Finite Group Databasebeta

← #29663330

Subgroups — #29663330

Order 15616.

592 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457161C211yesmaximal, characteristic, complemented
2#21457136C211yesmaximal, characteristic
3#21456144C211yesmaximal, characteristic
4#21454749C211yesmaximal, characteristic, complemented
5#21445404C212yesmaximal, complemented
6#21445390C212yesmaximal, complemented
7#21445252C212yesmaximal, complemented
8#21445238C212yesmaximal, complemented
9C2×#16839512C211yesmaximal, characteristic, complemented
10#21444497C211yesmaximal, characteristic
11C61×#1550C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13#16840312C2211yescharacteristic
14#16840119—21no—
15#16840119—21no—
16#16840114C2211yescharacteristic
17#16840114C2211yescharacteristic
18#16840110C2211yescharacteristic
19#16840110C2211yescharacteristic
20#16840057—21no—
21#16840054C2212yes—
22#16840053—21no—
23#16840048C2212yes—
24#16840001—22no—
25#16839991—22no—
26#16839972—21no—
27#16839969C2212yes—
28#16839968—21no—
29#16839963C2212yes—
30#16839889—21no—
31#16839889C2212yes—
32#16839886—21no—
33#16839886C2212yes—
34#16839790—21no—
35C2×C4⋊#92418C2211yescharacteristic
36#16839756—21no—
37#16839756—21no—
38#16839720—21no—
39C2×C4×#92418C2211yescharacteristic
40#16839512C2212yes—
41#16839512C2212yes—
42#16839509C2212yes—
43#16839505C2212yes—
44C61×#533C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#333C2212yes—
48C61×#332C2212yes—
49C2×#11758222—21no—
50C2×#11758202—22no—
51C2×#11758190—22no—
52C22×#92418—21no—
53C22×#92418D411yescharacteristic
54C22×#92418D411yescharacteristic
55#14519777D411yescharacteristic
56#14519777D411yescharacteristic
57#14519777C2311yescharacteristic
58#14519774C2311yescharacteristic
59#14519769—21no—
60#14519765—21no—
61#14519756—21no—
62#14519756D412yes—
63#14519751—21no—
64#14519751—21no—
65C4×Dic122—21no—
66C4×Dic122C2312yes—
67C4×D244—21no—
68C4×D244—21no—
69#14519736—21no—
70#14519710—22no—
71#14519709—22no—
72C4⋊#92418—21no—
73C4⋊#92418C2312yes—
74#14519706—22no—
75#14519706—22no—
76#14519702—22no—
77#14519701—22no—
78#14519700—22no—
79#14519700—21no—
80#14519700—21no—
81#14519700—21no—
82#14519700—21no—
83#14519699—22no—
84#14519697—21no—
85#14519697—21no—
86#14519697—21no—
87#14519697—21no—
88#14519687—22no—
89C4×#92418D411yescharacteristic
90C4×#92418D411yescharacteristic
91C4×#92418C2311yescharacteristic
92C4×#92418C2311yescharacteristic
93C8×Dic61—22no—
94C2×M16×C61—21no—
95C22×C8×C61—21no—
96C4⋊Q8×C61C2311yescharacteristic
97C4⋊Q8×C61C2311yescharacteristic
98#125×C61—21no—
99C22⋊Q8×C61—21no—
100C22⋊Q8×C61—21no—
101C4×Q8×C61—21no—
102C4×Q8×C61—21no—
103#117×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—21no—
106C4⋊C8×C61—21no—
107C4⋊C8×C61C2312yes—
108C4⋊C8×C61C2312yes—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C2×C22⋊D61—41no—
115#11758222—42no—
116#11758222—42no—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244—21no—
120C2×C4×D61—41no—
121C2×Dic122—21no—
122C2×Dic122—21no—
123C2×Dic122C2×D412yes—
124M16×C61—42no—
125C2×C8×C61—22no—
126C2×C8×C61—22no—
127C2×C8×C61—81no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—41no—
133C4⋊C4×C61—41no—
134C22⋊C4×C61—41no—
135C42×C61—41no—
136C42×C61C2×D411yescharacteristic
137C42×C61C2×D411yescharacteristic
138C42×C61#4111yescharacteristic
139C42×C61C2411yescharacteristic
140C22⋊Dic61—22no—
