Finite Group Databasebeta

← #30600900

Subgroups — #30600900

Order 15616.

470 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462512C211yesmaximal, characteristic, complemented
2#21461724C211yesmaximal, characteristic, complemented
3#21461255C211yesmaximal, characteristic
4#21461094C211yesmaximal, characteristic
5#21460719C211yesmaximal, characteristic, complemented
6#21460701C211yesmaximal, characteristic, complemented
7#21455757C211yesmaximal, characteristic
8#16840786—22no—
9#16840784—21no—
10#16840780—21no—
11#16840769—21no—
12#16840733C2211yescharacteristic
13D61⋊(C22⋊C8)C2211yescharacteristic
14D61⋊(C22⋊C8)C2211yescharacteristic
15#16840727—21no—
16#16840704C2211yescharacteristic
17#16840635—21no—
18#16840635—21no—
19#16840613—21no—
20#16840613—21no—
21#16840605—22no—
22C2×#14519748C2211yescharacteristic
23#117×D61C2211yescharacteristic
24#16840085—21no—
25#16840084—21no—
26#16840079—21no—
27#16840074—21no—
28#16839805C411yescharacteristic, complemented
29#16839804C411yescharacteristic, complemented
30#16839767C411yescharacteristic, complemented
31#16839725C2211yescharacteristic
32C61×#418C411yescharacteristic, complemented
33C2×#11758213—21no—
34C2×D61⋊C8—21no—
35C2×D61⋊C8—21no—
36C2×C4⋊#6923—21no—
37C2×C22⋊#6923—21no—
38#14519844—22no—
39C4⋊#92420—22no—
40#14519834—21no—
41#14519834—21no—
42#14519834D411yescharacteristic
43#14519834D411yescharacteristic
44C22⋊#92420—21no—
45C22⋊#92420—21no—
46C22⋊#92420D412yes—
47#14519832—22no—
48C4×#92420—22no—
49(C2×#6923)⋊C4—21no—
50C22×C22⋊D61—21no—
51C22×C4×D61C2311yescharacteristic
52C2×#11758194C2×C411yescharacteristic
53#14519748—21no—
54#14519748—21no—
55#14519748—21no—
56#14519748—21no—
57#14519736C2×C411yescharacteristic
58#14519735D411yescharacteristic
59#14519735D411yescharacteristic
60#14519734—21no—
61#14519733—21no—
62C2×C4⋊Dic61C2×C411yescharacteristic
63C4⋊C4×D61D411yescharacteristic
64C4⋊C4×D61D411yescharacteristic
65C2×C22⋊Dic61—21no—
66C22⋊C4×D61—21no—
67C42×D61—21no—
68#14519711—21no—
69#14519710—21no—
70#14519709—21no—
71#14519708—21no—
72#14519698—21no—
73#14519698—21no—
74#14519689—21no—
75#14519689—21no—
76C22×D4×C61C2×C411yescharacteristic
77C2×M16×C61C2×C411yescharacteristic
78#117×C61C2×C411yescharacteristic
79#102×C61—21no—
80#102×C61—21no—
81C24×C61—21no—
82C23×D61#4111yescharacteristic
83C22×#6923—41no—
84C2×D4×C61—21no—
85C2×D4×C61—21no—
86C2×D4×C61—41no—
87C2×D4×C61—41no—
88C22×C4×C61C22×C411yescharacteristic
89C2×C22⋊D61—41no—
90C2×C22⋊D61—41no—
91C2×C22⋊D61—81no—
92C2×C22⋊D61—81no—
93C22×Dic61—21no—
94C22×Dic61C2×D411yescharacteristic
95C2×C4×D61—21no—
96C2×C4×D61—21no—
97C2×C4×D61—41no—
98C2×C4×D61—41no—
99C2×C4×D61C2×D411yescharacteristic
100C2×C4×D61C2×D411yescharacteristic
101C2×C4×D61C2×D411yescharacteristic
102C2×C4×D61#4111yescharacteristic
103C22⋊#6923—81no—
104C2×#92420—21no—
105C2×#92420—22no—
106C2×#92420—21no—
107C2×#92420—41no—
108C4⋊#6923—41no—
109C4⋊#6923—41no—
110#11758213—41no—
111#11758213—41no—
112D61⋊C8—41no—
113D61⋊C8—41no—
114D61⋊C8—41no—
115D61⋊C8—41no—
116M16×C61—41no—
117C2×C8×C61—21no—
118C4⋊C4×C61C22⋊C411yescharacteristic
119C4⋊C4×C61C22⋊C411yescharacteristic
120C22⋊C4×C61—21no—
121C42×C61—21no—
122C22⋊Dic61—21no—
