Finite Group Databasebeta

← #30607404

Subgroups — #30607404

Order 15616.

378 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 90 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#21462481C211yesmaximal, characteristic
2#21462175C211yesmaximal, characteristic, complemented
3C2×#16840691C211yesmaximal, characteristic, complemented
4#21461706C211yesmaximal, characteristic
5#21460900C211yesmaximal, characteristic, complemented
6#21460881C211yesmaximal, characteristic, complemented
7#350×D61C211yesmaximal, characteristic
8#16840841—21no—
9#16840777—21no—
10#16840777—22no—
11C2×#14519848—21no—
12C2×#14519848C2211yescharacteristic
13C2×#14519848C2211yescharacteristic
14D61⋊#108—21no—
15D61⋊#108C2211yescharacteristic
16#16840772C2211yescharacteristic
17#16840759—21no—
18#16840691—22no—
19#16840691—21no—
20#16840690—21no—
21#16840645—21no—
22#131×D61C2211yescharacteristic
23M32×D61C2211yescharacteristic
24C2×C16×D61C2211yescharacteristic
25#16839547C411yescharacteristic, complemented
26#16839546C411yescharacteristic, complemented
27#16839545C411yescharacteristic, complemented
28C61×#350C411yescharacteristic, complemented
29#14519865—41no—
30#14519864—42no—
31#14519864—41no—
32C2×#11758213—22no—
33C2×#11758213—21no—
34C2×#11758213—21no—
35D61⋊M16—22no—
36D61⋊M16—21no—
37#14519856—21no—
38#14519848—22no—
39#14519848—21no—
40#14519848—21no—
41#14519848—21no—
42#14519848D411yescharacteristic
43#14519848D411yescharacteristic
44#14519848D412yes—
45#14519847—22no—
46#14519847—21no—
47#14519846—21no—
48#14519845—21no—
49#14519842—41no—
50#14519841—41no—
51#41×D61D411yescharacteristic
52#14519785—21no—
53#14519785Q811yescharacteristic
54#14519784Q811yescharacteristic
55#14519783—21no—
56#14519783C2×C411yescharacteristic
57M16×D61—21no—
58M16×D61D411yescharacteristic
59C2×C8×D61—21no—
60C2×C8×D61C2311yescharacteristic
61#14519717C2×C411yescharacteristic
62#14519716—21no—
63C2×#11758185C2×C411yescharacteristic
64C16×D61—21no—
65C16×D61—21no—
66C16×D61—21no—
67#131×C61C2×C411yescharacteristic
68M32×C61C2×C411yescharacteristic
69C2×C16×C61C2×C411yescharacteristic
70#41×C61C4⋊C411yescharacteristic
71#11758226D811yescharacteristic
72Q8×D61Q1611yescharacteristic
73#11758224—21no—
74#11758224D811yescharacteristic
75D4×D61—21no—
76D4×D61Q1611yescharacteristic
77#11758222—21no—
78#11758222C4⋊C411yescharacteristic
79C2×C4×D61—21no—
80C2×C4×D61C2×D411yescharacteristic
81C22⋊#6923—41no—
82#11758217—42no—
83#11758217—41no—
84C2×#92420—42no—
85C2×#92420—41no—
86C2×#92420—41no—
87C4⋊#6923—41no—
88C4×#6923—41no—
89#11758213—21no—
90#11758213—22no—
91#11758213—21no—
92#11758213—42no—
93#11758213—41no—
94#11758213—41no—
95#11758213—41no—
96D61⋊C8—42no—
97D61⋊C8—41no—
98M16×C61—21no—
99M16×C61C4⋊C411yescharacteristic
100C2×C8×C61—21no—
101C2×C8×C61C22×C411yescharacteristic
