Finite Group Databasebeta

← #29834924

Subgroups — #29834924

Order 15616.

534 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455475C211yesmaximal, characteristic
2#21454427C211yesmaximal, characteristic
3#21448040C218yesmaximal, complemented
4#21447591C211yesmaximal, characteristic, complemented
5C2×#16839579C211yesmaximal, characteristic, complemented
6C2×#16839577C211yesmaximal, characteristic, complemented
7#21447439C211yesmaximal, characteristic
8C61×#2155C211yesmaximal, characteristic, complemented
9C22×Dic244C2211yescharacteristic
10#16840528—21no—
11C2×#14519783C2211yescharacteristic
12#16840338—21no—
13C2×#14519717C2211yescharacteristic
14#16840021C2214yes—
15#16840004—22no—
16#16839810—21no—
17#16839809C2214yes—
18C2×#14519713C2211yescharacteristic
19#108×D61—22no—
20#16839770—21no—
21#16839584C2214yes—
22#16839579C2214yes—
23#16839577C2214yes—
24#16839543C2214yes—
25C2×M32×C61C2211yescharacteristic
26C2×#108×C61C2211yescharacteristic
27C61×#427C2211yescharacteristic
28C61×#365C2214yes—
29C2×#11758222D411yescharacteristic
30C22×Dic122—21no—
31C2×Dic244—21no—
32C2×Dic244—21no—
33C2×Dic244—21no—
34C2×Dic244—21no—
35C2×Dic244—21no—
36C2×Dic244C2312yes—
37#14519785—42no—
38#14519783—21no—
39#14519783—21no—
40#14519783—21no—
41#14519783—21no—
42#14519783—42no—
43C2×#11758194—21no—
44C2×#11758194D411yescharacteristic
45C2×#11758189C2312yes—
46M16×D61—24no—
47C2×C8×D61D412yes—
48#14519777—41no—
49#14519765—41no—
50#14519761—42no—
51#14519757—24no—
52#14519717—21no—
53#14519717C2312yes—
54C2×#11758185—21no—
55#14519713—21no—
56#14519713C2312yes—
57#14519712—24no—
58#14519700—24no—
59C2×M16×C61—21no—
60C22×C8×C61C2311yescharacteristic
61#117×C61—21no—
62M32×C61—21no—
63M32×C61C2312yes—
64C2×C16×C61—21no—
65#108×C61—21no—
66#108×C61C2312yes—
67#107×C61—21no—
68#106×C61C2312yes—
69C22×C4×C61C2×D411yescharacteristic
70C2×C22⋊D61SD1612yes—
71C22×Dic61—41no—
72#11758222—21no—
73#11758222—21no—
74#11758222—21no—
75#11758222—21no—
76C2×D244SD1611yescharacteristic
77C2×C4×D61C2×D412yes—
78C2×Dic122—22no—
79C2×Dic122—41no—
80C2×Dic122—41no—
81C2×Dic122—41no—
82C2×Dic122—41no—
83C2×Dic122—41no—
84C2×Dic122SD1611yescharacteristic
85M16×C61—22no—
86M16×C61—41no—
87C2×C8×C61—41no—
88C2×C8×C61C2×D411yescharacteristic
89C2×C8×C61#4111yescharacteristic
90C2×C8×C61#4111yescharacteristic
91C2×C8×C61#4111yescharacteristic
92C2×C8×C61#4111yescharacteristic
93C2×C8×C61C2411yescharacteristic
94C4⋊C4×C61—22no—
95C22⋊C4×C61—41no—
96C42×C61—41no—
97#11758198—44no—
98C4⋊Dic61—44no—
99#11758194—21no—
100#11758194—21no—
101#11758194—22no—
102#11758194—41no—
103C2×#92418—41no—
104C2×#92418C2×D412yes—
105Dic244—22no—
106Dic244—22no—
107Dic244—41no—
108Dic244—41no—
109#11758189—22no—
110#11758189—22no—
111#11758189—44no—
112C8×D61—22no—
113C8×D61—22no—
114C8×D61—44no—
115C16×C61—22no—
116#11758185—22no—
117C23×C61C2×SD1611yescharacteristic
118C22×D61C2×SD1612yes—
119C2×C4×C61—42no—
120C2×C4×C61#12311yescharacteristic
121C2×C4×C61#12311yescharacteristic
122C2×C4×C61C2×SD1611yescharacteristic
123C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
125C2×C4×C61C22×D411yescharacteristic
126C22⋊D61—22no—
127C22⋊D61—22no—
128C2×Dic61—42no—
129C2×Dic61—81no—
130C2×Dic61—81no—
131C2×Dic61C2×SD1612yes—
132D244—21no—
133D244—21no—
134C4×D61—22no—
135C4×D61—22no—
136Dic122—21no—
137Dic122—21no—
138Dic122—42no—
139Dic122—42no—
140Dic122—81no—
141Dic122—81no—
142C8×C61—42no—
143C8×C61#12311yescharacteristic
144C8×C61#12311yescharacteristic
145C8×C61C2×#4111yescharacteristic
146C8×C61C2×#4111yescharacteristic
147#92418—22no—
148#92418—42no—
149#33658—611nomaximal, Sylow
150C22×C61—41no—
151C22×C61#48111yescharacteristic
152C22×C61#48111yescharacteristic
153C22×C61C22×SD1611yescharacteristic
154C2×D61—22no—
155C2×D61—42no—
156C4×C61—82no—
157C4×C61#48111yescharacteristic
158C4×C61#48111yescharacteristic
159C4×C61C2×#12311yescharacteristic
160C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
161Dic61—22no—
162Dic61—82no—
163#3059—611no—
164#2916—611no—
165#2239—618no—
166#2155—611no—
