Finite Group Databasebeta

← #30600131

Subgroups — #30600131

Order 15616.

550 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 106 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#21462348C211yesmaximal, characteristic
2#21461998C211yesmaximal, characteristic
3#21461985C211yesmaximal, characteristic
4D61⋊#332C211yesmaximal, characteristic
5#21460654C211yesmaximal, characteristic
6#21460654C211yesmaximal, characteristic
7#498×D61C211yesmaximal, characteristic
8C2×Q8×#6923—21no—
9D61⋊(C4⋊C8)C2211yescharacteristic
10D61⋊(C4⋊C8)C2211yescharacteristic
11C2×#14519840—21no—
12C2×#14519840—21no—
13(C2×#6923)⋊Q8—21no—
14#16840713—21no—
15#16840711—22no—
16C4⋊C4×#6923—21no—
17C4×C4⋊#6923C2211yescharacteristic
18C4×C4⋊#6923C2211yescharacteristic
19#16840635—21no—
20#16840635—21no—
21#16840621—22no—
22C4⋊Q8×D61C2211yescharacteristic
23C4⋊Q8×D61C2211yescharacteristic
24#16840160C411yescharacteristic, complemented
25#16840159—22no—
26#16840158C411yescharacteristic, complemented
27#16840158C411yescharacteristic, complemented
28#106×D61—22no—
29C4×C8×D61C2211yescharacteristic
30C61×#498C411yescharacteristic, complemented
31C2×D61⋊C8—21no—
32C2×D61⋊C8—21no—
33Q8×#6923—42no—
34Q8×#6923—42no—
35C2×C4⋊#6923—22no—
36C2×C4⋊#6923D411yescharacteristic
37C2×C4⋊#6923D411yescharacteristic
38C2×C4⋊#6923D411yescharacteristic
39C2×C4⋊#6923D411yescharacteristic
40C2×C4×#6923—22no—
41C2×C4×#6923—21no—
42C2×C4×#6923—21no—
43#14519844—21no—
44#14519844—21no—
45C4⋊#92420—21no—
46C4⋊#92420—21no—
47#14519840—42no—
48#14519840—42no—
49(C2×#6923)⋊C4—22no—
50(C2×#6923)⋊C4—22no—
51(C2×#6923)⋊C4—21no—
52(C2×#6923)⋊C4—21no—
53C2×Q8×D61—21no—
54C2×Q8×D61—21no—
55C2×C8×D61—21no—
56#14519777C2×C411yescharacteristic
57#14519777C2×C411yescharacteristic
58#14519776—21no—
59#14519776—21no—
60#14519776—21no—
61#14519776D412yes—
62#14519775—21no—
63#14519775—21no—
64C4⋊C4×D61—21no—
65C4⋊C4×D61—21no—
66C4⋊C4×D61—21no—
67C4⋊C4×D61D412yes—
68C42×D61C2311yescharacteristic
69#14519709—22no—
70#14519708—22no—
71#14519708—22no—
72C4×#92418C2×C411yescharacteristic
73C8×Dic61—21no—
74C4⋊Q8×C61C2×C411yescharacteristic
75C4⋊Q8×C61C2×C411yescharacteristic
76#106×C61—22no—
77C4×C8×C61C2×C411yescharacteristic
78C22×#6923—21no—
79C22×#6923—21no—
80C22×#6923—21no—
81C22×#6923—21no—
82C2×Q8×C61—21no—
83C2×Q8×C61—21no—
84Q8×D61—41no—
85Q8×D61—41no—
86Q8×D61—41no—
87Q8×D61—41no—
88Q8×D61—41no—
89Q8×D61—41no—
90Q8×D61—41no—
91Q8×D61—41no—
92C2×C4×D61—22no—
93C2×C4×D61—41no—
94C2×C4×D61C2×D411yescharacteristic
95C2×C4×D61C2×D411yescharacteristic