123C22⋊Dic61—41no—
124C22⋊Dic61—41no—
125#11758198—21no—
126#11758198—21no—
127C4⋊Dic61—21no—
128C4⋊Dic61—21no—
129C4⋊Dic61C22⋊C411yescharacteristic
130C4⋊Dic61C22⋊C411yescharacteristic
131#11758196—21no—
132#11758196—21no—
133C4×Dic61—21no—
134C4×Dic61—21no—
135C4×Dic61—21no—
136#11758194—41no—
137C2×#92418—21no—
138C23×C61—42no—
139C23×C61—41no—
140C23×C61—81no—
141C23×C61—81no—
142C23×C61C4×D411yescharacteristic
143C22×D61—21no—
144C22×D61—21no—
145C22×D61—41no—
146C22×D61—41no—
147C22×D61#12111yescharacteristic
148C22×D61#12311yescharacteristic
149C2×#6923—81no—
150C2×#6923—81no—
151C2×#6923—81no—
152D4×C61—41no—
153D4×C61—41no—
154D4×C61—81no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61C2×C22⋊C411yescharacteristic
160C2×C4×C61C4×D411yescharacteristic
161C22⋊D61—82no—
162C22⋊D61—81no—
163C22⋊D61—81no—
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—81no—
169C2×Dic61#12011yescharacteristic
170C2×Dic61#12111yescharacteristic
171C4×D61—21no—
172C4×D61—21no—
173C4×D61—21no—
174C4×D61—21no—
175C4×D61—41no—
176C4×D61—41no—
177C4×D61—41no—
178C4×D61—41no—
179C4×D61—81no—
180#92420—41no—
181#92420—41no—
182#92420—42no—
183#92420—81no—
184C8×C61—41no—
185#92418—41no—
186#9348—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—81no—
191C22×C61—81no—
192C22×C61—81no—
193C22×C61—162no—
194C22×C61#38611yescharacteristic
195C2×D61—21no—
196C2×D61—21no—
197C2×D61—21no—
198C2×D61—21no—
199C2×D61—21no—
200C2×D61—81no—
201C2×D61—81no—
202C2×D61#47111yescharacteristic
203C2×D61#47211yescharacteristic
204#6923—161no—
205C4×C61—21no—
206C4×C61—41no—
207C4×C61—41no—
208C4×C61#41011yescharacteristic
209C4×C61#41011yescharacteristic
210Dic61—21no—
211Dic61—21no—
212Dic61—41no—
213Dic61—41no—
214Dic61—81no—
215#3094—611no—
216#2924—611no—
217#1825—611no—
218#1519—611no—
219#1471—611no—
220#1305—611no—
221#1298—611no—
222C2×C61—41no—
223C2×C61—161no—
224C2×C61#188111yescharacteristic
225C2×C61#189911yescharacteristic
226C2×C61#190711yescharacteristic
227D61—21no—
228D61—21no—
229D61—41no—
230C2×#121—611no—
231C2×#117—611no—
232#482—1221no—
233#482—1221no—
234#481—1221no—
235#480—1221no—
236#433—1222no—
237#432—1221no—
238#431—1221no—
239#428—611no—
240#428—611no—
241#424—1221no—
242#418—611no—
243#418#692311yescharacteristic, complemented
244#407—611no—
245#407—611no—
246C2×C22⋊C8—611no—
247C2×C22⋊C8—611no—
248#405—1221no—
249#380—611no—
250#340—1221no—
251#340—1221no—
252#327—1221no—
253#327—1221no—
254#323—1222no—
255C61#934811yescharacteristic, complemented, Sylow
256C22×D4—1221no—
257C22×D4C2×#692311yescharacteristic
258C23×C4—611no—
259C2×M16—611no—
260C2×M16—1221no—
261C2×M16C2×#692311yescharacteristic
262C22×C8—1221no—
263C22×C8—1221no—
264#121—1221no—
265#121—1221no—
266#121—1221no—
267#121—1221no—
268#117—611no—
269#117—611no—
270#117—611no—
271#117—1221no—
272#117—1221no—
273#117C2×#692311yescharacteristic
274C2×C4⋊C4—611no—
275C2×C4⋊C4—611no—
276C2×C4⋊C4—611no—
277C2×C4⋊C4—1221no—
278C2×C22⋊C4—1221no—
279C2×C22⋊C4—1221no—
280C2×C22⋊C4—1221no—
281C2×C42—1221no—
282#107—1221no—
283#107—1221no—
284#106—1221no—
285#106—1221no—
286C4⋊C8—1222no—
287C4⋊C8—1222no—
288#102—21no—
289#102—21no—
290#102—1221no—
291#102—1221no—
