Finite Group Databasebeta

← #30600085

Subgroups — #30600085

Order 15616.

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

normal subgroups only

This group has 468 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#21462686C211yesmaximal, characteristic
2#21461725C211yesmaximal, characteristic, complemented
3#21461713C211yesmaximal, characteristic, complemented
4#21461195C211yesmaximal, characteristic
5#21460702C211yesmaximal, characteristic, complemented
6#21460660C211yesmaximal, characteristic, complemented
7#21455727C211yesmaximal, characteristic
8#16840795—21no—
9#16840793—21no—
10#16840791—21no—
11D4×#92420—21no—
12#16840787—21no—
13#16840735—21no—
14#16840733C2211yescharacteristic
15C2×#14519834C2211yescharacteristic
16C2×#14519834C2211yescharacteristic
17C4×C22⋊#6923C2211yescharacteristic
18#16840632—21no—
19#16840632—21no—
20#16840628—21no—
21#16840628—21no—
22#16840615—21no—
23#16840613—21no—
24#16840272C2211yescharacteristic
25C2×#14519735C2211yescharacteristic
26#16840105—21no—
27#16840098C411yescharacteristic, complemented
28#16840092C411yescharacteristic, complemented
29#16840084—21no—
30#16840078C411yescharacteristic, complemented
31#16839799—21no—
32#16839755—21no—
33C22⋊C8×D61C2211yescharacteristic
34C61×#480C411yescharacteristic, complemented
35C2×#11758213D411yescharacteristic
36C2×#11758213D411yescharacteristic
37C2×#11758217—21no—
38C22×#92420—21no—
39C22×#92420—21no—
40C22×#92420D411yescharacteristic
41C22×#92420D411yescharacteristic
42C2×C22⋊#6923—21no—
43C2×C4×#6923—21no—
44#14519844—21no—
45C4⋊#92420—21no—
46#14519834—21no—
47#14519834—21no—
48#14519834—21no—
49#14519834—21no—
50#14519834—21no—
51C22⋊#92420—21no—
52C22⋊#92420—21no—
53C22⋊#92420—21no—
54#14519832—21no—
55C4×#92420—21no—
56(C2×#6923)⋊C4—21no—
57C2×#11758224D411yescharacteristic
58C22×C4×D61C2311yescharacteristic
59C2×C8×D61—21no—
60#14519759—21no—
61#14519757D411yescharacteristic
62#14519756C2×C411yescharacteristic
63#14519748D411yescharacteristic
64#14519745—21no—
65D4×Dic61—21no—
66#14519737—21no—
67#14519735—21no—
68#14519735—21no—
69#14519735—21no—
70#14519735—21no—
71#14519735D411yescharacteristic
72C2×C4⋊Dic61C2×C411yescharacteristic
73C2×#11758198—21no—
74C2×C4×Dic61—21no—
75#14519711—21no—
76#14519708—21no—
77#14519708—21no—
78#14519702—21no—
79#14519700—21no—
80#14519698—21no—
81C22⋊#92418C2×C411yescharacteristic
82#14519689—21no—
83#121×C61C2×C411yescharacteristic
84C2×C4⋊C4×C61C2×C411yescharacteristic
85#107×C61—21no—
86#102×C61—21no—
87C22⋊C8×C61C2×C411yescharacteristic
88C23×D61#4111yescharacteristic
89C22×#6923—41no—
90C2×D4×C61—21no—
91C2×D4×C61C22⋊C411yescharacteristic
92C22×C4×C61—21no—
93C22×C4×C61C22×C411yescharacteristic
94C2×C22⋊D61—21no—
95C2×C22⋊D61—21no—
96C22×Dic61—21no—
97C22×Dic61—21no—
98C22×Dic61C2×D411yescharacteristic
99#11758224—41no—
100#11758224—41no—
101C2×C4×D61—41no—
102C2×C4×D61—41no—
103C2×C4×D61C2×D411yescharacteristic
104C2×C4×D61C2×D411yescharacteristic
105C2×C4×D61C2×D411yescharacteristic
106C2×C4×D61#4111yescharacteristic
107C2×Dic122C22⋊C411yescharacteristic
108C22⋊#6923—22no—
109C22⋊#6923—41no—
110#11758217—81no—
111C2×#92420—21no—
112C2×#92420—21no—
113C2×#92420—21no—
114C2×#92420—21no—
115C2×#92420—41no—
116C2×#92420—41no—
117C2×#92420—41no—
118C2×#92420—41no—
119C2×#92420—41no—
120C2×#92420—81no—
121C4×#6923—41no—
122C4×#6923—41no—
123#11758213—41no—
124#11758213—41no—
