Finite Group Databasebeta

← #29576289

Subgroups — #29576289

Order 15616.

550 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 111 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#21455823C211yesmaximal, characteristic
2#21448280C211yesmaximal, characteristic
3#21446797C211yesmaximal, characteristic, complemented
4#21444245C211yesmaximal, characteristic
5#21443722C211yesmaximal, characteristic, complemented
6#21443651C211yesmaximal, characteristic, complemented
7C61×#1503C211yesmaximal, characteristic, complemented
8C2×C4×D244C2211yescharacteristic
9C2×#14519734—21no—
10#16840090—21no—
11#16840049C412yescomplemented
12#16839964C412yescomplemented
13#16839912—21no—
14C4×#11758201—22no—
15#16839815C2211yescharacteristic
16#16839786C2211yescharacteristic
17#16839663—21no—
18#16839658—21no—
19#16839651—22no—
20C4×#11758198—21no—
21C42×Dic61C2211yescharacteristic
22#16839521—21no—
23#16839521—21no—
24#16839491—22no—
25C61×#533C2211yescharacteristic
26C61×#431C2211yescharacteristic
27C4×C4⋊C4×C61C2211yescharacteristic
28C61×#328—22no—
29C22×D244D411yescharacteristic
30C22×C4×D61—21no—
31C2×#11758201—22no—
32C2×#11758194—21no—
33C4×D244—21no—
34C4×D244—21no—
35C4×D244C2×C412yes—
36C4×D244D412yes—
37#14519734—42no—
38#14519734—42no—
39C2×C4⋊Dic61D411yescharacteristic
40C2×#11758196—22no—
41C2×#11758198—21no—
42C2×#11758198—22no—
43C2×#11758198—22no—
44C2×C4×Dic61—21no—
45C2×C4×Dic61—21no—
46C2×C4×Dic61D411yescharacteristic
47C2×C4×Dic61D411yescharacteristic
48#14519708—22no—
49C4⋊#92418—21no—
50C4⋊#92418—21no—
51C4⋊#92418C2×C412yes—
52#14519701—22no—
53#14519697—22no—
54C22⋊#92418—21no—
55C4×#92418—21no—
56#14519684—22no—
57C4⋊Q8×C61C2×C411yescharacteristic
58C4⋊Q8×C61C2×C411yescharacteristic
59#125×C61—21no—
60C22⋊Q8×C61—21no—
61C22⋊Q8×C61—21no—
62C4×Q8×C61—21no—
63C4×Q8×C61—21no—
64#117×C61—21no—
65C2×C4⋊C4×C61—21no—
66C2×C42×C61—21no—
67C2×C42×C61C2311yescharacteristic
68C4⋊C8×C61—21no—
69C22⋊C8×C61—21no—
70C4×C8×C61—21no—
71#95×C61D412yes—
72C23×D61—21no—
73C2×Q8×C61—21no—
74C2×Q8×C61—21no—
75C22×C4×C61—22no—
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C22×Dic61—21no—
80C22×Dic61—21no—
81C22×Dic61—21no—
82C2×D244—21no—
83C2×D244—42no—
84C2×D244C22⋊C412yes—
85C2×C4×D61—22no—
86C2×C4×D61—22no—
87C2×C4×D61—41no—
88C2×C4×D61—41no—
89C2×C4×D61—41no—
90C2×C4×D61—41no—
91C2×C8×C61—41no—
92C4⋊C4×C61—21no—
93C4⋊C4×C61—21no—
94C4⋊C4×C61—21no—
95C4⋊C4×C61—21no—
96C4⋊C4×C61—42no—
97C4⋊C4×C61—41no—
98C4⋊C4×C61—41no—
99C22⋊C4×C61—41no—
100C42×C61—41no—
101C42×C61—41no—
102C42×C61—41no—
103C42×C61C22×C411yescharacteristic
104C42×C61C2×D411yescharacteristic
105C42×C61#4111yescharacteristic
106C42×C61#4111yescharacteristic
107#11758201—44no—
108#11758198—22no—
109#11758198—22no—
110#11758198—44no—
111#11758198—44no—
112C4⋊Dic61—21no—
113C4⋊Dic61C22⋊C412yes—
114#11758196—44no—
115C4×Dic61—21no—
116C4×Dic61—21no—
117C4×Dic61—21no—
118C4×Dic61—42no—
119C4×Dic61—42no—
120C4×Dic61SD1612yescomplemented
121#11758194—41no—
122#11758194—41no—
123C2×#92418—22no—
124C2×#92418—41no—
