Finite Group Databasebeta

← #29575917

Subgroups — #29575917

Order 15616.

566 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 113 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#21455612C211yesmaximal, characteristic
2#21448727C211yesmaximal, characteristic
3#21444318C211yesmaximal, characteristic, complemented
4#21444211C211yesmaximal, characteristic
5#21443721C211yesmaximal, characteristic, complemented
6#21443677C211yesmaximal, characteristic, complemented
7C61×#1518C211yesmaximal, characteristic, complemented
8C2×C4×D244C2211yescharacteristic
9C2×#14519739—21no—
10#16840059—22no—
11#16840049C412yescomplemented
12#16840039C412yescomplemented
13#16839815C2211yescharacteristic
14#16839805—21no—
15#16839786C2211yescharacteristic
16#16839766—21no—
17#16839658—21no—
18#16839636—22no—
19#16839621—21no—
20C4×C4⋊Dic61C2211yescharacteristic
21#95×D61—21no—
22#16839499—21no—
23#16839499—21no—
24#16839491—22no—
25C61×#533C2211yescharacteristic
26C61×#431C2211yescharacteristic
27C61×#382C2211yescharacteristic
28C61×#328—22no—
29C22×D244D411yescharacteristic
30C22×C4×D61—21no—
31C22×C4×D61—22no—
32C2×#11758194—21no—
33C4×D244—21no—
34C4×D244—21no—
35C4×D244C2×C412yes—
36C4×D244D412yes—
37#14519739—42no—
38#14519739—42no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61D411yescharacteristic
41C2×#11758196—22no—
42C2×#11758198—21no—
43C2×#11758198—22no—
44C2×C4×Dic61—21no—
45#14519710—22no—
46#14519708—22no—
47C4⋊#92418—21no—
48C4⋊#92418—21no—
49C4⋊#92418C2×C412yescomplemented
50#14519697—22no—
51#14519697—22no—
52C22⋊#92418—21no—
53C4×#92418—21no—
54#14519684—22no—
55#14519684D411yescharacteristic
56#14519684D411yescharacteristic
57C4⋊Q8×C61C2×C411yescharacteristic, complemented
58C4⋊Q8×C61C2×C411yescharacteristic, complemented
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×C61C2311yescharacteristic
67C4⋊C8×C61—21no—
68C22⋊C8×C61—21no—
69C4×C8×C61—21no—
70#95×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—21no—
84C2×D244—21no—
85C2×D244C22⋊C412yes—
86C2×D244SD1612yes—
87C2×D244Q1612yes—
88C2×C4×D61—22no—
89C2×C4×D61—22no—
90C2×C4×D61—44no—
91C2×C4×D61—44no—
92C2×C4×D61—82no—
93C2×C4×D61—82no—
94C2×C8×C61—41no—
95C4⋊C4×C61—21no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61—21no—
99C4⋊C4×C61—41no—
100C4⋊C4×C61—41no—
101C4⋊C4×C61—41no—
102C4⋊C4×C61—41no—
103C22⋊C4×C61—41no—
104C42×C61—41no—
105C42×C61C22×C411yescharacteristic
106C42×C61C2×D411yescharacteristic
107C42×C61#4111yescharacteristic
108C42×C61#4111yescharacteristic
109#11758198—22no—
110#11758198—22no—
111#11758198—44no—
112C4⋊Dic61—21no—
113C4⋊Dic61—42no—
114C4⋊Dic61C22⋊C412yes—
115#11758196—82no—
116C4×Dic61—41no—
117C4×Dic61—41no—
118#11758194—41no—
119#11758194—41no—
120C2×#92418—22no—
121C2×#92418—41no—
122C23×C61#12011yescharacteristic
123C22×D61—22no—
124C22×D61—22no—
125C22×D61—22no—
126C22×D61—42no—
127C22×D61—42no—
128Q8×C61—81no—
