Finite Group Databasebeta

← #29568950

Subgroups — #29568950

Order 15616.

472 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 83 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#21452835C211yesmaximal, characteristic
2#21446957C211yesmaximal, characteristic, complemented
3#21446952C211yesmaximal, characteristic
4#21443644C211yesmaximal, characteristic, complemented
5#21443642C211yesmaximal, characteristic, complemented
6#21443639C211yesmaximal, characteristic
7C61×#2043C211yesmaximal, characteristic, complemented
8C2×#14519755—21no—
9C2×#14519700—21no—
10C2×#14519697—21no—
11#16839697—21no—
12#16839696—21no—
13#16839684—21no—
14#16839668C2211yescharacteristic
15#16839664C2211yescharacteristic
16#16839640—21no—
17#16839633—21no—
18#16839524—21no—
19#16839524—21no—
20#16839517C2211yescharacteristic
21#16839517C2211yescharacteristic
22C2×C4⋊Q8×C61C2211yescharacteristic
23C2×#103×C61—21no—
24C2×#103×C61—21no—
25C61×#389C2211yescharacteristic
26C61×#336C2211yescharacteristic
27C22×Dic122D411yescharacteristic
28C22×D244D411yescharacteristic
29C22×C4×D61—21no—
30C22×#92418—21no—
31C22×#92418—21no—
32#14519755—42no—
33#14519755—42no—
34C2×C4⋊Dic61D411yescharacteristic
35C2×C4⋊Dic61D411yescharacteristic
36C2×#11758196—21no—
37C2×#11758196—21no—
38C2×#11758196—21no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×#11758198—21no—
44#14519700—42no—
45#14519697—42no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49C22×Q8×C61—21no—
50C22×C8×C61—21no—
51C4⋊Q8×C61—42no—
52C2×C4⋊C4×C61—21no—
53C2×C4⋊C4×C61—21no—
54C2×C4⋊C4×C61—21no—
55C2×C4⋊C4×C61D411yescharacteristic
56C2×C4⋊C4×C61D411yescharacteristic
57C2×C42×C61C2311yescharacteristic
58#103×C61—42no—
59#103×C61—42no—
60#95×C61—21no—
61C23×D61—21no—
62C2×Q8×C61—22no—
63C2×Q8×C61—41no—
64C2×Q8×C61—41no—
65C2×Q8×C61—41no—
66C2×Q8×C61—41no—
67C2×Q8×C61—41no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—41no—
71C22×C4×C61C2×D411yescharacteristic
72C22×C4×C61C2×D411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C22×Dic61—21no—
75C22×Dic61—21no—
76C22×Dic61—21no—
77C2×D244—21no—
78C2×D244—41no—
79C2×D244—41no—
80C2×D244Q1612yes—
81C2×C4×D61—42no—
82C2×C4×D61—81no—
83C2×C4×D61—81no—
84C2×Dic122—21no—
85C2×Dic122—41no—
86C2×Dic122—41no—
87C2×Dic122SD1612yes—
88C2×C8×C61—41no—
89C2×C8×C61—41no—
90C2×C8×C61—41no—
91C4⋊C4×C61—22no—
92C4⋊C4×C61—22no—
93C4⋊C4×C61—21no—
94C4⋊C4×C61—21no—
95C4⋊C4×C61—41no—
96C4⋊C4×C61—41no—
97C4⋊C4×C61—41no—
98C4⋊C4×C61—41no—
99C4⋊C4×C61—41no—
100C42×C61—41no—
101#11758198—42no—
102#11758198—41no—
103#11758198—41no—
104#11758198—41no—
105#11758198—41no—
106#11758198—81no—
107#11758198—81no—
108C4⋊Dic61—41no—
109C4⋊Dic61—41no—
110#11758196—42no—
111#11758196—42no—
112#11758196—81no—
113C2×#92418—41no—
114C2×#92418—41no—
115C2×#92418—41no—
116C2×#92418—41no—
117C2×#92418—41no—
118C2×#92418—41no—
119C23×C61#12011yescharacteristic
120C22×D61—22no—
121C22×D61—41no—
122C22×D61—41no—
123C22×D61—41no—
124C22×D61—41no—
125C22×D61—41no—
126C22×D61—41no—
127Q8×C61—42no—
128Q8×C61—42no—
129Q8×C61—81no—
130Q8×C61—81no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—42no—
143C2×C4×C61—41no—
144C2×C4×C61—81no—
145C2×C4×C61—81no—
146C2×C4×C61C2×SD1611yescharacteristic
147C2×C4×C61C2×Q1611yescharacteristic
148C2×C4×C61#13511yescharacteristic
149C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C61#13711yescharacteristic
152C2×Dic61—22no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161D244—42no—
162D244—81no—
163C4×D61—84no—
164Dic122—42no—
165Dic122—81no—
166C8×C61—81no—
167#92418—81no—
168#92418—81no—
169#7774—611nomaximal, Sylow
170C22×C61#44911yescharacteristic
171C22×C61#45011yescharacteristic
172C22×C61#45111yescharacteristic
173C22×C61#45411yescharacteristic
174C22×C61#45511yescharacteristic
175C22×C61#45511yescharacteristic
176C22×C61B(4,𝔽2)11yescharacteristic
177C2×D61—42no—
178C2×D61—42no—
179C2×D61—42no—
180C2×D61—81no—
181C2×D61—81no—
182C2×D61—81no—
183C2×D61—81no—
184C4×C61—21no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61—21no—
189C4×C61—21no—
190C4×C61—82no—
191C4×C61—81no—
192C4×C61—81no—
193Dic61—42no—
194Dic61—81no—
195Dic61—81no—
196#2732—611no—
197#2043D6111yescharacteristic, complemented
198#2024—611no—
199#2023—611no—
200#1272—611no—
201#1271—611no—
