Finite Group Databasebeta

← #29596930

Subgroups — #29596930

Order 15616.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840180C211yesmaximal, characteristic, complemented
2#21454723C211yesmaximal, characteristic
3#21447448C211yesmaximal, characteristic, complemented
4C2×#16839547C211yesmaximal, characteristic, complemented
5#21447439C211yesmaximal, characteristic, complemented
6#21443908C211yesmaximal, characteristic, complemented
7C61×#1378C211yesmaximal, characteristic, complemented
8#16840562C411yescharacteristic
9C2×#14519783C2211yescharacteristic
10C2×#14519783C2211yescharacteristic
11#16840180—22no—
12#16840180—22no—
13C2×#14519717C2211yescharacteristic
14C2×#14519716—21no—
15C22×#11758185C2211yescharacteristic
16C2×C16×D61—21no—
17#16839885—22no—
18#16839880—22no—
19#16839811C411yescharacteristic
20#16839780C411yescharacteristic
21#16839780C411yescharacteristic
22#16839547—22no—
23#16839547—21no—
24#16839547—21no—
25#16839543—22no—
26C2×M32×C61C2211yescharacteristic
27C2×M32×C61C2211yescharacteristic
28C61×#405C2211yescharacteristic
29C2×#11758222C2×C411yescharacteristic
30C2×#11758222C2×C411yescharacteristic
31#14519810—21no—
32#14519810C812yes—
33#14519805—21no—
34#14519805C812yes—
35#14519799—22no—
36#14519783—21no—
37#14519783—21no—
38#14519783—21no—
39#14519783—21no—
40#14519783—21no—
41#14519783—21no—
42#14519783D412yes—
43#14519783D412yes—
44C2×#11758194C2×C411yescharacteristic
45C2×#11758194C2×C411yescharacteristic
46C2×#11758189—21no—
47C2×#11758189—21no—
48C2×C8×D61—21no—
49C2×C8×D61—21no—
50#14519717—21no—
51#14519717—21no—
52#14519716—24no—
53C2×#11758185—21no—
54C2×#11758185—21no—
55C2×#11758185—21no—
56C2×#11758185D411yescharacteristic
57C2×#11758185D411yescharacteristic
58C16×D61—24no—
59#14519713—22no—
60#14519705—22no—
61#14519705—22no—
62C2×M16×C61C2×C411yescharacteristic
63C22×C8×C61C2311yescharacteristic
64#117×C61C2×C411yescharacteristic
65M32×C61—21no—
66M32×C61—21no—
67M32×C61—21no—
68M32×C61D412yes—
69C2×C16×C61—21no—
70C2×C16×C61—21no—
71#97×C61—21no—
72C4×C8×C61—21no—
73C22×C4×C61C22×C411yescharacteristic
74C2×C22⋊D61—21no—
75C2×C22⋊D61—21no—
76#11758222—21no—
77#11758222—21no—
78#11758222—21no—
79#11758222—21no—
80#11758222—21no—
81#11758222—21no—
82#11758222C22⋊C412yes—
83#11758222C2×C812yes—
84C2×D244C2×C811yescharacteristic
85C2×D244M1611yescharacteristic
86C2×C4×D61—21no—
87C2×C4×D61—21no—
88C2×Dic122C2×C811yescharacteristic
89C2×Dic122M1611yescharacteristic
90M16×C61—21no—
91M16×C61C2×C812yes—
92C2×C8×C61—21no—
93C2×C8×C61C2×D411yescharacteristic
94C2×C8×C61C2×D411yescharacteristic
95C2×C8×C61C2×D411yescharacteristic
96C2×C8×C61C2×D411yescharacteristic
97C2×C8×C61#4111yescharacteristic
98C2×C8×C61#4111yescharacteristic
99C4⋊C4×C61—21no—
100C22⋊C4×C61—21no—
101C42×C61—21no—
102#11758194—21no—
103#11758194—21no—
104#11758194—21no—
105#11758194C22⋊C412yes—
106C2×#92418—21no—
107C2×#92418—21no—
