Finite Group Databasebeta

← #29597436

Subgroups — #29597436

Order 15616.

456 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458922C211yesmaximal, characteristic, complemented
2C2×#16839572C211yesmaximal, characteristic, complemented
3#21447547C211yesmaximal, characteristic, complemented
4#21445878C211yesmaximal, characteristic, complemented
5#21443930C211yesmaximal, characteristic, complemented
6#21443919C211yesmaximal, characteristic, complemented
7C61×#3067C211yesmaximal, characteristic
8#16840179—22no—
9#16840179—22no—
10C2×#14519717—22no—
11C2×#14519717C2211yescharacteristic
12C2×#14519717C2211yescharacteristic
13C2×#14519713C2211yescharacteristic
14C2×#14519709C2211yescharacteristic
15#16839763—21no—
16#16839572—22no—
17#16839569—22no—
18#16839530—21no—
19C61×#575—21no—
20C2×#135×C61C2211yescharacteristic
21C2×#131×C61C2211yescharacteristic
22C61×#473C412yescomplemented
23C61×#472C412yescomplemented
24C61×#471—22no—
25C2×#108×C61C2211yescharacteristic
26C61×#413—21no—
27C2×#11758194—21no—
28C2×C4⋊Dic61—21no—
29#14519717—24no—
30#14519717—21no—
31#14519717—21no—
32#14519717—21no—
33#14519717D412yes—
34C2×#11758185—24no—
35C2×#11758185—21no—
36C2×#11758185D412yes—
37#14519713—21no—
38#14519713—21no—
39#14519709—21no—
40#14519709—21no—
41#14519701—41no—
42#14519698—41no—
43C2×#41×C61—21no—
44C2×#41×C61D411yescharacteristic
45#137×C61—22no—
46#137×C61—41no—
47AGL(1,ℤ/8)×C61—22no—
48AGL(1,ℤ/8)×C61—41no—
49#135×C61—21no—
50#135×C61—21no—
51#135×C61—21no—
52#135×C61—21no—
53#135×C61—42no—
54C2×Q16×C61C2×C411yescharacteristic
55C2×SD16×C61—21no—
56C2×D8×C61C2×C411yescharacteristic
57#131×C61—21no—
58#131×C61—21no—
59#131×C61C2×C412yes—
60#131×C61D412yes—
61C2×M16×C61—21no—
62C2×M16×C61—21no—
63C2×M16×C61D411yescharacteristic
64C22×C8×C61—21no—
65C22×C8×C61C2311yescharacteristic
66#108×C61—21no—
67#108×C61C2×C412yes—
68#101×C61—22no—
69#100×C61—22no—
70#41×C61—21no—
71#41×C61—21no—
72#41×C61—41no—
73#41×C61—41no—
74#41×C61—41no—
75#41×C61—41no—
76#41×C61C22⋊C412yes—
77#41×C61SD1612yes—
78C2×Q8×C61—21no—
79C2×Q8×C61D811yescharacteristic
80C2×D4×C61—21no—
81C2×D4×C61—21no—
82C2×D4×C61—41no—
83C2×D4×C61D811yescharacteristic
84C22×C4×C61—21no—
85C22×C4×C61—41no—
86C22×C4×C61C2×D411yescharacteristic
87C22×Dic61—41no—
88Q16×C61—21no—
89Q16×C61—21no—
90Q16×C61—42no—
91SD16×C61—42no—
92SD16×C61—42no—
93SD16×C61—41no—
94SD16×C61—41no—
95D8×C61—21no—
96D8×C61—21no—
97D8×C61—42no—
98M16×C61—21no—
99M16×C61—22no—
100M16×C61—22no—
101M16×C61—22no—
102M16×C61—41no—
103M16×C61C22⋊C412yes—
104C2×C8×C61—21no—
105C2×C8×C61—22no—
106C2×C8×C61—22no—
107C2×C8×C61—41no—
108C2×C8×C61C22×C411yescharacteristic
109C2×C8×C61C2×D411yescharacteristic
110C2×C8×C61C2×D411yescharacteristic
111C2×C8×C61C2×D411yescharacteristic
112C2×C8×C61#4111yescharacteristic
113C2×C8×C61#4111yescharacteristic
114C4⋊Dic61—41no—
115C4⋊Dic61—41no—