129Q8×C61—81no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—42no—
133C2×C4×C61—42no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—42no—
137C2×C4×C61C2×C22⋊C411yescharacteristic
138C2×C4×C61C4×D411yescharacteristic
139C2×C4×C61C4×D411yescharacteristic
140C2×C4×C61#12111yescharacteristic
141C2×C4×C61#12111yescharacteristic
142C2×C4×C61#12311yescharacteristic
143C2×C4×C61C2×SD1611yescharacteristic
144C2×C4×C61C2×Q1611yescharacteristic
145C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C61#13711yescharacteristic
147C2×Dic61—22no—
148C2×Dic61—22no—
149C2×Dic61—42no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155D244—22no—
156D244—22no—
157D244—22no—
158D244#10314yes—
159C4×D61—82no—
160C4×D61—82no—
161C4×D61—88no—
162C8×C61—81no—
163#92418—42no—
164#92418—81no—
165#9175—611nomaximal, Sylow
166C22×C61—21no—
167C22×C61—21no—
168C22×C61#38611yescharacteristic
169C22×C61#45011yescharacteristic
170C22×C61#45111yescharacteristic
171C2×D61—24no—
172C2×D61—44no—
173C2×D61—44no—
174C2×D61—42no—
175C2×D61—42no—
176C4×C61—21no—
177C4×C61—21no—
178C4×C61—21no—
179C4×C61—21no—
180C4×C61—82no—
181C4×C61—81no—
182C4×C61—81no—
183C4×C61C2×#10311yescharacteristic
184C4×C61#41711yescharacteristic
185C4×C61#48311yescharacteristic
186C4×C61#48411yescharacteristic
187Dic61—42no—
188Dic61—42no—
189Dic61—81no—
190#3077—611no—
191#2349—611no—
192#1518D6111yescharacteristic, complemented
193#1503—611no—
194#1478—611no—
195#1303—611no—
196#1286—611no—
197C2×C61—21no—
198C2×C61—21no—
199C2×C61#187211yescharacteristic
200C2×C61#190411yescharacteristic
201C2×C61#190611yescharacteristic
202D61—44no—
203#533C2×D6111yescharacteristic
204C2×C4×D4—611no—
205C2×C4×D4—1221no—
206#478—1222no—
207#476—612no—
208#474—612no—
209#431—611no—
210#431C2×D6111yescharacteristic
211#428—1221no—
212#423—611no—
213#418—1221no—
214#389—1221no—
215#385—1222no—
216#382C2×D6111yescharacteristic
217#380—1221no—
218C4×C4⋊C4—611no—
219C2×#95—1221no—
220#328—22no—
221#328—1221no—
222#328—1221no—
223D4⋊C8—1222no—
224C61#917511yescharacteristic, complemented, Sylow
225C22×D4—611no—
226C23×C4—1222no—
227C23×C4—1221no—
228C2×M16—1221no—
229C4⋊Q8C4×D6111yescharacteristic, complemented
230C4⋊Q8C4×D6111yescharacteristic, complemented
231#125—21no—
232C22⋊Q8—21no—
233C22⋊Q8—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×D4—612no—
237C4×D4—612no—
238C4×D4—1221no—
239C4×D4—1221no—
240C4×D4—2442no—
241C4×D4—2442no—
242#117—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—611no—
245C2×C4⋊C4—1222no—
246C2×C4⋊C4—1221no—
247C2×C22⋊C4—1221no—
248C2×C22⋊C4—1222no—
249C2×C42—1221no—
250C2×C42C22×D6111yescharacteristic
251#107—1222no—
252#106—1222no—
253C4⋊C8—21no—
254C4⋊C8—612no—
255C4⋊C8—1221no—
256C4⋊C8—1221no—
257#102—1222no—
258#102—1222no—
259C22⋊C8—21no—
260C22⋊C8—1221no—
261C4×C8—21no—
262C4×C8—1221no—
263#95—21no—
264#95—611no—
265#95—611no—
266#95—1222no—
267#95C22⋊D6112yes—
268C24—1221no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×D4—612no—
