C61—22no—
117C2×C8×C61—22no—
118C2×C8×C61—22no—
119C2×C8×C61—41no—
120C2×C8×C61—82no—
121C2×C8×C61C2×D411yescharacteristic
122C2×C8×C61C2×D411yescharacteristic
123C2×C8×C61C2×D411yescharacteristic
124C2×C8×C61C2×D411yescharacteristic
125C42×C61C2×D411yescharacteristic
126C42×C61C2×D411yescharacteristic
127C42×C61#4111yescharacteristic
128C42×C61C2411yescharacteristic
129C22⋊Dic61—21no—
130#11758198—21no—
131#11758198—21no—
132#11758198—21no—
133#11758198—21no—
134#11758198—42no—
135#11758198—42no—
136C4⋊Dic61—22no—
137C4⋊Dic61—22no—
138C4⋊Dic61C2×D411yescharacteristic
139C4⋊Dic61C2×D411yescharacteristic
140#11758196—21no—
141#11758196—21no—
142#11758196—21no—
143#11758196—21no—
144C4×Dic61—21no—
145C2×#92418—21no—
146C2×#92418—21no—
147D488—42no—
148D488—42no—
149D488—82no—
150#11758190—82no—
151#11758189—42no—
152C8×D61—42no—
153C23×C61#12111yescharacteristic
154C22×D61—21no—
155C22×D61—42no—
156C2×C4×C61—22no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61C22×D411yescharacteristic
165C2×C4×C61C22×D411yescharacteristic
166C2×C4×C61C2×#4111yescharacteristic
167C22⋊D61—42no—
168C22⋊D61—84no—
169C2×Dic61—21no—
170C2×Dic61—21no—
171C2×Dic61—21no—
172C2×Dic61—42no—
173D244—21no—
174D244—21no—
175D244—42no—
176D244—42no—
177D244—42no—
178D244—42no—
179D244—82no—
180C4×D61—41no—
181C4×D61—41no—
182C4×D61—84no—
183Dic122—21no—
184Dic122—21no—
185Dic122—82no—
186C8×C61—21no—
187C8×C61—21no—
188C8×C61—21no—
189C8×C61—42no—
190C8×C61—42no—
191C8×C61#13512yes—
192#92418—41no—
193#92418—41no—
194#19222—611nomaximal, Sylow
195C22×C61—21no—
196C22×C61—21no—
197C22×C61#47212yes—
198C22×C61C2×#12111yescharacteristic
199C2×D61—41no—
200C2×D61—41no—
201C2×D61—41no—
202C2×D61—82no—
203C2×D61—82no—
204C2×D61—82no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—21no—
208C4×C61—21no—
209C4×C61C2×#13511yescharacteristic
210C4×C61#57511yescharacteristic
211C4×C61#57611yescharacteristic
212C4×C61#57811yescharacteristic
213Dic61—41no—
214Dic61—41no—
215Dic61—41no—
216Dic61—82no—
217#3247—612no—
218#3169—611no—
219#2966—611no—
220#1698—612no—
221#1697—612no—
222#1676—611no—
223#1675—611no—
224#1668—611no—
225#1667—611no—
226C2×#335D6111yescharacteristic, complemented
227#1551—612no—
228C2×C61—22no—
229C2×C61#303511yescharacteristic
230C2×C61#306111yescharacteristic
231C2×C61C2×#47211yescharacteristic
232D61—81no—
233D61—162no—
234#532—612no—
235#529—611no—
236#500—611no—
237#493—611no—
238#486—612no—
239#478—1222no—
240#477—612no—
241#476—612no—
242#466—1222no—
243#466—1221no—
244#466—1221no—
245#465—1222no—
246#463—1222no—
247#460—612no—
248#459—612no—
249#446—611no—
250C8×Q8—611no—
251C4×D8—1221no—
252C4×D8—1221no—
253#436—611no—
254#435—1221no—
255C8×D4—611no—
256C8×D4—1221no—
257#433—1221no—
258#432—611no—
259C2×C4⋊C8C2×D6112yes—
260#405—1221no—
261C2×C4×C8C2×D6111yescharacteristic
262#335C2×D6111yescharacteristic
263#335C2×D6111yescharacteristic
264#335C2×D6112yes—
265#333—612no—
266#333—612no—
267#331—612no—
268#331—612no—
269C61#1922211yescharacteristic, complemented, Sylow
270C2×#41—1222no—
271C2×SD16—1222no—
272C2×D8—1222no—
273C2×D8—1221no—
274C2×D8—1221no—
275C2×M16—1221no—
276C22×C8—22no—
277C22×C8—1221no—
278C22×C8D24411yescharacteristic
279C22×C8D24411yescharacteristic
280C4⋊Q8—612no—
281#127—612no—
282#126—1221no—
283#125—611no—
284#124—611no—
285#124—1222no—
286#123—1221no—
287C22⋊Q8—611no—
288#121—611no—
289#121—1222no—
290#121—1222no—
291C4×Q8—611no—
292C4×D4—611no—
293C4×D4—1221no—
294C4×D4—1222no—
295C4×D4—1222no—
296#117—1221no—
297C2×C42C22×D6111yescharacteristic
298#107—1222no—
299#107—1221no—
300#107—1221no—
