Finite Group Databasebeta

← #30600641

Subgroups — #30600641

Order 15616.

478 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 80 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#21462509C211yesmaximal, characteristic, complemented
2#21461725C211yesmaximal, characteristic, complemented
3#21461216C211yesmaximal, characteristic
4#21461096C211yesmaximal, characteristic
5#21460691C211yesmaximal, characteristic, complemented
6#21460661C211yesmaximal, characteristic, complemented
7C2×#16840088C211yesmaximal, characteristic
8#16840787—22no—
9#16840782—21no—
10#16840778—21no—
11#16840769—21no—
12#16840733C2211yescharacteristic
13#16840733C2211yescharacteristic
14D61⋊(C22⋊C8)C2211yescharacteristic
15#16840727—21no—
16C4×C22⋊#6923C2211yescharacteristic
17#16840632—21no—
18#16840632—21no—
19#16840614—22no—
20#16840606—21no—
21#16840606—21no—
22C2×#14519752C2211yescharacteristic
23C2×#14519735C2211yescharacteristic
24#16840088—21no—
25#16840088—21no—
26#16840088—21no—
27#16840088—21no—
28C2×#14519710C411yescharacteristic, complemented
29C2×#14519699C411yescharacteristic, complemented
30C2×#14519697C411yescharacteristic, complemented
31C2×#14519689C2211yescharacteristic
32C2×#106×C61C411yescharacteristic, complemented
33C2×#11758213—21no—
34C2×#11758213—21no—
35C2×D61⋊C8—21no—
36C2×C22⋊#6923—21no—
37C2×C4×#6923—21no—
38#14519844—22no—
39C4⋊#92420—22no—
40#14519834—21no—
41#14519834—21no—
42#14519834D412yes—
43C22⋊#92420—21no—
44C22⋊#92420—21no—
45C22⋊#92420D411yescharacteristic
46C22⋊#92420D411yescharacteristic
47#14519832—22no—
48C4×#92420—22no—
49(C2×#6923)⋊C4—21no—
50C22×D244—21no—
51C22×D244C2×C411yescharacteristic
52C22×C4×D61C2311yescharacteristic
53C22×#92418C2×C411yescharacteristic
54#14519752—21no—
55#14519752—21no—
56#14519752—21no—
57#14519752—21no—
58#14519735—21no—
59#14519735—21no—
60#14519735D411yescharacteristic
61#14519735D411yescharacteristic
62#14519735D411yescharacteristic
63#14519735D411yescharacteristic
64C2×C4⋊Dic61C2×C411yescharacteristic
65C2×#11758198—21no—
66C2×#11758198—21no—
67C2×C4×Dic61—21no—
68#14519710—21no—
69#14519710—21no—
70#14519699—21no—
71#14519699—21no—
72#14519697—21no—
73#14519697—21no—
74#14519689—21no—
75#14519689—21no—
76C22×C8×C61C2×C411yescharacteristic
77C2×C4⋊C4×C61C2×C411yescharacteristic
78C2×C4⋊C4×C61C2×C411yescharacteristic
79#106×C61—21no—
80#106×C61—21no—
81C23×D61—21no—
82C23×D61#4111yescharacteristic
83C22×#6923—41no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61C22×C411yescharacteristic
87C22×Dic61—21no—
88C22×Dic61C2×D411yescharacteristic
89C2×D244—21no—
90C2×D244—21no—
91C2×D244—41no—
92C2×D244—41no—
93C2×D244—41no—
94C2×D244—41no—
95C2×D244—82no—
96C2×C4×D61—41no—
97C2×C4×D61—41no—
98C2×C4×D61C2×D411yescharacteristic
99C2×C4×D61C2×D411yescharacteristic
100C2×C4×D61C2×D411yescharacteristic
101C2×C4×D61#4111yescharacteristic
102C22⋊#6923—22no—
103C22⋊#6923—41no—
104C2×#92420—21no—
105C2×#92420—22no—
106C2×#92420—21no—
107C2×#92420—41no—
108C4×#6923—41no—
109C4×#6923—41no—
110#11758213—41no—
111#11758213—41no—
112#11758213—41no—
113#11758213—41no—
114D61⋊C8—41no—
115D61⋊C8—41no—
116C2×C8×C61—21no—
117C2×C8×C61—41no—
118C4⋊C4×C61—21no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61—21no—
121C4⋊C4×C61C22⋊C411yescharacteristic