272C2×D4—612no—
273C2×D4—612no—
274C2×D4—1221no—
275C2×D4—1221no—
276C2×D4—1221no—
277C22×C4—22no—
278C22×C4—1221no—
279C22×C4—1221no—
280C22×C4—1222no—
281C22×C4—1222no—
282C22×C4—1221no—
283C22×C4—4882no—
284C22×C4—4882no—
285C22×C4—2444no—
286C22×C4—2444no—
287C22×C4D4×D6111yescharacteristic
288C22×C4D4×D6111yescharacteristic
289C22×C4C2×C22⋊D6111yescharacteristic
290M16—2441no—
291M16—2441no—
292C2×C8—41no—
293C2×C8—1222no—
294C2×C8—2441no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—612no—
304C4⋊C4—1221no—
305C4⋊C4—4882no—
306C4⋊C4—2442no—
307C22⋊C4—41no—
308C22⋊C4—1222no—
309C22⋊C4—1222no—
310C22⋊C4—2444no—
311C42—41no—
312C42—2441no—
313C42—2441no—
314C42C2×C4×D6111yescharacteristic
315C42#1175822211yescharacteristic
316C42D4×D6111yescharacteristic
317C42#1175822411yescharacteristic
318C23—1222no—
319C23—1222no—
320C23—1222no—
321C23—2442no—
322C23—2442no—
323C23#1451974511yescharacteristic
324Q8—81no—
325Q8—81no—
326D4—614no—
327D4—1222no—
328D4—1222no—
329D4—1222no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—1222no—
338C2×C4—1222no—
339C2×C4—4882no—
340C2×C4—4882no—
341C2×C4—4888no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4—2441no—
345C2×C4—2442no—
346C2×C4—2441no—
347C2×C4—2441no—
348C2×C4C22⋊C4×D6111yescharacteristic
349C2×C4#1451973711yescharacteristic
350C2×C4C4×C22⋊D6111yescharacteristic
351C2×C4#1451974811yescharacteristic
352C2×C4#1451975011yescharacteristic
353C2×C4#1451975911yescharacteristic
354C2×C4SD16×D6111yescharacteristic
355C2×C4Q16×D6111yescharacteristic
356C2×C4#1451980411yescharacteristic
357C2×C4#1451980911yescharacteristic
358C8—81no—
359C8—4881no—
360C8—2442no—
361C22—21no—
362C22—21no—
363C22—1224no—
364C22—2444no—
365C22—2444no—
366C22—2442no—
367C22—2442no—
368C22#1683963811yescharacteristic
369C22#1683991111yescharacteristic
370C22#1683991611yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—82no—
376C4—81no—
377C4—81no—
378C4—4881no—
379C4—2442no—
380C4—2442no—
381C4#103×D6111yescharacteristic
382C4#1683976211yescharacteristic
383C4#1684009111yescharacteristic
384C4#1684009611yescharacteristic
385C2—21no—
386C2—21no—
387C2—2444no—
388C2#2144612311yescharacteristic
389C2#2144630311yescharacteristic
390C2#2144631711yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1518C2
socleC22×C61#386
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenter#1518D61
hypocenterC61#9175
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
3. #21444318C2semidirect
5. #21443721C2semidirect
6. #21443677C2semidirect
7. C61×#1518C2semidirect
11. #16840049C4semidirect
12. #16840039C4semidirect
49. C4⋊#92418C2×C4semidirect
57. C4⋊Q8×C61C2×C4semidirect
58. C4⋊Q8×C61C2×C4semidirect
192. #1518D61semidirect
224. C61#9175semidirect
229. C4⋊Q8C4×D61semidirect
230. C4⋊Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214556122
#214487272
#214443182
#214442112
#214437212
#214436772
C61×#15182
#917561