Finite Group Databasebeta

← #29596884

Subgroups — #29596884

Order 15616.

456 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458921C211yesmaximal, characteristic, complemented
2#21454723C211yesmaximal, characteristic
3#21447446C211yesmaximal, characteristic, complemented
4C2×#16839547C211yesmaximal, characteristic, complemented
5#21447434C211yesmaximal, characteristic, complemented
6#21443908C211yesmaximal, characteristic, complemented
7C61×#1378C211yesmaximal, characteristic, complemented
8#16840562C411yescharacteristic
9C2×#14519783C2211yescharacteristic
10C2×#14519783C2211yescharacteristic
11#16840180—22no—
12#16840178—22no—
13C2×#14519717C2211yescharacteristic
14M32×D61—22no—
15C22×#11758185C2211yescharacteristic
16#16839885—22no—
17#16839880—22no—
18#16839811C411yescharacteristic
19#16839780C411yescharacteristic
20#16839780C411yescharacteristic
21#16839547—22no—
22#16839545—22no—
23#16839541—22no—
24C2×M32×C61C2211yescharacteristic
25C2×M32×C61C2211yescharacteristic
26C61×#405C2211yescharacteristic
27C2×#11758222C2×C411yescharacteristic
28C2×#11758222C2×C411yescharacteristic
29#14519810—21no—
30#14519810C812yes—
31#14519805—21no—
32#14519805C812yes—
33#14519799—22no—
34#14519783—21no—
35#14519783—21no—
36#14519783—21no—
37#14519783—21no—
38#14519783—21no—
39#14519783—21no—
40#14519783D412yes—
41#14519783D412yes—
42C2×#11758194C2×C411yescharacteristic
43C2×#11758194C2×C411yescharacteristic
44C2×#11758189—21no—
45C2×#11758189—21no—
46C2×C8×D61—21no—
47C2×C8×D61—21no—
48#14519717—21no—
49#14519717D412yes—
50#14519716—24no—
51C2×#11758185—21no—
52C2×#11758185—21no—
53C2×#11758185—21no—
54C2×#11758185—21no—
55C16×D61—24no—
56#14519713—22no—
57#14519705—22no—
58#14519705—22no—
59C2×M16×C61C2×C411yescharacteristic
60C22×C8×C61C2311yescharacteristic
61#117×C61C2×C411yescharacteristic
62M32×C61—21no—
63M32×C61—21no—
64M32×C61—21no—
65M32×C61—21no—
66C2×C16×C61—21no—
67C2×C16×C61D412yes—
68#97×C61—21no—
69C4×C8×C61—21no—
70C22×C4×C61C22×C411yescharacteristic
71C2×C22⋊D61—21no—
72C2×C22⋊D61—21no—
73#11758222—21no—
74#11758222—21no—
75#11758222—21no—
76#11758222—21no—
77#11758222—21no—
78#11758222—21no—
79#11758222C22⋊C412yes—
80#11758222C2×C812yes—
81C2×D244C2×C811yescharacteristic
82C2×D244M1611yescharacteristic
83C2×C4×D61—21no—
84C2×C4×D61—21no—
85C2×Dic122C2×C811yescharacteristic
86C2×Dic122M1611yescharacteristic
87M16×C61—21no—
88M16×C61C2×C812yes—
89C2×C8×C61—21no—
90C2×C8×C61C2×D411yescharacteristic
91C2×C8×C61C2×D411yescharacteristic
92C2×C8×C61C2×D411yescharacteristic
93C2×C8×C61C2×D411yescharacteristic
94C2×C8×C61#4111yescharacteristic
95C2×C8×C61#4111yescharacteristic
96C4⋊C4×C61—21no—
97C22⋊C4×C61—21no—
98C42×C61—21no—
99#11758194—21no—
100#11758194—21no—
101#11758194—21no—
102#11758194C22⋊C412yes—
103C2×#92418—21no—
104C2×#92418—21no—
105Dic244—22no—
106D488—22no—
107#11758190—22no—