116#11758194—41no—
117#11758194—41no—
118C2×#92418—41no—
119#11758185—22no—
120#11758185—22no—
121C23×C61—21no—
122C23×C61—41no—
123C23×C61#13511yescharacteristic
124Q8×C61—21no—
125Q8×C61—41no—
126Q8×C61—41no—
127Q8×C61#10212yes—
128D4×C61—22no—
129D4×C61—22no—
130D4×C61—21no—
131D4×C61—41no—
132D4×C61—41no—
133D4×C61—81no—
134D4×C61—81no—
135D4×C61#10212yes—
136C2×C4×C61—21no—
137C2×C4×C61—22no—
138C2×C4×C61—22no—
139C2×C4×C61—41no—
140C2×C4×C61—81no—
141C2×C4×C61—81no—
142C2×C4×C61C2×C22⋊C411yescharacteristic
143C2×C4×C61#12111yescharacteristic
144C2×C4×C61#12111yescharacteristic
145C2×C4×C61C2×D811yescharacteristic
146C2×C4×C61C2×SD1611yescharacteristic
147C2×C4×C61#13511yescharacteristic
148C2×Dic61—81no—
149C2×Dic61—161no—
150C8×C61—22no—
151C8×C61—42no—
152C8×C61C4×D411yescharacteristic
153C8×C61C4×D411yescharacteristic
154C8×C61#12011yescharacteristic
155C8×C61#12311yescharacteristic
156#92418—81no—
157#13070—611nomaximal, Sylow
158C22×C61—22no—
159C22×C61—41no—
160C22×C61—81no—
161C22×C61—81no—
162C22×C61—161no—
163C22×C61#41611yescharacteristic
164C22×C61#46511yescharacteristic
165C22×C61#46611yescharacteristic
166C4×C61—22no—
167C4×C61—81no—
168C4×C61#38611yescharacteristic
169C4×C61C2×#10211yescharacteristic
170C4×C61#46511yescharacteristic
171C4×C61#46611yescharacteristic
172Dic61—161no—
173#3407—611no—
174#3067D6111yescharacteristic
175C2×#360—611no—
176#2143—611no—
177#1815—611no—
178#1388—611no—
179#1383—611no—
180C2×C61—21no—
181C2×C61—21no—
182C2×C61—21no—
183C2×C61—42no—
184C2×C61—161no—
185C2×C61#189311yescharacteristic
186#575—21no—
187C2×#135C2×D6111yescharacteristic
188C2×#131C2×D6111yescharacteristic
189#504—1222no—
190#504—1222no—
191C2×M32—611no—
192C2×M32—611no—
193C2×M32—1222no—
194#473Dic6112yescomplemented
195#472Dic6112yescomplemented
196#471—22no—
197C2×#108—611no—
198C2×#108C2×D6111yescharacteristic
199C2×#106—611no—
200#417—1221no—
201#413—21no—
202#360—1222no—
203#359—1222no—
204#343—1221no—
205C61#1307011yescharacteristic, complemented, Sylow
206C2×#41—21no—
207C2×#41C22⋊D6111yescharacteristic
208#137—22no—
209#137—41no—
210AGL(1,ℤ/8)—22no—
211AGL(1,ℤ/8)—41no—
212#135—21no—
213#135—21no—
214#135—21no—
215#135—21no—
216#135—42no—
217C2×Q16C2×Dic6111yescharacteristic
218C2×SD16—21no—
219C2×D8C2×Dic6111yescharacteristic
220#131—21no—
221#131—21no—
222#131C2×Dic6112yes—
223#131C22⋊D6112yes—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—1221no—
227C2×M16C22⋊D6111yescharacteristic
228C22×C8—21no—
229C22×C8C22×D6111yescharacteristic
230C2×C4⋊C4—1221no—
231M32—612no—
232M32—1224no—
233M32—1221no—
234M32—1221no—
235M32—1221no—
236C2×C16—612no—
237C2×C16—1224no—
238C2×C16—1221no—
239#108—21no—
240#108—1221no—
241#108—1221no—
242#108C2×Dic6112yes—
243#106—1221no—
244#106—1221no—
245#103—2441no—
246#102—2441no—
247#101—22no—
248#100—22no—
249#41—21no—
250#41—21no—
251#41—41no—
252#41—41no—
