Finite Group Databasebeta

← #29663454

Subgroups — #29663454

Order 15616.

628 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457139C212yesmaximal, complemented
2#21454794C211yesmaximal, characteristic, complemented
3#21445344C211yesmaximal, characteristic, complemented
4#21445339C211yesmaximal, characteristic, complemented
5#21445300C211yesmaximal, characteristic, complemented
6#21445295C211yesmaximal, characteristic, complemented
7#21445232C212yesmaximal, complemented
8#21445225C212yesmaximal, complemented
9C2×#16839513C211yesmaximal, characteristic, complemented
10#21444514C212yesmaximal, complemented
11C61×#3169C211yesmaximal, characteristic, complemented
12#16840296C2211yescharacteristic
13#16840062—22no—
14#16840054C2212yes—
15#16840049C2212yes—
16#16839996—22no—
17#16839989—22no—
18#16839977—22no—
19#16839959C2212yes—
20#16839954C2212yes—
21#16839892C2211yescharacteristic
22C8×Dic122C2211yescharacteristic
23#16839841C2211yescharacteristic
24#16839833—21no—
25C8×D244C2211yescharacteristic
26C8×C22⋊D61—21no—
27#16839821—21no—
28#16839819C2211yescharacteristic
29C2×C4⋊#92418C2212yes—
30#16839716—21no—
31#16839513C2212yes—
32#16839513C2211yescharacteristic
33#16839513C2211yescharacteristic
34#16839511C2212yes—
35#16839511C2212yes—
36#16839506C2212yes—
37#16839506C2212yes—
38C61×#532C2212yes—
39C61×#500C2211yescharacteristic
40C61×#493C2211yescharacteristic
41C61×#486C2212yes—
42C61×#466—21no—
43C61×#466—21no—
44C4×D8×C61—21no—
45C4×D8×C61—21no—
46C2×C4×C8×C61C2211yescharacteristic
47C2×#11758200—22no—
48C2×#11758199—22no—
49C2×#11758189—21no—
50C22×#92418—22no—
51C2×C8×D61—21no—
52#14519771—21no—
53#14519769D411yescharacteristic
54#14519765D411yescharacteristic
55#14519760—21no—
56#14519756D411yescharacteristic
57#14519751D411yescharacteristic
58C4×Dic122C2311yescharacteristic
59C4×D244C2311yescharacteristic
60C4×C22⋊D61—21no—
61#14519734—21no—
62#14519710—22no—
63#14519708—22no—
64C4⋊#92418—22no—
65C4⋊#92418C2312yes—
66C4⋊#92418C2312yes—
67C4⋊#92418C2311yescharacteristic
68C4⋊#92418C2311yescharacteristic
69#14519706—21no—
70#14519706—21no—
71#14519700—22no—
72#14519698—22no—
73#14519698—22no—
74#14519697—22no—
75C22⋊#92418—21no—
76#14519689—21no—
77#14519689—21no—
78#14519688—21no—
79#14519687—21no—
80C8×Dic61—21no—
81C2×#41×C61—22no—
82C2×D8×C61—21no—
83C2×D8×C61—21no—
84C22×C8×C61D411yescharacteristic
85C22×C8×C61D411yescharacteristic
86C4⋊Q8×C61C2312yes—
87#127×C61C2312yes—
88#124×C61—22no—
89#121×C61—22no—
90#121×C61—22no—
91C4×D4×C61—22no—
92C4×D4×C61—22no—
93C2×C42×C61C2311yescharacteristic
94#107×C61—21no—
95#107×C61—21no—
96#102×C61—22no—
97#102×C61—22no—
98C4×C8×C61D411yescharacteristic
99C4×C8×C61D411yescharacteristic
100C4×C8×C61C2311yescharacteristic
101C4×C8×C61C2311yescharacteristic
102#41×C61—84no—
103C2×Q8×C61—22no—
104C2×D4×C61—22no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—42no—
108C22×C4×C61—42no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C2×C22⋊D61—21no—
113C2×D244C2×D411yescharacteristic
114C2×C4×D61—21no—
115C2×Dic122C2×D411yescharacteristic
116D8×C61—42no—
117D8×C61—42no—
118C2×C8×C61—22no—
119C2×C8×C61—41no—
120C2×C8×C61C2×D411yescharacteristic
121C2×C8×C61C2×D411yescharacteristic
122C2×C8×C61C2×D411yescharacteristic
123C2×C8×C61C2×D411yescharacteristic
124C4⋊C4×C61—22no—
125C4⋊C4×C61—22no—
126C22⋊C4×C61—42no—
127C22⋊C4×C61—42no—
128C42×C61C2×D411yescharacteristic