125C23×C61#12011yescharacteristic
126C22×D61—22no—
127C22×D61—42no—
128C22×D61—42no—
129C22×D61—41no—
130C22×D61—41no—
131Q8×C61—42no—
132Q8×C61—81no—
133C2×C4×C61—22no—
134C2×C4×C61—41no—
135C2×C4×C61—42no—
136C2×C4×C61—42no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—42no—
140C2×C4×C61C2×C22⋊C411yescharacteristic
141C2×C4×C61C4×D411yescharacteristic
142C2×C4×C61C4×D411yescharacteristic
143C2×C4×C61#12111yescharacteristic
144C2×C4×C61#12111yescharacteristic
145C2×C4×C61#12311yescharacteristic
146C2×C4×C61C2×SD1611yescharacteristic
147C2×C4×C61#13511yescharacteristic
148C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C61#13711yescharacteristic
150C2×Dic61—22no—
151C2×Dic61—22no—
152C2×Dic61—22no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159D244—22no—
160D244—22no—
161D244—42no—
162C4×D61—42no—
163C4×D61—42no—
164C4×D61—42no—
165C4×D61—42no—
166C8×C61—81no—
167#92418—42no—
168#92418—81no—
169#9235—611nomaximal, Sylow
170C22×C61—21no—
171C22×C61—21no—
172C22×C61#38611yescharacteristic
173C22×C61#45011yescharacteristic
174C22×C61#45211yescharacteristic
175C2×D61—42no—
176C2×D61—42no—
177C2×D61—42no—
178C2×D61—82no—
179C2×D61—82no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—21no—
183C4×C61—82no—
184C4×C61—81no—
185C4×C61—81no—
186C4×C61#42112yes—
187C4×C61C4×SD1611yescharacteristic
188C4×C61#44011yescharacteristic
189C4×C61#46111yescharacteristic
190C4×C61#46311yescharacteristic
191Dic61—42no—
192Dic61—42no—
193Dic61—42no—
194#3101—611no—
195#2293—611no—
196#1994—611no—
197#1503D6111yescharacteristic, complemented
198#1486—611no—
199#1303—611no—
200#1276—611no—
201C2×C61—21no—
202C2×C61—21no—
203C2×C61#187711yescharacteristic
204C2×C61#189711yescharacteristic
205C2×C61#190611yescharacteristic
206D61—82no—
207#533C2×D6111yescharacteristic
208C2×C4×D4—611no—
209C2×#117—1221no—
210#483—1221no—
211#476—612no—
212#461—612no—
213#450—1221no—
214C4×SD16—1222no—
215#431—611no—
216#431C2×D6111yescharacteristic
217#423—611no—
218#390—1221no—
219#389—1221no—
220#388—1222no—
221C4×C4⋊C4C2×D6111yescharacteristic
222C4×C22⋊C4—1221no—
223C43—611no—
224#339—1221no—
225#339—1221no—
226#328—22no—
227D4⋊C8—1222no—
228C61#923511yescharacteristic, complemented, Sylow
229C22×D4—611no—
230C23×C4—1221no—
231C2×SD16—1222no—
232C2×M16—1221no—
233C4⋊Q8C4×D6111yescharacteristic
234C4⋊Q8C4×D6111yescharacteristic
235#125—21no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×D4—612no—
241C4×D4—612no—
242C4×D4—1221no—
243C4×D4—1221no—
244#117—21no—
245#117—2442no—
246#117—2442no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—611no—
249C2×C4⋊C4—1222no—
250C2×C22⋊C4—1221no—
251C2×C22⋊C4—1222no—
252C2×C22⋊C4—1222no—
253C2×C42—21no—
254C2×C42—611no—
255C2×C42—611no—
256C2×C42—1221no—
257C2×C42—1221no—
258C2×C42C22×D6111yescharacteristic
259#106—1222no—
260C4⋊C8—21no—
261C4⋊C8—612no—
262C4⋊C8—1221no—
263C4⋊C8—1221no—
264#103—1222no—
265#102—1222no—
266C22⋊C8—21no—
267C22⋊C8—1221no—
268C4×C8—21no—
269C4×C8—1221no—
270#95—1222no—
271#95C22⋊D6112yes—