108#11758190—22no—
109#11758189—22no—
110#11758189—22no—
111#11758189—41no—
112#11758189—41no—
113C8×D61—22no—
114C8×D61—22no—
115C8×D61—41no—
116C8×D61—41no—
117C16×C61—22no—
118C16×C61—41no—
119#11758185—22no—
120#11758185—41no—
121C23×C61#13111yescharacteristic
122C22×D61—21no—
123C22×D61—21no—
124C2×C4×C61—41no—
125C2×C4×C61C2×C22⋊C411yescharacteristic
126C2×C4×C61C4×D411yescharacteristic
127C2×C4×C61C4×D411yescharacteristic
128C2×C4×C61C22×C811yescharacteristic
129C2×C4×C61C2×M1611yescharacteristic
130C2×C4×C61#13111yescharacteristic
131C22⋊D61—22no—
132C22⋊D61—22no—
133C22⋊D61—41no—
134C22⋊D61—41no—
135C2×Dic61—21no—
136C2×Dic61—21no—
137D244—21no—
138D244—21no—
139D244—21no—
140D244C22⋊C812yes—
141C4×D61—22no—
142C4×D61—22no—
143C4×D61—41no—
144C4×D61—41no—
145Dic122—21no—
146Dic122—21no—
147Dic122—21no—
148Dic122C22⋊C812yes—
149C8×C61—22no—
150C8×C61#12011yescharacteristic
151C8×C61#12111yescharacteristic
152C8×C61#12111yescharacteristic
153C8×C61#12311yescharacteristic
154#92418—22no—
155#92418—41no—
156#12979—611nomaximal, Sylow
157C22×C61—41no—
158C22×C61#40811yescharacteristic
159C22×C61C8×D411yescharacteristic
160C22×C61#43511yescharacteristic
161C2×D61—22no—
162C2×D61—42no—
163C2×D61—41no—
164C2×D61—81no—
165C4×C61—81no—
166C4×C61#38611yescharacteristic
167C4×C61C2×C22⋊C811yescharacteristic
168C4×C61C8×D411yescharacteristic
169C4×C61#43511yescharacteristic
170Dic61—22no—
171Dic61—41no—
172#3407—611no—
173#2955—611no—
174#2117—611no—
175C2×#350—611no—
176#2113—611no—
177#1379—611no—
178#1378D6111yescharacteristic, complemented
179C2×C61—21no—
180C2×C61—21no—
181C2×C61—21no—
182C2×C61#186911yescharacteristic
183D61—42no—
184D61—81no—
185#575—611no—
186C2×#131—611no—
187C2×#131—611no—
188#504—1222no—
189#504—1222no—
190C2×M32—611no—
191C2×M32—1222no—
192C2×M32C2×D6111yescharacteristic
193C2×M32C2×D6111yescharacteristic
194C22×C16—611no—
195#444—1222no—
196#443—1222no—
197#430—611no—
198#421—611no—
199#421—611no—
200#405C2×D6111yescharacteristic
201#350—1222no—
202#350—1222no—
203C22⋊C16—1222no—
204C61#1297911yescharacteristic, complemented, Sylow
205C2×#41—611no—
206C2×#41—611no—
207#137—612no—
208#137—1221no—
209AGL(1,ℤ/8)—612no—
210AGL(1,ℤ/8)—1221no—
211#135—1222no—
212#131—612no—
213#131—612no—
214#131—1221no—
215#131—1221no—
216#131—1221no—
217#131—1221no—
218#131—1221no—
219#131—1221no—
220C2×M16—611no—
221C2×M16—611no—
222C2×M16—1221no—
223C2×M16—1221no—
224C2×M16C4×D6111yescharacteristic
225C22×C8—1221no—
226C22×C8—1221no—
227C22×C8C22×D6111yescharacteristic
228#117C4×D6111yescharacteristic
229M32—21no—
230M32—21no—
231M32—21no—
232M32—21no—
233M32—612no—
234M32—1224no—
235M32—1221no—
236C2×C16—21no—
237C2×C16—1224no—
238C2×C16—1221no—
239C2×C16—1221no—
240C2×C16—1221no—
241C2×C16—1221no—
242C2×C16D24412yes—
243#108—1222no—
244#104—1222no—
245#104—1222no—
246#97—21no—
247C4×C8—21no—