129C42×C61C2×D411yescharacteristic
130C42×C61#4111yescharacteristic
131C42×C61C2411yescharacteristic
132C22⋊Dic61—21no—
133#11758200—82no—
134#11758199—82no—
135#11758198—21no—
136#11758198—21no—
137#11758198—21no—
138#11758198—21no—
139C4⋊Dic61C2×D411yescharacteristic
140C4⋊Dic61C2×D411yescharacteristic
141#11758196—21no—
142#11758196—21no—
143#11758196—21no—
144#11758196—21no—
145C4×Dic61—21no—
146C2×#92418—21no—
147C2×#92418—21no—
148C2×#92418—22no—
149C2×#92418—22no—
150C2×#92418—82no—
151#11758189—42no—
152C8×D61—42no—
153C23×C61—42no—
154C23×C61#12111yescharacteristic
155C22×D61—21no—
156Q8×C61—82no—
157D4×C61—42no—
158D4×C61—42no—
159D4×C61—42no—
160D4×C61—42no—
161D4×C61—82no—
162D4×C61—84no—
163C2×C4×C61—22no—
164C2×C4×C61—21no—
165C2×C4×C61—21no—
166C2×C4×C61—22no—
167C2×C4×C61—42no—
168C2×C4×C61—84no—
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61#12111yescharacteristic
172C2×C4×C61C22×D411yescharacteristic
173C2×C4×C61C22×D411yescharacteristic
174C2×C4×C61C2×#4111yescharacteristic
175C22⋊D61—42no—
176C2×Dic61—21no—
177C2×Dic61—21no—
178C2×Dic61—21no—
179D244—22no—
180C4×D61—41no—
181C4×D61—41no—
182Dic122—22no—
183C8×C61—21no—
184C8×C61—21no—
185C8×C61—21no—
186C8×C61#13512yes—
187#92418—41no—
188#92418—41no—
189#92418—42no—
190#92418—42no—
191#19227—611nomaximal, Sylow
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—82no—
195C22×C61—82no—
196C22×C61—82no—
197C22×C61#47212yes—
198C22×C61C2×#12111yescharacteristic
199C2×D61—41no—
200C2×D61—41no—
201C2×D61—41no—
202C4×C61—21no—
203C4×C61—21no—
204C4×C61—21no—
205C4×C61—21no—
206C4×C61—82no—
207C4×C61C2×#13511yescharacteristic
208C4×C61#57511yescharacteristic
209C4×C61#57712yes—
210Dic61—41no—
211Dic61—41no—
212Dic61—41no—
213#3247—612no—
214#3169D6111yescharacteristic, complemented
215#2966—611no—
216#1696—611no—
217#1695—611no—
218#1688—611no—
219#1687—611no—
220#1678—612no—
221#1677—612no—
222C2×#334—611no—
223#1551—612no—
224C2×C61—22no—
225C2×C61—162no—
226C2×C61#303511yescharacteristic
227C2×C61#306011yescharacteristic
228C2×C61C2×#47211yescharacteristic
229D61—81no—
230#532C2×D6112yes—
231#529—611no—
232#500C2×D6111yescharacteristic
233#493C2×D6111yescharacteristic
234#486C2×D6112yes—
235#478—1222no—
236#477—612no—
237#476—612no—
238#466—21no—
239#466—21no—
240#466—1222no—
241#465—1222no—
242#463—1222no—
243#460—612no—
244#459—612no—
245#446—611no—
246C8×Q8—611no—
247C4×D8—21no—
248C4×D8—21no—
249#436—611no—
250#435—1221no—
251C8×D4—611no—
252C8×D4—1221no—
253#433—1221no—
254#432—611no—
255C2×C4⋊C8—612no—
256#405—1221no—
257C2×C4×C8C2×D6111yescharacteristic
258#334—612no—
259#334—611no—
260#334—611no—
261#333—612no—
262#333—612no—
263#331—612no—
264#331—612no—
265C61#1922711yescharacteristic, complemented, Sylow
266C2×#41—22no—
267C2×SD16—1222no—
268C2×D8—21no—
269C2×D8—21no—
270C2×D8—1222no—
271C2×M16—1221no—
272C22×C8—1222no—
273C22×C8—1221no—
274C22×C8C22⋊D6111yescharacteristic
275C22×C8C22⋊D6111yescharacteristic
276C4⋊Q8C22×D6112yes—
277#127C22×D6112yes—
278#126—1221no—
279#125—611no—
280#124—22no—
281#124—611no—
282#123—1221no—
283C22⋊Q8—611no—
284#121—22no—
285#121—22no—
286#121—611no—
287C4×Q8—611no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—611no—
291C4×D4—1221no—
292#117—1221no—
293C2×C42C22×D6111yescharacteristic
294#107—21no—
295#107—21no—
296#107—1222no—
297#106—1222no—
298C4⋊C8—611no—
299C4⋊C8—611no—
300C4⋊C8—612no—
301C4⋊C8—612no—
302C4⋊C8—1221no—