292C22⋊C8—612no—
293C22⋊C8—611no—
294C22⋊C8—611no—
295C22⋊C8—1221no—
296C22⋊C8—1221no—
297C22⋊C8—1221no—
298C22⋊C8—1221no—
299C22⋊C8—1221no—
300C22⋊C8—1221no—
301#97—1222no—
302C4×C8—1222no—
303#95—1221no—
304C24—21no—
305C24—611no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—4881no—
311C2×D4—4881no—
312C2×D4—2441no—
313C2×D4—2441no—
314C22×C4—611no—
315C22×C4—611no—
316C22×C4—611no—
317C22×C4—611no—
318C22×C4—611no—
319C22×C4—1221no—
320C22×C4—1221no—
321C22×C4—1221no—
322C22×C4—2441no—
323C22×C4—2441no—
324C22×C4—2441no—
325C22×C4C22×#692311yescharacteristic
326M16—41no—
327M16—2441no—
328M16—2441no—
329M16—2441no—
330C2×C8—21no—
331C2×C8—1221no—
332C2×C8—1222no—
333C2×C8—1221no—
334C2×C8—1221no—
335C2×C8—2441no—
336C2×C8—2441no—
337C2×C8—2441no—
338C2×C8—2441no—
339C2×C8—2441no—
340C4⋊C4—611no—
341C4⋊C4—611no—
342C4⋊C4—1221no—
343C4⋊C4—1221no—
344C4⋊C4—1221no—
345C4⋊C4—1221no—
346C4⋊C4—2441no—
347C4⋊C4—2441no—
348C4⋊C4C22⋊#692311yescharacteristic
349C4⋊C4C22⋊#692311yescharacteristic
350C22⋊C4—21no—
351C22⋊C4—1221no—
352C22⋊C4—1221no—
353C22⋊C4—1221no—
354C22⋊C4—4881no—
355C22⋊C4—2441no—
356C22⋊C4—2441no—
357C42—21no—
358C42—1221no—
359C42—1221no—
360C42—1221no—
361C23—42no—
362C23—41no—
363C23—81no—
364C23—81no—
365C23—611no—
366C23—611no—
367C23—1221no—
368C23—1221no—
369C23—2441no—
370C23—2441no—
371C23D4×#692311yescharacteristic
372D4—41no—
373D4—41no—
374D4—81no—
375D4—4881no—
376D4—4881no—
377D4—4882no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—611no—
383C2×C4—611no—
384C2×C4—1221no—
385C2×C4—1221no—
386C2×C4—1221no—
387C2×C4—1221no—
388C2×C4—1221no—
389C2×C4—1221no—
390C2×C4—4881no—
391C2×C4—4881no—
392C2×C4—4881no—
393C2×C4—4881no—
394C2×C4—4881no—
395C2×C4—2441no—
396C2×C4—2441no—
397C2×C4—2441no—
398C2×C4—2441no—
399C2×C4—2441no—
400C2×C4—2441no—
401C2×C4C2×C22⋊#692311yescharacteristic
402C2×C4D4×#692311yescharacteristic
403C8—41no—
404C8—4881no—
405C8—2441no—
406C8—2441no—
407C8—2442no—
408C8—2441no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—162no—
416C22—611no—
417C22—611no—
418C22—1221no—
419C22—1221no—
420C22—1221no—
421C22—1221no—
422C22—1221no—
423C22—4881no—
424C22—4881no—
425C22#1684071011yescharacteristic
426C4—21no—
427C4—41no—
428C4—41no—
429C4—1221no—
430C4—1221no—
431C4—9761no—
432C4—4881no—
433C4—2441no—
434C4—2441no—
435C4#1684074111yescharacteristic
436C4#1684074211yescharacteristic
437C2—41no—
438C2—161no—
439C2—1221no—
440C2—1221no—
441C2—2441no—
442C2#2146181711yescharacteristic
443C2#2146184211yescharacteristic
444C2#2146185811yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#418C4
socleC22×C61#386
prime radicalC2×#14519748C22
cosocleC22×C4×D61C23
hypercenter#418#6923
hypocenterC61#9348
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #21462512C2semidirect
2. #21461724C2semidirect
5. #21460719C2semidirect
6. #21460701C2semidirect
28. #16839805C4semidirect
29. #16839804C4semidirect
30. #16839767C4semidirect
32. C61×#418C4semidirect
243. #418#6923semidirect
255. C61#9348semidirect

Maximal subgroups

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

subgroupindex
#214625122
#214617242
#214612552
#214610942
#214607192
#214607012
#214557572
#934861