108Dic244—22no—
109D488—22no—
110#11758190—22no—
111#11758190—22no—
112#11758189—22no—
113#11758189—22no—
114#11758189—41no—
115#11758189—41no—
116C8×D61—22no—
117C8×D61—22no—
118C8×D61—41no—
119C8×D61—41no—
120C16×C61—22no—
121C16×C61—41no—
122#11758185—21no—
123#11758185—21no—
124#11758185—41no—
125C23×C61#13111yescharacteristic
126C22×D61—21no—
127C22×D61—21no—
128C2×C4×C61—41no—
129C2×C4×C61C2×C22⋊C411yescharacteristic
130C2×C4×C61C4×D411yescharacteristic
131C2×C4×C61C4×D411yescharacteristic
132C2×C4×C61C22×C811yescharacteristic
133C2×C4×C61C2×M1611yescharacteristic
134C2×C4×C61#13111yescharacteristic
135C22⋊D61—22no—
136C22⋊D61—22no—
137C22⋊D61—41no—
138C22⋊D61—41no—
139C2×Dic61—21no—
140C2×Dic61—21no—
141D244—21no—
142D244—21no—
143D244—21no—
144D244C22⋊C812yes—
145C4×D61—22no—
146C4×D61—22no—
147C4×D61—41no—
148C4×D61—41no—
149Dic122—21no—
150Dic122—21no—
151Dic122—21no—
152Dic122C22⋊C812yes—
153C8×C61—22no—
154C8×C61#12011yescharacteristic
155C8×C61#12111yescharacteristic
156C8×C61#12111yescharacteristic
157C8×C61#12311yescharacteristic
158#92418—22no—
159#92418—41no—
160#12987—611nomaximal, Sylow
161C22×C61—41no—
162C22×C61#40811yescharacteristic
163C22×C61C8×D411yescharacteristic
164C22×C61#43511yescharacteristic
165C2×D61—22no—
166C2×D61—42no—
167C2×D61—41no—
168C2×D61—81no—
169C4×C61—81no—
170C4×C61#38611yescharacteristic
171C4×C61C2×C22⋊C811yescharacteristic
172C4×C61C8×D411yescharacteristic
173C4×C61#43511yescharacteristic
174Dic61—22no—
175Dic61—41no—
176C2×#504—611no—
177#2955—611no—
178#2117—611no—
179C2×#350—611no—
180#2115—611no—
181#1379—611no—
182#1378D6111yescharacteristic, complemented
183C2×C61—21no—
184C2×C61—21no—
185C2×C61—21no—
186C2×C61#186911yescharacteristic
187D61—42no—
188D61—81no—
189#575—611no—
190C2×#131—611no—
191C2×#131—611no—
192#504—1222no—
193#504—1222no—
194C2×M32—611no—
195C2×M32—1221no—
196C2×M32C2×D6111yescharacteristic
197C2×M32C2×D6111yescharacteristic
198C22×C16—611no—
199C22×C16—1221no—
200#444—1222no—
201#443—1222no—
202#430—611no—
203#421—611no—
204#421—611no—
205#405C2×D6111yescharacteristic
206#350—1221no—
207#350—1221no—
208#350—1222no—
209#349—1222no—
210C61#1298711yescharacteristic, complemented, Sylow
211C2×#41—611no—
212C2×#41—611no—
213#137—612no—
214#137—1221no—
215AGL(1,ℤ/8)—612no—
216AGL(1,ℤ/8)—1221no—
217#135—1222no—
218#131—612no—
219#131—612no—
220#131—1221no—
221#131—1221no—
222#131—1221no—
223#131—1221no—
224#131—1221no—
225#131—1221no—
226C2×M16—611no—
227C2×M16—611no—
228C2×M16—1221no—
229C2×M16—1221no—
230C2×M16C4×D6111yescharacteristic
231C22×C8—1221no—
232C22×C8—1221no—
233C22×C8C22×D6111yescharacteristic
234#117C4×D6111yescharacteristic
235M32—21no—
236M32—21no—
237M32—21no—
238M32—1224no—
239M32—1221no—
240M32—1221no—
241M32D24412yes—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—611no—
245C2×C16—611no—
246C2×C16—1224no—
247C2×C16—1221no—