303C4⋊C8—1221no—
304C4⋊C8—1222no—
305#103—1222no—
306#102—22no—
307#102—22no—
308#102—1222no—
309#102—1222no—
310#102—1222no—
311C22⋊C8—1221no—
312C22⋊C8—1221no—
313C22⋊C8—1221no—
314#97—1221no—
315#97—1221no—
316C4×C8—1221no—
317C4×C8C22⋊D6111yescharacteristic
318C4×C8C22⋊D6111yescharacteristic
319C4×C8C22×D6111yescharacteristic
320C4×C8C22×D6111yescharacteristic
321#41—84no—
322C2×Q8—22no—
323C2×Q8—611no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—42no—
328C2×D4—611no—
329C2×D4—1221no—
330C22×C4—42no—
331C22×C4—1221no—
332C22×C4D4×D6111yescharacteristic
333C22×C4#1175822411yescharacteristic
334C22×C4C2×C22⋊D6111yescharacteristic
335SD16—4882no—
336D8—42no—
337D8—42no—
338D8—4882no—
339M16—2442no—
340C2×C8—22no—
341C2×C8—41no—
342C2×C8—1222no—
343C2×C8—1222no—
344C2×C8—1221no—
345C2×C8—1221no—
346C2×C8—4882no—
347C2×C8—2442no—
348C2×C8C2×C22⋊D6111yescharacteristic
349C2×C8C2×C22⋊D6111yescharacteristic
350C2×C8C2×C22⋊D6111yescharacteristic
351C2×C8C2×C22⋊D6111yescharacteristic
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—611no—
355C4⋊C4—611no—
356C4⋊C4—1221no—
357C4⋊C4—1221no—
358C4⋊C4—1221no—
359C4⋊C4—1221no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—1221no—
363C22⋊C4—1221no—
364C22⋊C4—1221no—
365C22⋊C4—1221no—
366C22⋊C4—1221no—
367C42—1221no—
368C42D4×D6111yescharacteristic
369C42#1175822411yescharacteristic
370C42C2×C22⋊D6111yescharacteristic
371C42C23×D6111yescharacteristic
372C23—42no—
373C23—1221no—
374C23#1451974811yescharacteristic
375Q8—82no—
376Q8—1222no—
377D4—42no—
378D4—42no—
379D4—42no—
380D4—42no—
381D4—82no—
382D4—84no—
383D4—1222no—
384D4—2442no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—42no—
390C2×C4—84no—
391C2×C4—1221no—
392C2×C4—1221no—
393C2×C4—1221no—
394C2×C4—2441no—
395C2×C4—2441no—
396C2×C4#1451974811yescharacteristic
397C2×C4#1451974811yescharacteristic
398C2×C4#1451974811yescharacteristic
399C2×C4C2×D4×D6111yescharacteristic
400C2×C4C22×C22⋊D6111yescharacteristic
401C2×C4C2×#1175822411yescharacteristic
402C8—21no—
403C8—21no—
404C8—21no—
405C8—2441no—
406C8—2441no—
407C8—2442no—
408C8—2442no—
409C8#1451979712yes—
410C22—21no—
411C22—21no—
412C22—82no—
413C22—82no—
414C22—82no—
415C22—2441no—
416C22—2441no—
417C22—2441no—
418C22#1684002912yes—
419C22C2×#1451974811yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—82no—
425C4—2441no—
426C4—2441no—
427C4—2441no—
428C4C2×#1451979711yescharacteristic
429C4#1684056011yescharacteristic
430C4#1684056912yes—
431C2—22no—
432C2—162no—
433C2—4881no—
434C2#2145526811yescharacteristic
435C2#2145547711yescharacteristic
436C2C2×#1684002911yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3169C2
socleC22×C61C2×#121
hypercenter#3169D61
hypocenterC61#19227
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3169

Complements

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

NH ≅ G/Ntype
1. #21457139C2semidirect
2. #21454794C2semidirect
3. #21445344C2semidirect
4. #21445339C2semidirect
5. #21445300C2semidirect
6. #21445295C2semidirect
7. #21445232C2semidirect
8. #21445225C2semidirect
9. C2×#16839513C2semidirect
10. #21444514C2semidirect
11. C61×#3169C2semidirect
214. #3169D61semidirect
265. C61#19227semidirect

Maximal subgroups

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

subgroupindex
#214571392
#214547942
#214453442
#214453392
#214453002
#214452952
#214452322
#214452252
C2×#168395132
#214445142
C61×#31692
#1922761