141#11758202—82no—
142#11758198—41no—
143#11758198—41no—
144C4⋊Dic61—21no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61C2×D412yes—
150#11758196—22no—
151#11758196—22no—
152C4×Dic61—22no—
153C2×#92418—22no—
154C2×#92418—22no—
155C2×#92418—22no—
156C2×#92418—81no—
157C2×#92418C2×D411yescharacteristic
158C2×#92418C2×D411yescharacteristic
159C2×#92418C2×D411yescharacteristic
160C2×#92418C2×D411yescharacteristic
161#11758190—82no—
162C23×C61#12111yescharacteristic
163C22×D61—41no—
164Q8×C61—41no—
165Q8×C61—41no—
166Q8×C61—81no—
167C2×C4×C61—21no—
168C2×C4×C61—21no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61#12111yescharacteristic
173C2×C4×C61#12111yescharacteristic
174C2×C4×C61#12111yescharacteristic
175C2×C4×C61#13512yes—
176C2×C4×C61#13712yes—
177C2×C4×C61C22×D411yescharacteristic
178C2×C4×C61C22×D411yescharacteristic
179C2×C4×C61C2×#4111yescharacteristic
180C22⋊D61—82no—
181C2×Dic61—22no—
182C2×Dic61—22no—
183C2×Dic61—41no—
184D244—41no—
185D244—41no—
186D244—41no—
187D244—41no—
188D244—81no—
189C4×D61—82no—
190Dic122—22no—
191Dic122—22no—
192Dic122—42no—
193Dic122—41no—
194Dic122—41no—
195C8×C61—42no—
196C8×C61—42no—
197#92418—21no—
198#92418—21no—
199#92418—21no—
200#92418—21no—
201#92418—42no—
202#19216—611nomaximal, Sylow
203C22×C61—21no—
204C22×C61—21no—
205C22×C61#46412yes—
206C22×C61C2×#12111yescharacteristic
207C2×D61—81no—
208C2×D61—81no—
209C2×D61—81no—
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—81no—
214C4×C61—81no—
215C4×C61#46412yes—
216C4×C61C2×#13511yescharacteristic
217C4×C61C2×#13711yescharacteristic
218C4×C61#57711yescharacteristic
219C4×C61#57711yescharacteristic
220Dic61—42no—
221Dic61—42no—
222Dic61—81no—
223#3249—611no—
224#3246—611no—
225#3131—611no—
226#2960—611no—
227#1704—612no—
228#1702—612no—
229#1680—612no—
230#1678—612no—
231C2×#334—611no—
232#1550D6111yescharacteristic, complemented
233#1547—611no—
234C2×C61—22no—
235C2×C61C2×#46411yescharacteristic
236C2×C61#306011yescharacteristic
237C2×C61#306711yescharacteristic
238D61—161no—
239#533—611no—
240#533C2×D6111yescharacteristic
241#532—611no—
242#488—1221no—
243#488—1221no—
244#487—611no—
245#487—611no—
246#486—611no—
247#486—611no—
248#477—612no—
249#477—1221no—
250#476—1221no—
251#475—612no—
252#467—1222no—
253#465—1222no—
254#462—612no—
255#462—1221no—
256#461—1221no—
257#460—612no—
258#446—612no—
259#446—1221no—
260C8×Q8—612no—
261C8×Q8—1221no—
262#424—1221no—
263#423C2×D6111yescharacteristic
264C2×C4⋊C8—611no—
265C2×C4⋊C8C2×D6111yescharacteristic
266#416—1221no—
267#416—1221no—
268#405—1221no—
269C2×C4×C8—611no—
270#334—612no—
271#334—612no—
272#333C2×D6112yes—
273#332—612no—
274#332C2×D6112yes—
275#331—612no—
276C61#1921611yescharacteristic, complemented, Sylow
277C2×#41—1221no—
278C2×Q16—1222no—
279C2×SD16—1222no—
280C2×M16—21no—
281C22×C8—21no—
282C22×C8—611no—
283C22×C8—611no—
284C22×C8—1221no—
285C4⋊Q8—611no—
286C4⋊Q8—611no—
287C4⋊Q8—611no—
288C4⋊Q8C22×D6111yescharacteristic
289C4⋊Q8C22×D6111yescharacteristic
290#127—611no—
291#125—21no—
292#125—1221no—
293#124—1221no—
294C22⋊Q8—21no—
295C22⋊Q8—21no—
296C22⋊Q8—612no—
297C22⋊Q8—1221no—
298#121—1221no—
299#121—1221no—
300C4×Q8—21no—
301C4×Q8—21no—
302C4×Q8—612no—
303C4×Q8—1221no—
304C4×D4—1221no—
305C4×D4—1221no—
306#117—21no—
307#117—1221no—
308C2×C42C22×D6111yescharacteristic
309#107—1222no—
310#106—1222no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—612no—