248C2×C16—1221no—
249C2×C16—1221no—
250#108—1222no—
251#104—1222no—
252#104—1222no—
253#97—21no—
254C4×C8—21no—
255#41—612no—
256#41—612no—
257#41—1221no—
258#41—1221no—
259#41—1221no—
260#41—1221no—
261#41—1221no—
262#41—1221no—
263C2×Q8—611no—
264C2×Q8—611no—
265C2×D4—611no—
266C2×D4—611no—
267C2×D4—1221no—
268C2×D4—1221no—
269C22×C4—1221no—
270C22×C4—1221no—
271C22×C4C2×C4×D6111yescharacteristic
272Q16—1222no—
273SD16—1222no—
274SD16—1222no—
275D8—1222no—
276M16—21no—
277M16—612no—
278M16—1221no—
279M16—1222no—
280M16—1222no—
281M16—1221no—
282M16—1221no—
283M16—2441no—
284M16—2441no—
285M16C8×D6112yes—
286C2×C8—21no—
287C2×C8—1221no—
288C2×C8—1222no—
289C2×C8—1222no—
290C2×C8—1221no—
291C2×C8—2441no—
292C2×C8—2441no—
293C2×C8C2×D24411yescharacteristic
294C2×C8#1175822211yescharacteristic
295C2×C8D4×D6111yescharacteristic
296C2×C8D4×D6111yescharacteristic
297C2×C8D4×D6111yescharacteristic
298C2×C8#1175822611yescharacteristic
299C4⋊C4—21no—
300C22⋊C4—21no—
301C42—21no—
302C16—22no—
303C16—41no—
304C16—1221no—
305C16—1221no—
306C16—2441no—
307C23—1221no—
308C23—1221no—
309C23#1451978311yescharacteristic
310Q8—612no—
311Q8—1221no—
312Q8—1221no—
313Q8—1221no—
314D4—612no—
315D4—1222no—
316D4—1222no—
317D4—1221no—
318D4—1221no—
319D4—1221no—
320D4—2441no—
321D4—2441no—
322C2×C4—41no—
323C2×C4—1221no—
324C2×C4—1222no—
325C2×C4—1222no—
326C2×C4—1221no—
327C2×C4—2441no—
328C2×C4—2441no—
329C2×C4C22⋊C4×D6111yescharacteristic
330C2×C4#1451973911yescharacteristic
331C2×C4C4×D24411yescharacteristic
332C2×C4C2×C8×D6111yescharacteristic
333C2×C4M16×D6111yescharacteristic
334C2×C4#1451978511yescharacteristic
335C8—22no—
336C8—1222no—
337C8—2441no—
338C8#1451974711yescharacteristic
339C8#1451975011yescharacteristic
340C8#1451975211yescharacteristic
341C8#1451976211yescharacteristic
342C22—41no—
343C22—1222no—
344C22—4881no—
345C22—2442no—
346C22—2441no—
347C22#1683972711yescharacteristic
348C22C8×D24411yescharacteristic
349C22#1683983211yescharacteristic
350C4—81no—
351C4—1222no—
352C4—2441no—
353C4#1683964011yescharacteristic
354C4C22⋊C8×D6111yescharacteristic
355C4#1683982711yescharacteristic
356C4#1683983511yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—4881no—
361C2—2442no—
362C2#2144611011yescharacteristic
subgroupG/H
centerC8#14519750
commutatorC2×C4×C61C22×C8
FrattiniC22×C8C22×D61
FittingC61×#1378C2
socleC2×C61#1869
prime radical#16840562C4
cosocleC22×C8×C61C23
hypercenter#1378D61
hypocenterC61#12987
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. C2×#16840180C2semidirect
3. #21447448C2semidirect
4. C2×#16839547C2semidirect
5. #21447439C2semidirect
6. #21443908C2semidirect
7. C61×#1378C2semidirect
182. #1378D61semidirect
210. C61#12987semidirect

Maximal subgroups

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

subgroupindex
C2×#168401802
#214547232
#214474482
C2×#168395472
#214474392
#214439082
C61×#13782
#1298761