Finite Group Databasebeta

← #30600486

Subgroups — #30600486

Order 15616.

566 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 94 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#21462332C211yesmaximal, characteristic
2#21461251C211yesmaximal, characteristic, complemented
3#21461203C211yesmaximal, characteristic, complemented
4#21461119C211yesmaximal, characteristic
5#21460715C211yesmaximal, characteristic, complemented
6#21460674C211yesmaximal, characteristic, complemented
7#21455611C211yesmaximal, characteristic
8C2×D4×#6923—21no—
9#16840781C2211yescharacteristic
10#16840780C2211yescharacteristic
11#16840713—21no—
12#16840709—22no—
13#16840706C2211yescharacteristic
14#16840704—21no—
15C4×C4⋊#6923C2211yescharacteristic
16C2×(C2×#6923)⋊C4—21no—
17#16840616—21no—
18#16840616—21no—
19#16840615—22no—
20#16840610—21no—
21D4⋊#92420—21no—
22#16840317C2211yescharacteristic
23C4×D4×D61C2211yescharacteristic
24#16840065—21no—
25#16840064—21no—
26#16840053C411yescharacteristic, complemented
27#16840052C411yescharacteristic, complemented
28#16840039C411yescharacteristic, complemented
29#16839803—21no—
30#16839784C2211yescharacteristic
31#16839765—21no—
32C61×#474C411yescharacteristic, complemented
33C23×#6923—22no—
34D4×#6923—42no—
35D4×#6923—42no—
36C2×C4⋊#6923—21no—
37C2×C4⋊#6923—21no—
38C2×C4⋊#6923—22no—
39C2×C22⋊#6923—22no—
40C2×C4×#6923—21no—
41#14519844—21no—
42C4⋊#92420—21no—
43#14519834—21no—
44#14519834—21no—
45C4×#92420—21no—
46C4×#92420—21no—
47(C2×#6923)⋊C4—22no—
48(C2×#6923)⋊C4—21no—
49(C2×#6923)⋊C4D411yescharacteristic
50(C2×#6923)⋊C4D412yes—
51(C2×#6923)⋊C4D411yescharacteristic
52C2×D4×D61D411yescharacteristic
53C22×C4×D61—21no—
54C2×#11758189—21no—
55#14519777C2×C411yescharacteristic, complemented
56#14519768—21no—
57#14519757—21no—
58#14519753—21no—
59Q8×Dic61—21no—
60#14519742—21no—
61C4×D244C2×C411yescharacteristic
62D4×Dic61D411yescharacteristic
63#14519739D411yescharacteristic
64C4×C22⋊D61—21no—
65#14519737—21no—
66#14519735—21no—
67C4⋊C4×D61D411yescharacteristic
68C22⋊C4×D61—21no—
69C42×D61C2311yescharacteristic
70#14519710—21no—
71#14519709—21no—
72#14519708—21no—
73C4⋊#92418C2×C411yescharacteristic, complemented
74#14519706—21no—
75#14519699—21no—
76#14519698—21no—
77#14519697—21no—
78C4⋊Q8×C61C2×C411yescharacteristic, complemented
79C4×D4×C61C2×C411yescharacteristic
80#107×C61—21no—
81C4⋊C8×C61C2×C411yescharacteristic, complemented
82#102×C61—21no—
83C23×D61—21no—
84C22×#6923—21no—
85C22×#6923—22no—
86C22×#6923—21no—
87C22×#6923—44no—
88C22×#6923—44no—
89C22×#6923—82no—
90C2×Q8×C61—21no—
91C2×D4×C61C22⋊C411yescharacteristic
92C22×C4×C61—21no—
93C2×C22⋊D61—21no—
94C22×Dic61—21no—
95D4×D61—21no—
96D4×D61—21no—
97D4×D61SD1612yes—
98D4×D61Q1612yes—
99C2×D244C22⋊C411yescharacteristic
100C2×C4×D61—21no—
101C2×C4×D61—21no—
102C2×C4×D61—21no—
103C2×C4×D61—82no—
104C2×C4×D61C2×D411yescharacteristic
105C2×C4×D61C2×D411yescharacteristic
106C2×C4×D61C2×D411yescharacteristic
107C2×Dic122—21no—
108C22⋊#6923—44no—
109C2×#92420—41no—
110C2×#92420—41no—
111C4⋊#6923—42no—
112C4⋊#6923—41no—
113C4⋊#6923—41no—
114C4⋊#6923—82no—
115C4×#6923—41no—
116C4×#6923—41no—
117C2×C8×C61—21no—
118C4⋊C4×C61—21no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61C22⋊C411yescharacteristic
121C22⋊C4×C61—21no—
122C42×C61C22×C411yescharacteristic
123C22⋊Dic61—21no—
124#11758198—21no—
125#11758198—21no—
126#11758198—41no—
127C4⋊Dic61—21no—
128C4⋊Dic61—21no—