202#1270—611no—
203C2×C61#159711yescharacteristic
204C2×C61#160111yescharacteristic
205C2×C61#160211yescharacteristic
206C2×C61#163711yescharacteristic
207C2×C61#164211yescharacteristic
208C2×C61#164411yescharacteristic
209C2×C61#166211yescharacteristic
210D61—82no—
211C2×C4⋊Q8C2×D6111yescharacteristic
212C2×C22⋊Q8—1221no—
213C2×#103—21no—
214C2×#103—21no—
215C2×#103—1221no—
216C2×#102—1221no—
217#399—1221no—
218#398—1221no—
219#396—1221no—
220#391—611no—
221#390—611no—
222#389C2×D6111yescharacteristic
223#386—1221no—
224#385—1221no—
225#340—1221no—
226#340—1221no—
227#336—611no—
228#336—611no—
229#336C2×D6111yescharacteristic
230C61#777411yescharacteristic, complemented, Sylow
231C22×Q8—21no—
232C22×Q8—611no—
233C22×D4—611no—
234C23×C4—1221no—
235C22×C8—21no—
236C22×C8—1221no—
237C22×C8—1221no—
238C4⋊Q8—42no—
239C22⋊Q8—2442no—
240C22⋊Q8—2442no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—611no—
245C2×C4⋊C4—611no—
246C2×C4⋊C4—1221no—
247C2×C4⋊C4—1221no—
248C2×C4⋊C4—1221no—
249C2×C4⋊C4C22⋊D6111yescharacteristic
250C2×C4⋊C4C22⋊D6111yescharacteristic
251C2×C22⋊C4—1221no—
252C2×C22⋊C4—1221no—
253C2×C22⋊C4—1221no—
254C2×C22⋊C4—1221no—
255C2×C22⋊C4—1221no—
256C2×C42C22×D6111yescharacteristic
257#103—42no—
258#103—42no—
259#103—2442no—
260#102—2442no—
261#95—21no—
262#95—1221no—
263#95—1221no—
264#95—1221no—
265C24—1221no—
266C2×Q8—22no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—612no—
273C2×Q8—1221no—
274C2×Q8—2441no—
275C2×Q8—2441no—
276C2×D4—612no—
277C2×D4—1221no—
278C2×D4—2441no—
279C2×D4—2441no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—41no—
283C22×C4—1221no—
284C22×C4—1221no—
285C22×C4—1221no—
286C22×C4—4881no—
287C22×C4—4881no—
288C22×C4—2442no—
289C22×C4D4×D6111yescharacteristic
290C22×C4D4×D6111yescharacteristic
291C22×C4C2×C22⋊D6111yescharacteristic
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—2441no—
296C2×C8—2441no—
297C2×C8—2441no—
298C2×C8—2441no—
299C2×C8—2441no—
300C2×C8—2441no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—4881no—
311C4⋊C4—2441no—
312C4⋊C4—2441no—
313C4⋊C4—2442no—
314C4⋊C4—2442no—
315C22⋊C4—4881no—
316C22⋊C4—4881no—
317C22⋊C4—2442no—
318C22⋊C4—2441no—
319C22⋊C4—2441no—
320C22⋊C4—2441no—
321C22⋊C4—2441no—
322C42—41no—
323C23—1222no—
324C23—2441no—
325C23—2441no—
326C23—2441no—
327C23—2441no—
328C23—2441no—
329C23—2441no—
330C23#1451974511yescharacteristic
331Q8—42no—
332Q8—42no—
333Q8—81no—
334Q8—81no—
335Q8—4881no—
336Q8—2442no—
337D4—4881no—
338D4—2442no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—1222no—
355C2×C4—4884no—
356C2×C4—2441no—
357C2×C4—2441no—
358C2×C4—2441no—
359C2×C4—2441no—
360C2×C4—2441no—
361C2×C4—2441no—
362C2×C4—2441no—
363C2×C4—2441no—
364C2×C4SD16×D6111yescharacteristic
365C2×C4Q16×D6111yescharacteristic
366C2×C4#1451979711yescharacteristic
367C2×C4#1451980311yescharacteristic
368C2×C4#1451980411yescharacteristic
369C2×C4#1451980911yescharacteristic
370C8—81no—
371C8—4881no—
372C8—4881no—
373C22—4881no—
374C22—4881no—
375C22—4881no—
376C22—4881no—
377C22—2442no—
378C22—2442no—
379C22—2442no—
380C22#1683990511yescharacteristic
381C22#1683991111yescharacteristic
382C22#1683991611yescharacteristic
383C22#1683993411yescharacteristic
384C22#1683993811yescharacteristic
385C22#1683993811yescharacteristic
386C22#1683994811yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—82no—
394C4—81no—
395C4—81no—
396C4—4881no—
397C4—4881no—
398C4—2442no—
399C2—4882no—
400C2#2144474611yescharacteristic
401C2#2144476911yescharacteristic
402C2#2144477411yescharacteristic
403C2#2144498111yescharacteristic
404C2#2144501711yescharacteristic
405C2#2144503111yescharacteristic
406C2#2144514011yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2043C2
socleC23×C61#120
prime radicalC22×D244D4
hypercenter#2043D61
hypocenterC61#7774
LCSG ▷ C2×C42×C61 ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2043

Complements

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

NH ≅ G/Ntype
2. #21446957C2semidirect
4. #21443644C2semidirect
5. #21443642C2semidirect
7. C61×#2043C2semidirect
197. #2043D61semidirect
230. C61#7774semidirect

Maximal subgroups

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

subgroupindex
#214528352
#214469572
#214469522
#214436442
#214436422
#214436392
C61×#20432
#777461