122C4⋊C4×C61C22⋊C411yescharacteristic
123#11758198—21no—
124#11758198—21no—
125#11758198—21no—
126#11758198—21no—
127#11758198—41no—
128#11758198—41no—
129C4⋊Dic61—21no—
130C4⋊Dic61C22⋊C411yescharacteristic
131C4⋊Dic61C22⋊C411yescharacteristic
132C4×Dic61—21no—
133C4×Dic61—21no—
134C4×Dic61—21no—
135C4×Dic61—21no—
136C2×#92418—21no—
137C2×#92418—41no—
138C23×C61C4×D411yescharacteristic
139C22×D61—21no—
140C22×D61—21no—
141C22×D61—41no—
142C22×D61—41no—
143C22×D61—41no—
144C22×D61—41no—
145C22×D61—41no—
146C22×D61—81no—
147C22×D61—81no—
148C22×D61#12111yescharacteristic
149C22×D61#12311yescharacteristic
150C2×#6923—42no—
151C2×#6923—81no—
152C2×#6923—81no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—81no—
161C2×C4×C61C2×C22⋊C411yescharacteristic
162C2×C4×C61C4×D411yescharacteristic
163C2×Dic61—21no—
164C2×Dic61—21no—
165C2×Dic61—41no—
166C2×Dic61—81no—
167C2×Dic61#12011yescharacteristic
168C2×Dic61#12111yescharacteristic
169D244—41no—
170D244—41no—
171D244—81no—
172D244—82no—
173D244—82no—
174C4×D61—21no—
175C4×D61—21no—
176C4×D61—21no—
177C4×D61—21no—
178C4×D61—41no—
179C4×D61—41no—
180#92420—41no—
181#92420—41no—
182#92420—42no—
183#92420—81no—
184C8×C61—41no—
185#92418—41no—
186#9242—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61#38611yescharacteristic
190C22×C61#44311yescharacteristic
191C22×C61#44411yescharacteristic
192C2×D61—21no—
193C2×D61—21no—
194C2×D61—21no—
195C2×D61—21no—
196C2×D61—21no—
197C2×D61—21no—
198C2×D61—81no—
199C2×D61—81no—
200C2×D61—81no—
201C2×D61—81no—
202C2×D61—81no—
203C2×D61—161no—
204C2×D61—161no—
205#6923—82no—
206C4×C61—21no—
207C4×C61—41no—
208C4×C61—41no—
209C4×C61—81no—
210C4×C61#41011yescharacteristic
211C4×C61#41011yescharacteristic
212Dic61—21no—
213Dic61—21no—
214Dic61—41no—
215Dic61—41no—
216C2×#482—611no—
217#2924—611no—
218#1826—611no—
219#1507—611no—
220#1471—611no—
221#1293—611no—
222#1282—611no—
223C2×C61—21no—
224C2×C61—21no—
225C2×C61#187611yescharacteristic
226C2×C61#189911yescharacteristic
227C2×C61#190311yescharacteristic
228D61—41no—
229D61—41no—
230D61—161no—
231C2×#121—611no—
232C2×#117—611no—
233#482—1221no—
234#482—1221no—
235#482—1221no—
236#482—1221no—
237#433—1222no—
238#432—1221no—
239#431—1221no—
240C2×#107—611no—
241C2×#106#692311yescharacteristic, complemented
242#424—1221no—
243C2×#102—611no—
244C2×#102—611no—
245#407—611no—
246#407—611no—
247C2×C22⋊C8—611no—
248C2×C22⋊C8—611no—
249#405—1221no—
250C4×C22⋊C4—611no—
251#339—1221no—
252#339—1221no—
253#327—1222no—
254#323—1221no—
255#323—1221no—
256C61#924211yescharacteristic, complemented, Sylow
257C22×D4—611no—
258C22×D4—1221no—
259C23×C4—611no—
260C2×M16—1221no—
261C2×M16—1221no—
262C22×C8—611no—
263C22×C8—1221no—
264C22×C8C2×#692311yescharacteristic
265#121—1221no—
266#121—1221no—
267#121—1221no—
268#121—1221no—
269#117—611no—
270#117—611no—
271#117—611no—
272#117—611no—
273#117—1221no—
274#117—1221no—
275C2×C4⋊C4—611no—
276C2×C4⋊C4C2×#692311yescharacteristic
277C2×C4⋊C4C2×#692311yescharacteristic
278C2×C22⋊C4—1221no—
279C2×C22⋊C4—1221no—
280C2×C22⋊C4—1221no—
281C2×C42—1221no—
282C2×C42—1221no—
283#107—1221no—
284#107—1221no—
285#106—21no—
286#106—21no—
287C4⋊C8—1222no—
288C4⋊C8—1222no—
289#102—1221no—
290#102—1221no—
291#102—1221no—
292#102—1221no—
293C22⋊C8—611no—
294C22⋊C8—611no—