129C4⋊Dic61—41no—
130C4⋊Dic61C22⋊C411yescharacteristic
131#11758196—21no—
132#11758196—41no—
133C4×Dic61—41no—
134C4×Dic61C2×D411yescharacteristic
135C4×Dic61#4111yescharacteristic
136C4×Dic61#4111yescharacteristic
137C2×#92418—21no—
138#11758189—41no—
139#11758189—41no—
140C23×C61—21no—
141C22×D61—21no—
142C22×D61—22no—
143C22×D61—22no—
144C22×D61—42no—
145C22×D61—42no—
146C22×D61#12011yescharacteristic
147C2×#6923—22no—
148C2×#6923—41no—
149C2×#6923—41no—
150C2×#6923—41no—
151C2×#6923—41no—
152C2×#6923—42no—
153C2×#6923—42no—
154C2×#6923—41no—
155C2×#6923—42no—
156C2×#6923—88no—
157Q8×C61—81no—
158D4×C61—21no—
159D4×C61#10312yes—
160C2×C4×C61—21no—
161C2×C4×C61—41no—
162C2×C4×C61—81no—
163C2×C4×C61C2×C22⋊C411yescharacteristic
164C2×C4×C61C4×D411yescharacteristic
165C2×C4×C61C4×D411yescharacteristic
166C22⋊D61—22no—
167C22⋊D61—22no—
168C2×Dic61—21no—
169C2×Dic61—41no—
170C2×Dic61—81no—
171C2×Dic61#12111yescharacteristic
172C2×Dic61#12111yescharacteristic
173C2×Dic61#12311yescharacteristic
174D244—21no—
175D244#10312yes—
176C4×D61—41no—
177C4×D61—41no—
178C4×D61—42no—
179C4×D61—81no—
180C4×D61—81no—
181C4×D61C2×SD1611yescharacteristic
182C4×D61C2×Q1611yescharacteristic
183C4×D61AGL(1,ℤ/8)11yescharacteristic
184C4×D61#13711yescharacteristic
185Dic122—81no—
186#92420—81no—
187#92420—81no—
188C8×C61—41no—
189#92418—41no—
190#9175—611nomaximal, Sylow
191C22×C61—22no—
192C22×C61—41no—
193C22×C61—41no—
194C22×C61#38611yescharacteristic
195C2×D61—21no—
196C2×D61—21no—
197C2×D61—22no—
198C2×D61—44no—
199C2×D61—44no—
200C2×D61—41no—
201C2×D61—41no—
202C2×D61#45011yescharacteristic
203C2×D61#45111yescharacteristic
204#6923—42no—
205#6923—81no—
206#6923—82no—
207C4×C61—21no—
208C4×C61—21no—
209C4×C61—41no—
210C4×C61—81no—
211C4×C61C2×#10311yescharacteristic
212C4×C61#41711yescharacteristic
213Dic61—21no—
214Dic61—21no—
215Dic61—41no—
216Dic61—81no—
217Dic61#48311yescharacteristic
218Dic61#48411yescharacteristic
219#3077—611no—
220#2349—611no—
221#1518—611no—
222#1503—611no—
223#1478—611no—
224#1303—611no—
225#1286—611no—
226C2×C61—42no—
227C2×C61#187211yescharacteristic
228C2×C61#190411yescharacteristic
229C2×C61#190611yescharacteristic
230D61—21no—
231D61—21no—
232D61—42no—
233#533—611no—
234C2×C4×D4—611no—
235C2×C4×D4—1221no—
236#478—1221no—
237#478—1221no—
238#476—611no—
239#476—611no—
240#474—611no—
241#474#692311yescharacteristic, complemented
242#431—611no—
243#431—611no—
244#428—1221no—
245#423—611no—
246#418—1221no—
247#389—1221no—
248#385—1222no—
249#382—611no—
250#380—1221no—
251C4×C4⋊C4—611no—
252C2×#95—1221no—
253#328—1222no—
254#328—1221no—
255#328—1221no—
256D4⋊C8—1221no—
257D4⋊C8—1221no—
258C61#917511yescharacteristic, complemented, Sylow
259C22×D4—611no—
260C23×C4—1221no—
261C23×C4—1222no—
262C2×M16—1221no—
263C4⋊Q8—611no—
264C4⋊Q8C2×#692311yescharacteristic, complemented
265#125—1221no—
266C22⋊Q8—1221no—
267C22⋊Q8—1221no—
268C4×Q8—1221no—
269C4×Q8—1221no—
270C4×D4—611no—
271C4×D4—611no—
272C4×D4—611no—
273C4×D4—1221no—
274C4×D4—1221no—
275C4×D4—2442no—
276C4×D4—2442no—
277C4×D4C2×#692311yescharacteristic
278#117—1221no—
279C2×C4⋊C4—611no—
280C2×C4⋊C4—1222no—
281C2×C4⋊C4—1221no—
282C2×C4⋊C4—1221no—
283C2×C22⋊C4—1221no—
284C2×C22⋊C4—1222no—
285C2×C42—611no—
286C2×C42—1221no—
287#107—21no—
288#107—1221no—
289#106—1221no—
290#106—1221no—
291C4⋊C8—611no—
292C4⋊C8—1221no—
293C4⋊C8—1221no—
294C4⋊C8—1221no—
295C4⋊C8C2×#692311yescharacteristic, complemented