102#11758194—21no—
103#11758194C4⋊C411yescharacteristic
104C2×#92418—21no—
105C2×#92418C2×Q811yescharacteristic
106#11758189—21no—
107#11758189—21no—
108#11758189—21no—
109C8×D61—21no—
110C8×D61—21no—
111C8×D61—21no—
112C8×D61C2×D411yescharacteristic
113C8×D61C2×D411yescharacteristic
114C8×D61#4111yescharacteristic
115C8×D61#4111yescharacteristic
116C16×C61—21no—
117C16×C61—21no—
118#11758185—21no—
119#11758185—21no—
120C22×D61—21no—
121C22×D61#13711yescharacteristic
122C2×#6923—81no—
123Q8×C61#10711yescharacteristic
124D4×C61—21no—
125D4×C61#10711yescharacteristic
126C2×C4×C61—21no—
127C2×C4×C61C2×C4⋊C411yescharacteristic
128C22⋊D61—21no—
129C22⋊D61—21no—
130C22⋊D61—21no—
131C2×Dic61—21no—
132C2×Dic61AGL(1,ℤ/8)11yescharacteristic
133D244—21no—
134D244#10711yescharacteristic
135C4×D61—21no—
136C4×D61—21no—
137C4×D61—21no—
138C4×D61#12011yescharacteristic
139C4×D61#12311yescharacteristic
140C4×D61C2×D811yescharacteristic
141C4×D61C2×Q1611yescharacteristic
142Dic122—21no—
143Dic122#10711yescharacteristic
144#92420—41no—
145#92420—42no—
146#92420—41no—
147#92420—81no—
148C8×C61—21no—
149C8×C61C4×D411yescharacteristic
150C8×C61C4×D411yescharacteristic
151#92418—21no—
152#92418C22⋊Q811yescharacteristic
153#92418C22⋊Q811yescharacteristic
154#12927—611nomaximal, Sylow
155C22×C61—21no—
156C22×C61#42811yescharacteristic
157C2×D61—21no—
158C2×D61—41no—
159C2×D61—81no—
160C2×D61#45111yescharacteristic
161C2×D61#48011yescharacteristic
162#6923—161no—
163C4×C61—21no—
164C4×C61#38511yescharacteristic
165C4×C61C2×#10711yescharacteristic
166Dic61—21no—
167Dic61#44711yescharacteristic
168Dic61#48411yescharacteristic
169#2916—611no—
170#2161—611no—
171C2×#368—611no—
172C2×#350—611no—
173#1814—611no—
174#1389—611no—
175#1382—611no—
176C2×C61—21no—
177C2×C61—41no—
178C2×C61#185211yescharacteristic
179D61—21no—
180D61—21no—
181D61—41no—
182#568—1221no—
183C2×#131—611no—
184C2×M32—611no—
185C22×C16—611no—
186#430—1222no—
187#430—1221no—
188C2×#108—611no—
189C2×#108—611no—
190C2×#108—611no—
191C2×#108—1221no—
192C2×#108—1221no—
193#427—611no—
194#421—1221no—
195#368—1222no—
196#368—1221no—
197#368—1221no—
198#350—611no—
199#350—611no—
200#350—611no—
201#350#692311yescharacteristic, complemented
202#344—1221no—
203C61#1292711yescharacteristic, complemented, Sylow
204C2×#41—611no—
205#131—611no—
206#131—611no—
207#131—611no—
208#131—1221no—
209#131—1221no—
210#131—2441no—
211#131—2441no—
212#131—2442no—
213#131C2×#692311yescharacteristic
214C2×M16—611no—
215C2×M16—1221no—
216C2×M16—1221no—
217C2×M16—1221no—
218C2×M16—1222no—
219C2×M16—1222no—
220C2×M16—1221no—
221C22×C8—611no—
222C22×C8—1221no—
223#117—1221no—
224M32—611no—
225M32—1221no—
226M32C2×#692311yescharacteristic
227C2×C16—611no—
228C2×C16—1221no—
229C2×C16—1221no—
230C2×C16—1221no—