96C2×C4×D61C2×D411yescharacteristic
97C2×Dic122—21no—
98C2×Dic122—22no—
99C2×Dic122—21no—
100C2×Dic122—41no—
101C2×#92420—41no—
102C2×#92420—41no—
103C4⋊#6923—21no—
104C4⋊#6923—21no—
105C4⋊#6923—21no—
106C4⋊#6923—21no—
107C4⋊#6923—21no—
108C4⋊#6923—21no—
109C4⋊#6923—44no—
110C4⋊#6923SD1612yes—
111C4⋊#6923SD1612yes—
112C4×#6923—44no—
113C4×#6923—41no—
114C4×#6923—41no—
115C4×#6923—41no—
116C4×#6923—41no—
117D61⋊C8—41no—
118D61⋊C8—41no—
119D61⋊C8—41no—
120D61⋊C8—41no—
121C2×C8×C61—21no—
122C4⋊C4×C61—21no—
123C4⋊C4×C61—21no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61C22⋊C412yes—
126C42×C61C22×C411yescharacteristic
127C4⋊Dic61—21no—
128C4⋊Dic61—21no—
129C4⋊Dic61—21no—
130C4⋊Dic61C22⋊C412yes—
131#11758196—22no—
132#11758196—22no—
133#11758196—41no—
134#11758196—41no—
135C4×Dic61C2×D411yescharacteristic
136C4×Dic61#4111yescharacteristic
137C4×Dic61#4111yescharacteristic
138C2×#92418—21no—
139C8×D61—22no—
140C8×D61—22no—
141C22×D61#12011yescharacteristic
142C2×#6923—22no—
143C2×#6923—22no—
144C2×#6923—41no—
145C2×#6923—41no—
146C2×#6923—41no—
147C2×#6923—41no—
148C2×#6923—41no—
149C2×#6923—41no—
150C2×#6923—41no—
151C2×#6923—41no—
152C2×#6923—41no—
153C2×#6923—41no—
154Q8×C61—41no—
155Q8×C61—41no—
156Q8×C61—41no—
157Q8×C61—41no—
158C2×C4×C61—22no—
159C2×C4×C61—41no—
160C2×C4×C61C2×C22⋊C411yescharacteristic
161C2×C4×C61C4×D411yescharacteristic
162C2×C4×C61C4×D411yescharacteristic
163C2×Dic61—22no—
164C2×Dic61—41no—
165C2×Dic61#12111yescharacteristic
166C2×Dic61#12111yescharacteristic
167C2×Dic61#12311yescharacteristic
168C4×D61—21no—
169C4×D61—21no—
170C4×D61—42no—
171C4×D61—42no—
172C4×D61—81no—
173C4×D61—81no—
174C4×D61C2×SD1611yescharacteristic
175C4×D61C2×SD1611yescharacteristic
176C4×D61#13511yescharacteristic
177C4×D61#13511yescharacteristic
178C4×D61AGL(1,ℤ/8)11yescharacteristic
179C4×D61AGL(1,ℤ/8)11yescharacteristic
180C4×D61#13711yescharacteristic
181C4×D61#13711yescharacteristic
182Dic122—41no—
183Dic122—41no—
184Dic122—42no—
185Dic122—42no—
186Dic122—41no—
187Dic122—41no—
188Dic122—81no—
189Dic122—81no—
190#92420—81no—
191#92420—81no—
192C8×C61—22no—
193#92418—22no—
194#9059—611nomaximal, Sylow
195C22×C61#38611yescharacteristic
196C2×D61#44811yescharacteristic
197C2×D61#44911yescharacteristic
198C2×D61#45011yescharacteristic
199C2×D61#45211yescharacteristic
200C2×D61#45611yescharacteristic
201C2×D61#45611yescharacteristic
202#6923—42no—
203#6923—42no—
204#6923—81no—
205#6923—81no—
206C4×C61—21no—
207C4×C61—42no—
208C4×C61—81no—
209C4×C61C4×SD1611yescharacteristic