253#41—41no—
254#41—41no—
255#41#1175820012yes—
256#41C22⋊Dic6112yes—
257C2×Q8—21no—
258C2×Q8#1175819911yescharacteristic
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—41no—
262C2×D4#1175819911yescharacteristic
263C22×C4—21no—
264C22×C4—41no—
265C22×C4—2441no—
266C22×C4C2×C22⋊D6111yescharacteristic
267Q16—21no—
268Q16—21no—
269Q16—42no—
270SD16—42no—
271SD16—42no—
272SD16—41no—
273SD16—41no—
274D8—21no—
275D8—21no—
276D8—42no—
277M16—21no—
278M16—22no—
279M16—22no—
280M16—22no—
281M16—41no—
282M16—2441no—
283M16—2441no—
284M16C22⋊Dic6112yes—
285C2×C8—21no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—2441no—
290C2×C8D4×D6111yescharacteristic
291C2×C8D4×D6111yescharacteristic
292C2×C8#1175822411yescharacteristic
293C2×C8#1175822411yescharacteristic
294C2×C8C22×Dic6111yescharacteristic
295C2×C8C2×C22⋊D6111yescharacteristic
296C4⋊C4—2441no—
297C4⋊C4—2441no—
298C16—1222no—
299C16—1222no—
300C23—21no—
301C23—41no—
302C23#1451979711yescharacteristic
303Q8—21no—
304Q8—41no—
305Q8—41no—
306Q8#1451969812yes—
307D4—22no—
308D4—22no—
309D4—21no—
310D4—41no—
311D4—41no—
312D4—81no—
313D4—81no—
314D4#1451969812yes—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—9761no—
322C2×C4—4881no—
323C2×C4C2×C22⋊Dic6111yescharacteristic
324C2×C4#1451974911yescharacteristic
325C2×C4#1451974911yescharacteristic
326C2×C4C2×#1175819911yescharacteristic
327C2×C4C2×#1175820011yescharacteristic
328C2×C4#1451979711yescharacteristic
329C8—22no—
330C8—42no—
331C8—4881no—
332C8D4×Dic6111yescharacteristic
333C8D4×Dic6111yescharacteristic
334C8#1451974511yescharacteristic
335C8#1451975811yescharacteristic
336C22—22no—
337C22—41no—
338C22—81no—
339C22—81no—
340C22—161no—
341C22#1683975811yescharacteristic
342C22#1683998911yescharacteristic
343C22#1683999611yescharacteristic
344C4—22no—
345C4—81no—
346C4—9761no—
347C4#1683964211yescharacteristic
348C4C2×#1451969811yescharacteristic
349C4#1683998911yescharacteristic
350C4#1683999611yescharacteristic
351C2—21no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—161no—
356C2#2144624811yescharacteristic
subgroupG/H
centerC2#21446248
commutatorC2×C8×C61C22×C4
FrattiniC22×C8C22×D61
Fitting, prime radicalC61×#3067C2
socleC2×C61#1893
cosocleC22×C8×C61C23
hypercenter#3067D61
hypocenterC61#13070
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #3067

Complements

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

NH ≅ G/Ntype
1. #21458922C2semidirect
2. C2×#16839572C2semidirect
3. #21447547C2semidirect
4. #21445878C2semidirect
5. #21443930C2semidirect
6. #21443919C2semidirect
22. C61×#473C4semidirect
23. C61×#472C4semidirect
194. #473Dic61semidirect
195. #472Dic61semidirect
205. C61#13070semidirect

Maximal subgroups

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

subgroupindex
#214589222
C2×#168395722
#214475472
#214458782
#214439302
#214439192
C61×#30672
#1307061