295C22⋊C8—612no—
296C22⋊C8—1221no—
297C22⋊C8—1221no—
298C22⋊C8—1221no—
299C22⋊C8—1221no—
300C22⋊C8—1221no—
301C22⋊C8—1221no—
302#97—1222no—
303C4×C8—1222no—
304#95—1221no—
305C24—611no—
306C24—1221no—
307C2×D4—1221no—
308C2×D4—1221no—
309C2×D4—4882no—
310C2×D4—2441no—
311C2×D4—2441no—
312C2×D4—2441no—
313C2×D4—2441no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—611no—
317C22×C4—611no—
318C22×C4—611no—
319C22×C4—611no—
320C22×C4—611no—
321C22×C4—1221no—
322C22×C4—2441no—
323C22×C4—2441no—
324C22×C4—2441no—
325C22×C4C22×#692311yescharacteristic
326M16—2441no—
327M16—2441no—
328M16—2441no—
329M16—2441no—
330C2×C8—21no—
331C2×C8—41no—
332C2×C8—1221no—
333C2×C8—1222no—
334C2×C8—1221no—
335C2×C8—1221no—
336C2×C8—2441no—
337C2×C8—2441no—
338C2×C8—2441no—
339C2×C8—2441no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—611no—
344C4⋊C4—611no—
345C4⋊C4—1221no—
346C4⋊C4C22⋊#692311yescharacteristic
347C4⋊C4C22⋊#692311yescharacteristic
348C22⋊C4—1221no—
349C22⋊C4—1221no—
350C22⋊C4—1221no—
351C22⋊C4—1221no—
352C22⋊C4—1222no—
353C22⋊C4—2441no—
354C22⋊C4—2441no—
355C22⋊C4—2441no—
356C42—1221no—
357C42—1221no—
358C42—1221no—
359C42—1221no—
360C42—2441no—
361C42—2441no—
362C23—611no—
363C23—611no—
364C23—1221no—
365C23—1221no—
366C23—4881no—
367C23—4881no—
368C23—2441no—
369C23—2441no—
370C23—2441no—
371C23—2441no—
372C23—2441no—
373C23D4×#692311yescharacteristic
374D4—4881no—
375D4—4882no—
376D4—4882no—
377D4—2441no—
378D4—2441no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—81no—
387C2×C4—611no—
388C2×C4—611no—
389C2×C4—1221no—
390C2×C4—1221no—
391C2×C4—1221no—
392C2×C4—1221no—
393C2×C4—1221no—
394C2×C4—1221no—
395C2×C4—4881no—
396C2×C4—4881no—
397C2×C4—4881no—
398C2×C4—2441no—
399C2×C4—2441no—
400C2×C4—2441no—
401C2×C4—2442no—
402C2×C4C2×C22⋊#692311yescharacteristic
403C2×C4D4×#692311yescharacteristic
404C8—41no—
405C8—4881no—
406C8—2441no—
407C8—2441no—
408C8—2442no—
409C8—2441no—
410C22—21no—
411C22—21no—
412C22—1221no—
413C22—1221no—
414C22—1221no—
415C22—1221no—
416C22—1221no—
417C22—1221no—
418C22—9761no—
419C22—9761no—
420C22—4881no—
421C22—4881no—
422C22—4881no—
423C22—4881no—
424C22—4881no—
425C22#1684071011yescharacteristic
426C22#1684080511yescharacteristic
427C22#1684080711yescharacteristic
428C4—21no—
429C4—41no—
430C4—41no—
431C4—81no—
432C4—1221no—
433C4—1221no—
434C4—4882no—
435C4—2441no—
436C4—2441no—
437C4#1684074111yescharacteristic
438C4#1684074311yescharacteristic
439C2—21no—
440C2—21no—
441C2—9761no—
442C2—2441no—
443C2—2441no—
444C2#2146180811yescharacteristic
445C2#2146184311yescharacteristic
446C2#2146185111yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#106×C61C4
socleC22×C61#386
prime radicalC2×#14519752C22
cosocleC22×C4×D61C23
hypercenterC2×#106#6923
hypocenterC61#9242
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
1. #21462509C2semidirect
2. #21461725C2semidirect
5. #21460691C2semidirect
6. #21460661C2semidirect
28. C2×#14519710C4semidirect
29. C2×#14519699C4semidirect
30. C2×#14519697C4semidirect
32. C2×#106×C61C4semidirect
241. C2×#106#6923semidirect
256. C61#9242semidirect

Maximal subgroups

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

subgroupindex
#214625092
#214617252
#214612162
#214610962
#214606912
#214606612
C2×#168400882
#924261