231C2×C16C2×#692311yescharacteristic
232#108—611no—
233#108—611no—
234#108—612no—
235#108—1221no—
236#108—1221no—
237#108—1221no—
238#108—1221no—
239#108—1222no—
240#108—1222no—
241#107—1221no—
242#106—1221no—
243#104—2441no—
244#104—2441no—
245#41—611no—
246#41—611no—
247#41—611no—
248#41—1221no—
249#41—1221no—
250#41C4⋊#692311yescharacteristic
251C2×Q8—611no—
252C2×D4—611no—
253C2×D4—1221no—
254C22×C4—611no—
255C22×C4—1221no—
256M16—21no—
257M16—611no—
258M16—1221no—
259M16—1221no—
260M16—1221no—
261M16—1221no—
262M16—1221no—
263M16—1221no—
264M16—1222no—
265M16—2441no—
266M16—2441no—
267M16—2441no—
268M16—2441no—
269M16—2442no—
270M16—2442no—
271M16C4⋊#692311yescharacteristic
272C2×C8—21no—
273C2×C8—611no—
274C2×C8—611no—
275C2×C8—611no—
276C2×C8—611no—
277C2×C8—611no—
278C2×C8—1221no—
279C2×C8—1221no—
280C2×C8—1221no—
281C2×C8—1221no—
282C2×C8—2441no—
283C2×C8—2441no—
284C2×C8—2441no—
285C2×C8—2442no—
286C2×C8—2442no—
287C2×C8C22×#692311yescharacteristic
288C4⋊C4—2441no—
289C22⋊C4—2441no—
290C42—2441no—
291C16—21no—
292C16—21no—
293C16—1221no—
294C16—1221no—
295C23—611no—
296C23—1221no—
297Q8—611no—
298Q8—1221no—
299Q8#1451984611yescharacteristic
300D4—21no—
301D4—611no—
302D4—1221no—
303D4—1221no—
304D4—1221no—
305D4—1221no—
306D4#1451984611yescharacteristic
307C2×C4—21no—
308C2×C4—611no—
309C2×C4—611no—
310C2×C4—611no—
311C2×C4—611no—
312C2×C4—611no—
313C2×C4—1221no—
314C2×C4—1221no—
315C2×C4—1221no—
316C2×C4—1221no—
317C2×C4—4881no—
318C2×C4C2×C4⋊#692311yescharacteristic
319C8—21no—
320C8—611no—
321C8—611no—
322C8—1221no—
323C8—4881no—
324C8—2441no—
325C8—2441no—
326C8—2442no—
327C8D4×#692311yescharacteristic
328C8D4×#692311yescharacteristic
329C22—21no—
330C22—611no—
331C22—611no—
332C22—1221no—
333C22—4881no—
334C22—2441no—
335C22#1684077311yescharacteristic
336C4—21no—
337C4—611no—
338C4—611no—
339C4—1221no—
340C4—9761no—
341C4#1684070911yescharacteristic
342C4C2×#1451984611yescharacteristic
343C2—21no—
344C2—41no—
345C2—1221no—
346C2—1221no—
347C2—2441no—
348C2#2146176811yescharacteristic
subgroupG/H
centerC2#21461768
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC61×#350C4
socleC2×C61#1852
prime radical#41×D61D4
cosocleC2×C8×D61C23
hypercenter#350#6923
hypocenterC61#12927
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #350

Complements

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

NH ≅ G/Ntype
2. #21462175C2semidirect
3. C2×#16840691C2semidirect
5. #21460900C2semidirect
6. #21460881C2semidirect
25. #16839547C4semidirect
26. #16839546C4semidirect
27. #16839545C4semidirect
28. C61×#350C4semidirect
201. #350#6923semidirect
203. C61#12927semidirect

Maximal subgroups

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

subgroupindex
#214624812
#214621752
C2×#168406912
#214617062
#214609002
#214608812
#350×D612
#1292761