296#102—21no—
297#102—1221no—
298#102—1221no—
299#102—1221no—
300C22⋊C8—1221no—
301C22⋊C8—1221no—
302C4×C8—1221no—
303C4×C8—1221no—
304#95—611no—
305#95—611no—
306#95—612no—
307#95—1221no—
308#95—1222no—
309C24—1221no—
310C2×Q8—21no—
311C2×Q8—1221no—
312C2×D4—612no—
313C2×D4—612no—
314C2×D4—611no—
315C2×D4—1221no—
316C2×D4—1221no—
317C2×D4—1221no—
318C2×D4C22⋊#692311yescharacteristic
319C22×C4—21no—
320C22×C4—611no—
321C22×C4—611no—
322C22×C4—611no—
323C22×C4—1221no—
324C22×C4—1221no—
325C22×C4—1221no—
326C22×C4—1221no—
327C22×C4—1221no—
328C22×C4—1222no—
329C22×C4—1221no—
330C22×C4—4882no—
331C22×C4—4882no—
332C22×C4—2444no—
333C22×C4—2444no—
334M16—2441no—
335M16—2441no—
336C2×C8—21no—
337C2×C8—1221no—
338C2×C8—2441no—
339C2×C8—2441no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—611no—
343C4⋊C4—1221no—
344C4⋊C4—1221no—
345C4⋊C4—1221no—
346C4⋊C4—4882no—
347C4⋊C4—2441no—
348C4⋊C4—2441no—
349C4⋊C4—2441no—
350C4⋊C4—2441no—
351C4⋊C4—2442no—
352C4⋊C4C22⋊#692311yescharacteristic
353C22⋊C4—21no—
354C22⋊C4—1221no—
355C22⋊C4—1221no—
356C22⋊C4—1221no—
357C22⋊C4—2441no—
358C22⋊C4—2444no—
359C42—611no—
360C42—611no—
361C42—611no—
362C42—2441no—
363C42—2441no—
364C42—2441no—
365C42C22×#692311yescharacteristic
366C23—21no—
367C23—611no—
368C23—1222no—
369C23—1222no—
370C23—1221no—
371C23—2442no—
372C23—2442no—
373Q8—81no—
374Q8—4881no—
375D4—21no—
376D4—612no—
377D4—1222no—
378D4—1222no—
379D4—1221no—
380D4#1451984012yes—
381C2×C4—21no—
382C2×C4—41no—
383C2×C4—81no—
384C2×C4—611no—
385C2×C4—611no—
386C2×C4—611no—
387C2×C4—611no—
388C2×C4—611no—
389C2×C4—611no—
390C2×C4—611no—
391C2×C4—1222no—
392C2×C4—1221no—
393C2×C4—4881no—
394C2×C4—4881no—
395C2×C4—4881no—
396C2×C4—4888no—
397C2×C4—2441no—
398C2×C4—2441no—
399C2×C4—2442no—
400C2×C4—2441no—
401C2×C4—2442no—
402C2×C4—2441no—
403C2×C4—2442no—
404C2×C4—2441no—
405C2×C4—2442no—
406C2×C4—2441no—
407C2×C4—2441no—
408C2×C4—2441no—
409C2×C4C2×C22⋊#692311yescharacteristic
410C2×C4D4×#692311yescharacteristic
411C2×C4D4×#692311yescharacteristic
412C8—41no—
413C8—4881no—
414C8—4881no—
415C8—2441no—
416C22—22no—
417C22—41no—
418C22—41no—
419C22—611no—
420C22—611no—
421C22—1221no—
422C22—1221no—
423C22—1222no—
424C22—2444no—
425C22—2444no—
426C22—2441no—
427C22—2441no—
428C22#1684071011yescharacteristic
429C4—21no—
430C4—21no—
431C4—41no—
432C4—81no—
433C4—611no—
434C4—611no—
435C4—1221no—
436C4—1221no—
437C4—4882no—
438C4—4881no—
439C4—4881no—
440C4—2442no—
441C4—2441no—
442C4C2×#1451984011yescharacteristic
443C4#1684075311yescharacteristic
444C2—42no—
445C2—1221no—
446C2—1221no—
447C2—2442no—
448C2#2146180411yescharacteristic
449C2#2146185311yescharacteristic
450C2#2146185711yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#474C4
socleC22×C61#386
prime radicalC2×D4×D61D4
cosocleC42×D61C23
hypercenter#474#6923
hypocenterC61#9175
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #21461251C2semidirect
3. #21461203C2semidirect
5. #21460715C2semidirect
6. #21460674C2semidirect
26. #16840053C4semidirect
27. #16840052C4semidirect
28. #16840039C4semidirect
32. C61×#474C4semidirect
55. #14519777C2×C4semidirect
73. C4⋊#92418C2×C4semidirect
78. C4⋊Q8×C61C2×C4semidirect
81. C4⋊C8×C61C2×C4semidirect
241. #474#6923semidirect
258. C61#9175semidirect
264. C4⋊Q8C2×#6923semidirect
295. C4⋊C8C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214623322
#214612512
#214612032
#214611192
#214607152
#214606742
#214556112
#917561