125C2×C8×C61—21no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61C22⋊C411yescharacteristic
129C22⋊C4×C61—21no—
130C22⋊Dic61—21no—
131C22⋊Dic61—21no—
132#11758198—21no—
133#11758198—21no—
134#11758198—41no—
135#11758198—41no—
136C4⋊Dic61—21no—
137C4⋊Dic61—21no—
138C4⋊Dic61—21no—
139C4⋊Dic61C22⋊C411yescharacteristic
140#11758196—21no—
141C4×Dic61—21no—
142C4×Dic61—41no—
143C4×Dic61—41no—
144C2×#92418—21no—
145C8×D61—41no—
146C8×D61—41no—
147C23×C61—21no—
148C23×C61C4×D411yescharacteristic
149C22×D61—21no—
150C22×D61—21no—
151C22×D61—41no—
152C22×D61—41no—
153C22×D61#12111yescharacteristic
154C22×D61#12311yescharacteristic
155C2×#6923—42no—
156C2×#6923—81no—
157C2×#6923—81no—
158D4×C61—41no—
159D4×C61—41no—
160D4×C61—41no—
161C2×C4×C61—21no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61C2×C22⋊C411yescharacteristic
167C2×C4×C61C4×D411yescharacteristic
168C22⋊D61—81no—
169C22⋊D61—81no—
170C2×Dic61—21no—
171C2×Dic61—21no—
172C2×Dic61—21no—
173C2×Dic61—21no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176C2×Dic61—42no—
177C2×Dic61—41no—
178C2×Dic61#12011yescharacteristic
179C2×Dic61#12111yescharacteristic
180C4×D61—21no—
181C4×D61—21no—
182C4×D61—21no—
183C4×D61—21no—
184C4×D61—41no—
185C4×D61—41no—
186Dic122—41no—
187#92420—41no—
188#92420—41no—
189#92420—41no—
190#92420—41no—
191#92420—81no—
192C8×C61—41no—
193#92418—41no—
194#9038—611nomaximal, Sylow
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—41no—
198C22×C61—41no—
199C22×C61—41no—
200C22×C61#38611yescharacteristic
201C22×C61#44311yescharacteristic
202C22×C61#44411yescharacteristic
203C2×D61—21no—
204C2×D61—21no—
205C2×D61—21no—
206C2×D61—21no—
207C2×D61—21no—
208C2×D61—21no—
209C2×D61—81no—
210C2×D61—81no—
211#6923—82no—
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—41no—
215C4×C61—81no—
216Dic61—21no—
217Dic61—41no—
218Dic61—41no—
219Dic61—81no—
220Dic61#45811yescharacteristic
221Dic61#45811yescharacteristic
222#3091—611no—
223#2976—611no—
224#1826—611no—
225#1818—611no—
226#1501—611no—
227#1298—611no—
228#1282—611no—
229C2×C61—21no—
230C2×C61—21no—
231C2×C61—81no—
232C2×C61#187611yescharacteristic
233C2×C61#188111yescharacteristic
234C2×C61#189011yescharacteristic
235D61—41no—
236D61—41no—
237#525—611no—
238C2×#117—611no—
239#485—1221no—
240#484—611no—
241#483—611no—
242#482—1221no—
243#481—611no—
244#480#692311yescharacteristic, complemented
245#436—1221no—
246#435—1221no—
247#435—1221no—
248C8×D4—1221no—
249#433—1221no—
250#427—1221no—
251#416—1221no—
252#408—1221no—
253#407—611no—
254C2×C22⋊C8—611no—
255C2×C22⋊C8—611no—
256C2×C22⋊C8—611no—
257C4×C22⋊C4—611no—
258#339—1221no—
259#339—1221no—
260#336—1221no—
261#336—1221no—
262#328—1221no—
263#327—1221no—
264C61#903811yescharacteristic, complemented, Sylow
265C2×#41—611no—
266C23×C4—611no—
267C2×M16—611no—
268C2×M16—611no—
269C2×M16—1221no—
270C22×C8—611no—
271C22×C8—611no—
272C22×C8—1221no—
273C22×C8—1221no—
274C22×C8—1221no—
275#123—1221no—
276C22⋊Q8—611no—
277C22⋊Q8—611no—
278#121—611no—
279#121C2×#692311yescharacteristic
280#120—1221no—
281C4×D4—1221no—
282C4×D4—1221no—
283#117—611no—
284#117—1221no—
285#117—1221no—
286#117—1221no—
287#117—1221no—
288C2×C4⋊C4—611no—
289C2×C4⋊C4C2×#692311yescharacteristic
290C2×C22⋊C4—1221no—
291C2×C22⋊C4—1221no—
292C2×C42—1221no—
293C2×C42—1221no—
294#107—21no—
295#107—1221no—
296#106—1221no—
297#106—1221no—
298C4⋊C8—1221no—
299C4⋊C8—1221no—