210C4×C61C4×SD1611yescharacteristic
211C4×C61#44011yescharacteristic
212C4×C61#44011yescharacteristic
213Dic61—21no—
214Dic61—42no—
215Dic61—81no—
216Dic61#46011yescharacteristic
217Dic61#46111yescharacteristic
218Dic61#46311yescharacteristic
219Dic61#46411yescharacteristic
220C2×#498—611no—
221#2365—611no—
222#1988—611no—
223#1981—611no—
224C2×#332—611no—
225#1279—611no—
226#1279—611no—
227C2×C61#187711yescharacteristic
228C2×C61#188011yescharacteristic
229C2×C61#188611yescharacteristic
230D61#165111yescharacteristic
231D61#165211yescharacteristic
232D61#165311yescharacteristic
233D61#165411yescharacteristic
234C2×C4⋊Q8—611no—
235C2×C4⋊Q8—611no—
236C2×C4×Q8—1221no—
237#498—611no—
238#498—611no—
239#498—611no—
240#498—1222no—
241#498#692311yescharacteristic, complemented
242C2×#106—1222no—
243C2×C4⋊C8—611no—
244C2×C4⋊C8—611no—
245C2×#103—1221no—
246C2×#103—1221no—
247C2×C4×C8—611no—
248#391—1221no—
249#389—1221no—
250#387—1222no—
251C4×C4⋊C4—611no—
252C4×C4⋊C4—611no—
253C4×C4⋊C4—1221no—
254#340—1221no—
255#340—1221no—
256#332—1222no—
257C61#905911yescharacteristic, complemented, Sylow
258C22×Q8—1221no—
259C22×Q8—1221no—
260C22×C8—1221no—
261C22×C8—1221no—
262C22×C8—1221no—
263C4⋊Q8—612no—
264C4⋊Q8—611no—
265C4⋊Q8—611no—
266C4⋊Q8—1221no—
267C4⋊Q8—1221no—
268C4⋊Q8—1221no—
269C4⋊Q8—1221no—
270C4⋊Q8—1221no—
271C4⋊Q8C2×#692311yescharacteristic
272C4⋊Q8C2×#692311yescharacteristic
273C4×Q8—2442no—
274C4×Q8—2442no—
275C2×C4⋊C4—611no—
276C2×C4⋊C4—611no—
277C2×C4⋊C4—612no—
278C2×C4⋊C4—611no—
279C2×C4⋊C4—611no—
280C2×C4⋊C4—1221no—
281C2×C4⋊C4—1221no—
282C2×C4⋊C4—1221no—
283C2×C4⋊C4—1222no—
284C2×C42—611no—
285C2×C42—1221no—
286C2×C42—1221no—
287C2×C42—1222no—
288#106—22no—
289#106—1222no—
290#106—1222no—
291#106—1222no—
292C4⋊C8—1221no—
293C4⋊C8—1221no—
294C4⋊C8—1221no—
295C4⋊C8—1221no—
296#103—2442no—
297#103—2442no—
298C4×C8—611no—
299C4×C8—1221no—
300C4×C8C2×#692311yescharacteristic
301#95—1221no—
302#95—1221no—
303#95—1222no—
304#95—1222no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—1222no—
308C2×Q8—1221no—
309C2×Q8—1221no—
310C2×Q8—2441no—
311C2×Q8—2441no—
312C2×Q8—2441no—
313C2×Q8—2441no—
314C2×Q8—2441no—
315C2×Q8—2441no—
316C2×Q8—2441no—
317C2×Q8—2441no—
318C2×Q8—2441no—
319C22×C4—611no—
320C22×C4—611no—
321C22×C4—611no—
322C22×C4—1222no—
323C22×C4—1221no—
324C22×C4—1221no—
325C22×C4—1221no—
326C22×C4—1221no—
327C22×C4—2441no—
328C2×C8—21no—
329C2×C8—1222no—
330C2×C8—1222no—
331C2×C8—1221no—
332C2×C8—2441no—
333C2×C8—2441no—
334C2×C8—2441no—
335C2×C8—2441no—