272C24—1221no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×D4—612no—
276C2×D4—1221no—
277C2×D4—2442no—
278C22×C4—22no—
279C22×C4—1221no—
280C22×C4—1221no—
281C22×C4—1221no—
282C22×C4—1222no—
283C22×C4—1222no—
284C22×C4—2441no—
285C22×C4—2441no—
286C22×C4—2441no—
287C22×C4—2441no—
288C22×C4D4×D6111yescharacteristic
289C22×C4D4×D6111yescharacteristic
290C22×C4C2×C22⋊D6111yescharacteristic
291SD16—2444no—
292M16—2441no—
293M16—2441no—
294C2×C8—41no—
295C2×C8—1222no—
296C2×C8—2441no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—612no—
305C4⋊C4—1221no—
306C4⋊C4—2444no—
307C22⋊C4—41no—
308C22⋊C4—1222no—
309C22⋊C4—1222no—
310C22⋊C4—2444no—
311C22⋊C4—2444no—
312C42—41no—
313C42—41no—
314C42—41no—
315C42—612no—
316C42—1221no—
317C42—1221no—
318C42—1221no—
319C42—2442no—
320C42—2442no—
321C42C2×C4×D6111yescharacteristic
322C42#1175822211yescharacteristic
323C42D4×D6111yescharacteristic
324C42#1175822411yescharacteristic
325C23—1222no—
326C23—2442no—
327C23—2442no—
328C23—2441no—
329C23—2441no—
330C23#1451974511yescharacteristic
331Q8—42no—
332Q8—81no—
333D4—1222no—
334D4—1222no—
335D4—2442no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—1222no—
344C2×C4—1222no—
345C2×C4—1222no—
346C2×C4—2441no—
347C2×C4—2441no—
348C2×C4—2441no—
349C2×C4—2441no—
350C2×C4—2441no—
351C2×C4—2441no—
352C2×C4—2442no—
353C2×C4—2442no—
354C2×C4—2442no—
355C2×C4—2442no—
356C2×C4C22⋊C4×D6111yescharacteristic
357C2×C4#1451973711yescharacteristic
358C2×C4C4×C22⋊D6111yescharacteristic
359C2×C4#1451974811yescharacteristic
360C2×C4#1451975011yescharacteristic
361C2×C4#1451975911yescharacteristic
362C2×C4SD16×D6111yescharacteristic
363C2×C4#1451980011yescharacteristic
364C2×C4#1451980411yescharacteristic
365C2×C4#1451980911yescharacteristic
366C8—81no—
367C8—4881no—
368C8—2442no—
369C22—21no—
370C22—21no—
371C22—4882no—
372C22—4882no—
373C22—2442no—
374C22—2442no—
375C22—2442no—
376C22#1683963811yescharacteristic
377C22#1683991111yescharacteristic
378C22#1683992311yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—82no—
383C4—81no—
384C4—81no—
385C4—2442no—
386C4—2442no—
387C4—2442no—
388C4#1683977712yes—
389C4#1683984711yescharacteristic
390C4#1683986511yescharacteristic
391C4#1683996611yescharacteristic
392C4#1683998011yescharacteristic
393C2—21no—
394C2—21no—
395C2—4882no—
396C2#2144615011yescharacteristic
397C2#2144626411yescharacteristic
398C2#2144631711yescharacteristic
subgroupG/H
centerC2×C4#14519750
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1503C2
socleC22×C61#386
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenter#1503D61
hypocenterC61#9235
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
3. #21446797C2semidirect
5. #21443722C2semidirect
6. #21443651C2semidirect
7. C61×#1503C2semidirect
11. #16840049C4semidirect
12. #16839964C4semidirect
120. C4×Dic61SD16semidirect
197. #1503D61semidirect
228. C61#9235semidirect

Maximal subgroups

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

subgroupindex
#214558232
#214482802
#214467972
#214442452
#214437222
#214436512
C61×#15032
#923561