167#2155D6111yescharacteristic, complemented
168C2×#362—611no—
169C2×#362—611no—
170#2115—611no—
171C2×C61—21no—
172C2×C61—21no—
173C2×C61—21no—
174C2×C61C2×#48111yescharacteristic
175D61—42no—
176C22×Q16—611no—
177#568—1221no—
178C2×#131—611no—
179#536—1221no—
180C2×M32—611no—
181C2×M32C2×D6111yescharacteristic
182#470—614no—
183#467—1222no—
184#430—614no—
185#430—1221no—
186C2×#108—611no—
187C2×#108—1222no—
188C2×#108C2×D6111yescharacteristic
189#427C2×D6111yescharacteristic
190#419—1221no—
191#365—614no—
192#365C2×D6114yes—
193#362—614no—
194#362—614no—
195#349—614no—
196C61#3365811yescharacteristic, complemented, Sylow
197C2×#41—611no—
198C22×Q8—1221no—
199C2×Q16—612no—
200C2×Q16—1221no—
201C2×Q16—1221no—
202C2×Q16—1221no—
203C2×Q16—1221no—
204C2×Q16—1221no—
205#131—1221no—
206#131—1221no—
207#131—1221no—
208#131—1221no—
209#131—2442no—
210#131—2442no—
211C2×M16—21no—
212C2×M16—611no—
213C2×M16—612no—
214C2×M16—1224no—
215C2×M16—1221no—
216C22×C8—612no—
217C22×C8C22×D6111yescharacteristic
218C4⋊Q8—2441no—
219#124—2441no—
220#123—2442no—
221C22⋊Q8—1224no—
222#117—21no—
223M32—21no—
224M32—612no—
225M32—1221no—
226M32C22×D6112yes—
227C2×C16—21no—
228C2×C16—1221no—
229#108—21no—
230#108—612no—
231#108—1224no—
232#108—1221no—
233#108C22×D6112yes—
234#107—21no—
235#106C22×D6112yes—
236#103—1224no—
237#41—1221no—
238#41—1221no—
239#41—1221no—
240#41—1221no—
241C2×Q8—611no—
242C2×Q8—1222no—
243C2×Q8—2441no—
244C2×Q8—2441no—
245C2×Q8—2441no—
246C2×Q8—2441no—
247C2×Q8—2441no—
248C2×D4—612no—
249C2×D4—611no—
250C22×C4—612no—
251C22×C4—2441no—
252C22×C4D4×D6111yescharacteristic
253Q16—1222no—
254Q16—1222no—
255Q16—2441no—
256Q16—2441no—
257M16—22no—
258M16—41no—
259M16—1221no—
260M16—1221no—
261M16—1222no—
262M16—1222no—
263M16—1222no—
264M16—2444no—
265M16—2441no—
266C2×C8—41no—
267C2×C8—612no—
268C2×C8—1222no—
269C2×C8—1222no—
270C2×C8—2444no—
271C2×C8—2441no—
272C2×C8#1175822211yescharacteristic
273C2×C8#1175822211yescharacteristic
274C2×C8D4×D6111yescharacteristic
275C2×C8#1175822611yescharacteristic
276C2×C8#1175822611yescharacteristic
277C2×C8C23×D6111yescharacteristic
278C4⋊C4—22no—
279C4⋊C4—2444no—
280C22⋊C4—41no—
281C22⋊C4—2444no—
282C42—41no—
283C16—22no—
284C16—1222no—
285C23—612no—
286C23SD16×D6111yescharacteristic
287Q8—1221no—
288Q8—1221no—
289Q8—4881no—
290Q8—4881no—
291Q8—2442no—
292Q8—2442no—
293D4—1222no—
294D4—1222no—
295D4—1221no—
296D4—1221no—
297C2×C4—42no—
298C2×C4—612no—
299C2×C4—1222no—
300C2×C4—1222no—
301C2×C4—4881no—
302C2×C4—4881no—
303C2×C4—2442no—
304C2×C4#1451976211yescharacteristic
305C2×C4#1451976211yescharacteristic
306C2×C4SD16×D6111yescharacteristic
307C2×C4#1451980411yescharacteristic
308C2×C4#1451980411yescharacteristic
309C2×C4C2×D4×D6111yescharacteristic
310C8—42no—
311C8—1222no—
312C8—2442no—
313C8#1451976211yescharacteristic
314C8#1451976211yescharacteristic
315C8C2×#1175822211yescharacteristic
316C8C2×#1175822611yescharacteristic
317C22—41no—
318C22—1222no—
319C22—2442no—
320C22#1684008211yescharacteristic
321C22#1684008211yescharacteristic
322C22C2×SD16×D6111yescharacteristic
323C4—82no—
324C4—1222no—
325C4—4882no—
326C4#1684008211yescharacteristic
327C4#1684008211yescharacteristic
328C4C2×#1451976211yescharacteristic
329C4C2×#1451980411yescharacteristic
330C2—21no—
331C2—21no—
332C2—21no—
333C2—2442no—
334C2C2×#1684008211yescharacteristic
subgroupG/H
centerC2C2×#16840082
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2155C2
socleC2×C61C2×#481
prime radicalC2×#11758222D4
hypercenter#2155D61
hypocenterC61#33658
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ #2155

Complements

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

NH ≅ G/Ntype
3. #21448040C2semidirect
4. #21447591C2semidirect
5. C2×#16839579C2semidirect
6. C2×#16839577C2semidirect
8. C61×#2155C2semidirect
167. #2155D61semidirect
196. C61#33658semidirect

Maximal subgroups

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

subgroupindex
#214554752
#214544272
#214480402
#214475912
C2×#168395792
C2×#168395772
#214474392
C61×#21552
#3365861