248#41—612no—
249#41—612no—
250#41—1221no—
251#41—1221no—
252#41—1221no—
253#41—1221no—
254#41—1221no—
255#41—1221no—
256C2×Q8—611no—
257C2×Q8—611no—
258C2×D4—611no—
259C2×D4—611no—
260C2×D4—1221no—
261C2×D4—1221no—
262C22×C4—1221no—
263C22×C4—1221no—
264C22×C4C2×C4×D6111yescharacteristic
265Q16—1222no—
266SD16—1222no—
267SD16—1222no—
268D8—1222no—
269M16—21no—
270M16—612no—
271M16—1221no—
272M16—1222no—
273M16—1222no—
274M16—1221no—
275M16—1221no—
276M16—2441no—
277M16—2441no—
278M16C8×D6112yes—
279C2×C8—21no—
280C2×C8—1221no—
281C2×C8—1222no—
282C2×C8—1222no—
283C2×C8—1221no—
284C2×C8—2441no—
285C2×C8—2441no—
286C2×C8C2×D24411yescharacteristic
287C2×C8#1175822211yescharacteristic
288C2×C8D4×D6111yescharacteristic
289C2×C8D4×D6111yescharacteristic
290C2×C8D4×D6111yescharacteristic
291C2×C8#1175822611yescharacteristic
292C4⋊C4—21no—
293C22⋊C4—21no—
294C42—21no—
295C16—22no—
296C16—41no—
297C16—1222no—
298C16—2441no—
299C23—1221no—
300C23—1221no—
301C23#1451978311yescharacteristic
302Q8—612no—
303Q8—1221no—
304Q8—1221no—
305Q8—1221no—
306D4—612no—
307D4—1222no—
308D4—1222no—
309D4—1221no—
310D4—1221no—
311D4—1221no—
312D4—2441no—
313D4—2441no—
314C2×C4—41no—
315C2×C4—1221no—
316C2×C4—1222no—
317C2×C4—1222no—
318C2×C4—1221no—
319C2×C4—2441no—
320C2×C4—2441no—
321C2×C4C22⋊C4×D6111yescharacteristic
322C2×C4#1451973911yescharacteristic
323C2×C4C4×D24411yescharacteristic
324C2×C4C2×C8×D6111yescharacteristic
325C2×C4M16×D6111yescharacteristic
326C2×C4#1451978511yescharacteristic
327C8—22no—
328C8—1222no—
329C8—2441no—
330C8#1451974711yescharacteristic
331C8#1451975011yescharacteristic
332C8#1451975211yescharacteristic
333C8#1451976211yescharacteristic
334C22—41no—
335C22—1222no—
336C22—4881no—
337C22—2442no—
338C22—2441no—
339C22#1683972711yescharacteristic
340C22C8×D24411yescharacteristic
341C22#1683983211yescharacteristic
342C4—81no—
343C4—1222no—
344C4—2441no—
345C4#1683964011yescharacteristic
346C4C22⋊C8×D6111yescharacteristic
347C4#1683982711yescharacteristic
348C4#1683983511yescharacteristic
349C2—21no—
350C2—21no—
351C2—21no—
352C2—4881no—
353C2—2442no—
354C2#2144611011yescharacteristic
subgroupG/H
centerC8#14519747
commutatorC2×C4×C61C22×C8
FrattiniC22×C8C22×D61
FittingC61×#1378C2
socleC2×C61#1869
prime radical#16840562C4
cosocleC22×C8×C61C23
hypercenter#1378D61
hypocenterC61#12979
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C8 ◁ C22×C8 ◁ #1378

Complements

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

NH ≅ G/Ntype
1. #21458921C2semidirect
3. #21447446C2semidirect
4. C2×#16839547C2semidirect
5. #21447434C2semidirect
6. #21443908C2semidirect
7. C61×#1378C2semidirect
178. #1378D61semidirect
204. C61#12979semidirect

Maximal subgroups

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

subgroupindex
#214589212
#214547232
#214474462
C2×#168395472
#214474342
#214439082
C61×#13782
#1297961