314C4⋊C8—1222no—
315C4⋊C8—1222no—
316C4⋊C8—1221no—
317C4⋊C8C22×D6112yes—
318C4⋊C8C22×D6112yes—
319#103—1221no—
320#103—1221no—
321#103—1221no—
322#103—1221no—
323#103—1222no—
324#103—1222no—
325#103—1222no—
326#102—1221no—
327#102—1221no—
328#102—1221no—
329#102—1221no—
330#102—1222no—
331#97—1222no—
332C4×C8—611no—
333C4×C8—611no—
334C4×C8—611no—
335C4×C8—611no—
336C4×C8—1222no—
337#41—2442no—
338#41—2442no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—612no—
342C2×Q8—1221no—
343C2×Q8—1221no—
344C2×D4—1221no—
345C2×D4—1221no—
346C2×D4—1221no—
347C2×D4—2441no—
348C22×C4—2441no—
349C22×C4#1175822211yescharacteristic
350C22×C4D4×D6111yescharacteristic
351C22×C4D4×D6111yescharacteristic
352Q16—4882no—
353SD16—4882no—
354M16—42no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—81no—
358C2×C8—611no—
359C2×C8—611no—
360C2×C8—611no—
361C2×C8—611no—
362C2×C8—1222no—
363C2×C8—1222no—
364C2×C8—1222no—
365C2×C8—4881no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—612no—
373C4⋊C4—1222no—
374C4⋊C4—1222no—
375C4⋊C4—1221no—
376C4⋊C4—1221no—
377C4⋊C4—1221no—
378C4⋊C4—1221no—
379C4⋊C4—1221no—
380C22⋊C4—41no—
381C22⋊C4—1222no—
382C22⋊C4—2441no—
383C22⋊C4—2441no—
384C42—41no—
385C42—1222no—
386C42#1175822211yescharacteristic
387C42D4×D6111yescharacteristic
388C42D4×D6111yescharacteristic
389C42C23×D6111yescharacteristic
390C23—2441no—
391C23#1451975011yescharacteristic
392Q8—41no—
393Q8—41no—
394Q8—81no—
395Q8—1222no—
396Q8—1222no—
397Q8—2442no—
398Q8—2441no—
399Q8—2441no—
400D4—4881no—
401D4—4882no—
402D4—2441no—
403D4—2441no—
404D4—2441no—
405D4—2441no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—1222no—
412C2×C4—1222no—
413C2×C4—4882no—
414C2×C4—2441no—
415C2×C4#1451975011yescharacteristic
416C2×C4#1451975011yescharacteristic
417C2×C4#1451975011yescharacteristic
418C2×C4#1451979812yes—
419C2×C4#1451980912yes—
420C2×C4C2×D4×D6111yescharacteristic
421C2×C4C2×D4×D6111yescharacteristic
422C2×C4C2×#1175822211yescharacteristic
423C8—42no—
424C8—42no—
425C8—1221no—
426C8—1221no—
427C8—1221no—
428C8—1221no—
429C8—2442no—
430C22—21no—
431C22—21no—
432C22—4881no—
433C22—4881no—
434C22—4881no—
435C22#1683998712yes—
436C22C2×#1451975011yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—81no—
441C4—81no—
442C4—4881no—
443C4—2442no—
444C4—2442no—
445C4#1683998712yes—
446C4C2×#1451979811yescharacteristic
447C4C2×#1451980911yescharacteristic
448C4#1684057111yescharacteristic
449C4#1684057211yescharacteristic
450C2—22no—
451C2—9761no—
452C2C2×#1683998711yescharacteristic
453C2#2145548011yescharacteristic
454C2#2145553811yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1550C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#1550D61
hypocenterC61#19216
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1550

Complements

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

NH ≅ G/Ntype
1. #21457161C2semidirect
4. #21454749C2semidirect
5. #21445404C2semidirect
6. #21445390C2semidirect
7. #21445252C2semidirect
8. #21445238C2semidirect
9. C2×#16839512C2semidirect
11. C61×#1550C2semidirect
232. #1550D61semidirect
276. C61#19216semidirect

Maximal subgroups

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

subgroupindex
#214571612
#214571362
#214561442
#214547492
#214454042
#214453902
#214452522
#214452382
C2×#168395122
#214444972
C61×#15502
#1921661