300#103—1221no—
301#103—1221no—
302#102—21no—
303#102—1221no—
304C22⋊C8—611no—
305C22⋊C8—1221no—
306C22⋊C8—1221no—
307C22⋊C8—1221no—
308C22⋊C8—1221no—
309C22⋊C8—1221no—
310C22⋊C8—1221no—
311C22⋊C8—1221no—
312C22⋊C8—1221no—
313C22⋊C8—1221no—
314C22⋊C8C2×#692311yescharacteristic
315#97—1221no—
316C4×C8—1221no—
317#95—1221no—
318C24—611no—
319#41—2441no—
320#41—2441no—
321C2×Q8—611no—
322C2×D4—21no—
323C2×D4—1221no—
324C2×D4—1221no—
325C2×D4C22⋊#692311yescharacteristic
326C22×C4—21no—
327C22×C4—611no—
328C22×C4—611no—
329C22×C4—611no—
330C22×C4—611no—
331C22×C4—611no—
332C22×C4—1221no—
333C22×C4—1221no—
334C22×C4—2441no—
335C22×C4—2441no—
336C22×C4—2441no—
337C22×C4C22×#692311yescharacteristic
338M16—4881no—
339M16—2441no—
340M16—2441no—
341C2×C8—21no—
342C2×C8—1221no—
343C2×C8—1221no—
344C2×C8—1221no—
345C2×C8—1221no—
346C2×C8—1221no—
347C2×C8—4881no—
348C2×C8—2441no—
349C2×C8—2441no—
350C2×C8—2441no—
351C2×C8—2441no—
352C2×C8—2441no—
353C2×C8—2441no—
354C2×C8—2441no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—611no—
358C4⋊C4—1221no—
359C4⋊C4—1221no—
360C4⋊C4—1221no—
361C4⋊C4—1221no—
362C4⋊C4C22⋊#692311yescharacteristic
363C22⋊C4—21no—
364C22⋊C4—1221no—
365C22⋊C4—1221no—
366C22⋊C4—1221no—
367C22⋊C4—1221no—
368C22⋊C4—1222no—
369C22⋊C4—2441no—
370C22⋊C4—2441no—
371C22⋊C4—2441no—
372C42—1221no—
373C42—2441no—
374C42—2441no—
375C42—2441no—
376C42—2441no—
377C23—21no—
378C23—611no—
379C23—611no—
380C23—1221no—
381C23—1221no—
382C23—2441no—
383C23—2441no—
384C23D4×#692311yescharacteristic
385Q8—2441no—
386D4—41no—
387D4—41no—
388D4—41no—
389D4—4881no—
390D4—4881no—
391C2×C4—21no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—611no—
397C2×C4—611no—
398C2×C4—1221no—
399C2×C4—1221no—
400C2×C4—1221no—
401C2×C4—1221no—
402C2×C4—1221no—
403C2×C4—1221no—
404C2×C4—1221no—
405C2×C4—1221no—
406C2×C4—4881no—
407C2×C4—4881no—
408C2×C4—2441no—
409C2×C4—2441no—
410C2×C4—2441no—
411C2×C4—2441no—
412C2×C4—2442no—
413C2×C4—2442no—
414C2×C4—2441no—
415C2×C4C2×C22⋊#692311yescharacteristic
416C2×C4D4×#692311yescharacteristic
417C8—41no—
418C8—4881no—
419C8—2441no—
420C8—2441no—
421C8—2441no—
422C8—2441no—
423C8—2441no—
424C22—21no—
425C22—21no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—1221no—
430C22—1221no—
431C22—1221no—
432C22—1221no—
433C22—1221no—
434C22—1221no—
435C22—4881no—
436C22—4881no—
437C22#1684071011yescharacteristic
438C22#1684080411yescharacteristic
439C22#1684080911yescharacteristic
440C4—21no—
441C4—21no—
442C4—41no—
443C4—81no—
444C4—611no—
445C4—611no—
446C4—1221no—
447C4—4882no—
448C4—4881no—
449C4—2441no—
450C4—2441no—
451C2—21no—
452C2—21no—
453C2—81no—
454C2—2441no—
455C2—2441no—
456C2#2146181011yescharacteristic
457C2#2146181611yescharacteristic
458C2#2146182911yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#480C4
socleC22×C61#386
prime radical#16840272C22
cosocleC22×C4×D61C23
hypercenter#480#6923
hypocenterC61#9038
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. #21461725C2semidirect
3. #21461713C2semidirect
5. #21460702C2semidirect
6. #21460660C2semidirect
27. #16840098C4semidirect
28. #16840092C4semidirect
30. #16840078C4semidirect
34. C61×#480C4semidirect
244. #480#6923semidirect
264. C61#9038semidirect

Maximal subgroups

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

subgroupindex
#214626862
#214617252
#214617132
#214611952
#214607022
#214606602
#214557272
#903861