336C2×C8—2441no—
337C2×C8—2441no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—612no—
342C4⋊C4—612no—
343C4⋊C4—612no—
344C4⋊C4—1222no—
345C4⋊C4—1222no—
346C4⋊C4—1221no—
347C4⋊C4—1221no—
348C4⋊C4—1221no—
349C4⋊C4—1221no—
350C4⋊C4—1221no—
351C4⋊C4—1221no—
352C4⋊C4—1221no—
353C4⋊C4—1221no—
354C4⋊C4—1221no—
355C4⋊C4—2441no—
356C4⋊C4—2441no—
357C4⋊C4—2444no—
358C4⋊C4C22⋊#692312yes—
359C42—611no—
360C42—611no—
361C42—611no—
362C42—2441no—
363C42—2441no—
364C42—2441no—
365C42—2441no—
366C42—2444no—
367C42C22×#692311yescharacteristic
368C23—611no—
369Q8—41no—
370Q8—41no—
371Q8—41no—
372Q8—41no—
373Q8—4881no—
374Q8—4881no—
375Q8—2442no—
376Q8—2442no—
377Q8—2441no—
378Q8—2441no—
379Q8—2441no—
380Q8—2441no—
381C2×C4—22no—
382C2×C4—41no—
383C2×C4—611no—
384C2×C4—611no—
385C2×C4—611no—
386C2×C4—611no—
387C2×C4—611no—
388C2×C4—611no—
389C2×C4—611no—
390C2×C4—611no—
391C2×C4—611no—
392C2×C4—611no—
393C2×C4—611no—
394C2×C4—1221no—
395C2×C4—1221no—
396C2×C4—1222no—
397C2×C4—1222no—
398C2×C4—1222no—
399C2×C4—4881no—
400C2×C4—4881no—
401C2×C4—2442no—
402C2×C4—2442no—
403C2×C4—2441no—
404C2×C4—2441no—
405C2×C4—2441no—
406C2×C4—2441no—
407C2×C4—2441no—
408C2×C4—2441no—
409C2×C4—2441no—
410C2×C4—2441no—
411C2×C4—2441no—
412C2×C4—2441no—
413C2×C4—2441no—
414C2×C4C2×C22⋊#692311yescharacteristic
415C2×C4D4×#692311yescharacteristic
416C2×C4D4×#692311yescharacteristic
417C8—22no—
418C8—1222no—
419C8—4881no—
420C8—4881no—
421C22—611no—
422C22—611no—
423C22—611no—
424C22—611no—
425C22—611no—
426C22—611no—
427C22#1684071011yescharacteristic
428C4—21no—
429C4—42no—
430C4—81no—
431C4—611no—
432C4—611no—
433C4—611no—
434C4—611no—
435C4—1221no—
436C4—4881no—
437C4—4881no—
438C4—4881no—
439C4—2442no—
440C4—2442no—
441C4—2442no—
442C4SD16×#692311yescharacteristic
443C4SD16×#692311yescharacteristic
444C4#1684080111yescharacteristic
445C4#1684080111yescharacteristic
446C2—611no—
447C2—611no—
448C2—611no—
449C2—611no—
450C2#2146181211yescharacteristic
451C2#2146181511yescharacteristic
452C2#2146182511yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#498C4
socleC22×C61#386
prime radicalC22×D61#120
cosocleC42×D61C23
hypercenter#498#6923
hypocenterC61#9059
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
24. #16840160C4semidirect
26. #16840158C4semidirect
27. #16840158C4semidirect
30. C61×#498C4semidirect
241. #498#6923semidirect
257. C61#9059semidirect

Maximal subgroups

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

subgroupindex
#214623482
#214619982
#214619852
D61⋊#3322
#214606542
#498×D612
#905961