301#106—1222no—
302C4⋊C8—22no—
303C4⋊C8—611no—
304C4⋊C8—611no—
305C4⋊C8—1221no—
306C4⋊C8—1221no—
307C4⋊C8C22×D6112yes—
308C4⋊C8C22×D6112yes—
309#103—1222no—
310#102—1222no—
311#102—1222no—
312#102—1222no—
313#102—1222no—
314#102—1222no—
315C22⋊C8—1221no—
316C22⋊C8—1221no—
317C22⋊C8—1221no—
318#97—1221no—
319#97—1221no—
320C4×C8—1221no—
321C4×C8D24411yescharacteristic
322C4×C8D24411yescharacteristic
323C4×C8C22×D6111yescharacteristic
324C4×C8C22×D6111yescharacteristic
325#41—4884no—
326C2×Q8—611no—
327C2×Q8—1222no—
328C2×D4—611no—
329C2×D4—1221no—
330C2×D4—1222no—
331C2×D4—1222no—
332C2×D4—1222no—
333C2×D4—2442no—
334C22×C4—1221no—
335C22×C4—2442no—
336C22×C4C2×D24411yescharacteristic
337C22×C4D4×D6111yescharacteristic
338C22×C4#1175822611yescharacteristic
339SD16—4882no—
340D8—4882no—
341D8—2442no—
342D8—2442no—
343M16—2442no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—41no—
348C2×C8—82no—
349C2×C8—1221no—
350C2×C8—1221no—
351C2×C8—2442no—
352C2×C8C2×D24411yescharacteristic
353C2×C8C2×D24411yescharacteristic
354C2×C8C2×D24411yescharacteristic
355C2×C8C2×D24411yescharacteristic
356C4⋊C4—611no—
357C4⋊C4—611no—
358C4⋊C4—1221no—
359C4⋊C4—1221no—
360C4⋊C4—1221no—
361C4⋊C4—1221no—
362C4⋊C4—1222no—
363C4⋊C4—1222no—
364C22⋊C4—1221no—
365C22⋊C4—1221no—
366C22⋊C4—1221no—
367C22⋊C4—1221no—
368C22⋊C4—1221no—
369C22⋊C4—2442no—
370C22⋊C4—2442no—
371C42—1221no—
372C42C2×D24411yescharacteristic
373C42D4×D6111yescharacteristic
374C42#1175822611yescharacteristic
375C42C23×D6111yescharacteristic
376C23—1221no—
377C23—2442no—
378C23#1451975211yescharacteristic
379Q8—1221no—
380Q8—1221no—
381Q8—4882no—
382D4—1221no—
383D4—1221no—
384D4—4882no—
385D4—4884no—
386D4—2442no—
387D4—2442no—
388D4—2442no—
389D4—2442no—
390D4—2442no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—1221no—
397C2×C4—1221no—
398C2×C4—1221no—
399C2×C4—4884no—
400C2×C4—2442no—
401C2×C4—2441no—
402C2×C4—2441no—
403C2×C4#1451975211yescharacteristic
404C2×C4#1451975211yescharacteristic
405C2×C4#1451975211yescharacteristic
406C2×C4C2×D4×D6111yescharacteristic
407C2×C4C22×D24411yescharacteristic
408C2×C4C2×#1175822611yescharacteristic
409C8—21no—
410C8—21no—
411C8—21no—
412C8—42no—
413C8—42no—
414C8—2441no—
415C8—2441no—
416C8#1451979912yes—
417C22—21no—
418C22—21no—
419C22—4882no—
420C22—4882no—
421C22—4882no—
422C22—2441no—
423C22—2441no—
424C22—2441no—
425C22#1684003312yes—
426C22C2×#1451975211yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—4882no—
432C4—2441no—
433C4—2441no—
434C4—2441no—
435C4C2×#1451979911yescharacteristic
436C4#1684056211yescharacteristic
437C4#1684056311yescharacteristic
438C4#1684057511yescharacteristic
439C2—22no—
440C2—9762no—
441C2—4881no—
442C2#2145527611yescharacteristic
443C2#2145549111yescharacteristic
444C2C2×#1684003311yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#335C2
socleC22×C61C2×#121
hypercenterC2×#335D61
hypocenterC61#19222
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#335

Complements

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

NH ≅ G/Ntype
1. #21457146C2semidirect
2. #21456398C2semidirect
3. #21454794C2semidirect
4. #21445360C2semidirect
5. #21445353C2semidirect
6. #21445224C2semidirect
7. #21445219C2semidirect
8. #21445180C2semidirect
9. #21445175C2semidirect
10. #21444512C2semidirect
11. C2×C61×#335C2semidirect
226. C2×#335D61semidirect
269. C61#19222semidirect

Maximal subgroups

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

subgroupindex
#214571462
#214563982
#214547942
#214453602
#214453532
#214452242
#214452192
#214451802
#214451752
#214445122
C2×C61×#3352
#1922261