Finite Group Databasebeta

← #29580730

Subgroups — #29580730

Order 15616.

536 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 87 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#21456626C211yesmaximal, characteristic
2C2×#16840097C211yesmaximal, characteristic, complemented
3C2×#16840075C211yesmaximal, characteristic
4#21444184C211yesmaximal, characteristic, complemented
5#21444180C211yesmaximal, characteristic
6#21443706C211yesmaximal, characteristic, complemented
7C61×#1821C211yesmaximal, characteristic, complemented
8C2×#14519756C2211yescharacteristic
9C2×#14519751C2211yescharacteristic
10#16840098—21no—
11#16840097—22no—
12#16840097—22no—
13#16840076—21no—
14#16840075—22no—
15#16840075—22no—
16#16840008—22no—
17#16839915—21no—
18#16839903—21no—
19C2×#14519710—21no—
20C2×#14519710—21no—
21C2×#14519700—21no—
22C2×#14519697—21no—
23C2×C22⋊#92418C2211yescharacteristic
24C2×C22⋊#92418C2211yescharacteristic
25#16839487—22no—
26#16839485—22no—
27C22×C4⋊C4×C61C2211yescharacteristic
28C61×#407C2211yescharacteristic
29C61×#407C2211yescharacteristic
30C61×#343—22no—
31C22×Dic122D411yescharacteristic
32C22×D244D411yescharacteristic
33C22×C22⋊D61—21no—
34C2×#11758190—22no—
35C22×#92418—21no—
36C22×#92418—21no—
37#14519756—21no—
38#14519756—21no—
39#14519756—21no—
40#14519756—21no—
41#14519751—21no—
42#14519751—21no—
43#14519751—21no—
44#14519751—21no—
45C2×C4⋊Dic61D411yescharacteristic
46C2×C4⋊Dic61D411yescharacteristic
47C2×#11758196—21no—
48C2×C22⋊Dic61—21no—
49C2×#11758198—21no—
50#14519711—22no—
51#14519710—24no—
52#14519710—24no—
53#14519702—22no—
54#14519700—24no—
55#14519699—22no—
56#14519697—24no—
57C22⋊#92418—21no—
58C22⋊#92418—21no—
59C22⋊#92418D412yes—
60C22⋊#92418D412yes—
61C23×C4×C61—21no—
62C23×C4×C61C2311yescharacteristic
63C2×M16×C61—21no—
64C2×M16×C61D412yes—
65C2×C4⋊C4×C61—21no—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61D412yes—
71C22⋊C8×C61—21no—
72C22⋊C8×C61—21no—
73C22⋊C8×C61—21no—
74C22⋊C8×C61—21no—
75C24×C61#4111yescharacteristic
76C23×D61—21no—
77C22×C4×C61—22no—
78C22×C4×C61—42no—
79C22×C4×C61—41no—
80C22×C4×C61—41no—
81C22×C4×C61—82no—
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C2×C22⋊D61—41no—
89C2×C22⋊D61—41no—
90C2×C22⋊D61—82no—
91C22×Dic61—21no—
92C22×Dic61—21no—
93C22×Dic61—21no—
94C2×D244—21no—
95C2×D244—21no—
96C2×D244—42no—
97C2×Dic122—21no—
98C2×Dic122—21no—
99C2×Dic122—42no—
100M16×C61—42no—
101M16×C61—42no—
102C2×C8×C61—22no—
103C2×C8×C61—41no—
104C4⋊C4×C61—22no—
105C4⋊C4×C61—22no—
106C4⋊C4×C61—41no—
107C4⋊C4×C61—41no—
108C22⋊Dic61—41no—
109C22⋊Dic61—41no—
110#11758198—41no—
111#11758198—41no—
112C4⋊Dic61—21no—
113C4⋊Dic61—21no—
114C4⋊Dic61—21no—
115C4⋊Dic61—21no—
116#11758196—41no—
117#11758196—41no—
118C2×#92418—22no—
119C2×#92418—22no—
120C2×#92418—42no—
121C2×#92418—42no—
122#11758190—82no—
123C23×C61—22no—
124C23×C61—22no—
125C23×C61—41no—
126C23×C61#12111yescharacteristic
127C23×C61#12111yescharacteristic
128C23×C61#12111yescharacteristic
129C22×D61—41no—
130C22×D61—42no—
131C22×D61—81no—
132C22×D61—81no—
133C2×C4×C61—22no—
134C2×C4×C61—22no—
135C2×C4×C61—22no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—42no—
140C2×C4×C61—82no—
141C2×C4×C61—82no—
142C2×C4×C61—82no—
143C2×C4×C61#12011yescharacteristic
144C2×C4×C61#12011yescharacteristic
145C2×C4×C61#12011yescharacteristic
146C2×C4×C61#12711yescharacteristic
147C22⋊D61—82no—
148C22⋊D61—82no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—42no—
153C2×Dic61—81no—
154C2×Dic61—81no—
155D244—42no—
156D244—42no—
157Dic122—42no—
158Dic122—42no—
159C8×C61—42no—
160C8×C61—81no—
161#92418—42no—
162#92418—42no—
163#9994—611nomaximal, Sylow
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—21no—
167C22×C61—21no—
168C22×C61—21no—
169C22×C61—21no—
170C22×C61—21no—
171C22×C61—44no—
172C22×C61#39211yescharacteristic
173C22×C61#47212yes—
174C22×C61#47312yes—
175C2×D61—81no—
176C2×D61—81no—
177C2×D61—81no—
178C2×D61—162no—
179C4×C61—21no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—82no—
183C4×C61#45812yes—
184Dic61—81no—
185Dic61—81no—
186Dic61—81no—
187#3193—611no—
188C2×#484—611no—
189C2×#480—611no—
190#1821D6111yescharacteristic, complemented
191#1472—611no—
192#1471—611no—
193#1298—611no—
194C2×C61—22no—
195C2×C61—22no—
196C2×C61—41no—
197C2×C61#201811yescharacteristic
198C2×C61#202011yescharacteristic
199C2×C61#202211yescharacteristic
200D61—161no—
201C2×C22⋊Q8—611no—
202C2×#121—611no—
203C22×C4⋊C4C2×D6111yescharacteristic
204#484—1221no—
205#484—1222no—
206#484—1222no—
207#480—1221no—
208#480—1222no—
209#480—1222no—
210#468—1222no—
211#450—1221no—
212#448—1221no—
213C2×#107—1221no—
214C2×#107—1221no—
215C2×#103—1221no—
216C2×#102—1221no—
217#407C2×D6111yescharacteristic
218#407C2×D6111yescharacteristic
219C2×C22⋊C8—611no—
220C2×C22⋊C8—611no—
221#343—22no—
222#324—1222no—
223#323—1222no—
224C61#999411yescharacteristic, complemented, Sylow
225C22×Q8—611no—
226C22×D4—611no—
227C22×D4—1221no—
228C23×C4—21no—
229C23×C4C22×D6111yescharacteristic
230C2×SD16—1222no—
231C2×M16—21no—
232C2×M16C22⋊D6112yes—
233C22×C8—1221no—
234C22×C8—1221no—
235C22⋊Q8—1221no—
236C22⋊Q8—1221no—
237C22⋊Q8—1221no—
238C22⋊Q8—1221no—
239#121—1221no—
240#121—1221no—
241#121—1221no—
242#121—1221no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—611no—
249C2×C4⋊C4—611no—
250C2×C4⋊C4—1221no—
251C2×C4⋊C4D24412yes—
252C2×C22⋊C4—1221no—
253C2×C22⋊C4—1221no—
254#107—1224no—
255#107—1224no—
256#107—1222no—
257#103—1224no—
258#103—1222no—
259#102—1224no—
260#102—1222no—
261C22⋊C8—21no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264C22⋊C8—21no—
265C22⋊C8—612no—
266C22⋊C8—612no—
267C22⋊C8—1221no—
268C22⋊C8—1221no—
269C24—1221no—
270C24#1175822211yescharacteristic
271C2×Q8—1221no—
272C2×Q8—1221no—
273C2×Q8—2442no—
274C2×D4—1221no—
275C2×D4—1221no—
276C2×D4—4882no—
277C2×D4—2442no—
278C2×D4—2441no—
279C2×D4—2441no—
280C22×C4—22no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—82no—
285C22×C4—1221no—
286C22×C4—1221no—
287C22×C4—1221no—
288C22×C4C2×D24411yescharacteristic
289C22×C4D4×D6111yescharacteristic
290C22×C4D4×D6111yescharacteristic
291C22×C4D4×D6111yescharacteristic
292C22×C4D4×D6111yescharacteristic
293C22×C4C2×C22⋊D6111yescharacteristic
294SD16—4882no—
295M16—42no—
296M16—42no—
297C2×C8—22no—
298C2×C8—41no—
299C2×C8—1222no—
300C2×C8—1222no—
301C2×C8—2442no—
302C2×C8—2442no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—1221no—
308C4⋊C4—1221no—
309C4⋊C4—1221no—
310C4⋊C4—1221no—
311C4⋊C4—2441no—
312C4⋊C4—2441no—
313C22⋊C4—2441no—
314C22⋊C4—2441no—
315C22⋊C4—2441no—
316C22⋊C4—2441no—
317C23—22no—
318C23—22no—
319C23—41no—
320C23—4881no—
321C23—4881no—
322C23—2442no—
323C23—2441no—
324C23#1451975011yescharacteristic
325C23#1451975011yescharacteristic
326C23#1451975111yescharacteristic
327Q8—2442no—
328Q8—2442no—
329D4—4882no—
330D4—4882no—
331D4—2442no—
332D4—2442no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—4881no—
344C2×C4—4881no—
345C2×C4—2442no—
346C2×C4—2441no—
347C2×C4—2441no—
348C2×C4—2441no—
349C2×C4#1451974511yescharacteristic
350C2×C4#1451974711yescharacteristic
351C2×C4#1451974711yescharacteristic
352C2×C4#1451977311yescharacteristic
353C8—42no—
354C8—81no—
355C8—2442no—
356C8—2442no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—44no—
365C22—9762no—
366C22—4881no—
367C22—4881no—
368C22—4881no—
369C22#1683967111yescharacteristic
370C22#1684003112yes—
371C22#1684003612yes—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—82no—
376C4—4881no—
377C4—4881no—
378C4—4881no—
379C4#1683995112yes—
380C2—22no—
381C2—22no—
382C2—41no—
383C2—9761no—
384C2#2144692711yescharacteristic
385C2#2144693911yescharacteristic
386C2#2144695011yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1821C2
socleC22×C61#392
prime radicalC2×#14519751C22
hypercenter#1821D61
hypocenterC61#9994
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
2. C2×#16840097C2semidirect
4. #21444184C2semidirect
6. #21443706C2semidirect
7. C61×#1821C2semidirect
190. #1821D61semidirect
224. C61#9994semidirect

Maximal subgroups

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

subgroupindex
#214566262
C2×#168400972
C2×#168400752
#214441842
#214441802
#214437062
C61×#18212
#999461