Finite Group Databasebeta

← #29579338

Subgroups — #29579338

Order 15616.

510 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840109C211yesmaximal, characteristic
2#21446766C211yesmaximal, characteristic, complemented
3#21446745C211yesmaximal, characteristic
4#21445686C211yesmaximal, characteristic, complemented
5C2×#16839510C211yesmaximal, characteristic, complemented
6C2×#16839505C211yesmaximal, characteristic
7C2×C61×#335C211yesmaximal, characteristic, complemented
8C2×#14519777C2211yescharacteristic
9C2×#14519774C2211yescharacteristic
10#16840109—22no—
11#16840109C414yescomplemented
12C2×C4⋊#92418C2211yescharacteristic
13C2×#14519699—22no—
14C2×#14519699—21no—
15C2×#14519699—21no—
16C2×#14519687—21no—
17C4×C4⋊Dic61C2211yescharacteristic
18#16839525—22no—
19#16839516—21no—
20#16839510—22no—
21#16839505—22no—
22C2×C4⋊C8×C61C2211yescharacteristic
23C2×C4⋊C8×C61C2211yescharacteristic
24C2×C4×C8×C61C2211yescharacteristic
25C61×#335—22no—
26C22×Dic122—21no—
27C22×D244—21no—
28C22×D244—21no—
29C22×D244—21no—
30C22×#92418—21no—
31#14519777—21no—
32#14519777—21no—
33#14519777—21no—
34#14519777C2×C412yes—
35#14519774—21no—
36#14519774—21no—
37#14519774—21no—
38#14519774C2×C412yes—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61D412yes—
42C2×C4×Dic61—21no—
43C4⋊#92418—21no—
44C4⋊#92418D412yes—
45#14519699—24no—
46#14519699—24no—
47#14519699—44no—
48#14519687—42no—
49#14519684—21no—
50C22×C8×C61—21no—
51C22×C8×C61—21no—
52C22×C8×C61D411yescharacteristic
53C22×C8×C61D411yescharacteristic
54C2×C42×C61C2311yescharacteristic
55C4⋊C8×C61—21no—
56C4⋊C8×C61—21no—
57C4⋊C8×C61—21no—
58C4⋊C8×C61—21no—
59C4×C8×C61C2×C412yes—
60C4×C8×C61D412yes—
61C23×D61—41no—
62C22×C4×C61C2×D411yescharacteristic
63C22×C4×C61C2×D411yescharacteristic
64C22×C4×C61#4111yescharacteristic
65C22×Dic61—22no—
66C22×Dic61—41no—
67C2×D244—22no—
68C2×D244—22no—
69C2×D244—22no—
70C2×D244—41no—
71C2×D244—41no—
72C2×D244—41no—
73C2×D244—42no—
74C2×D244—41no—
75C2×D244—42no—
76C2×D244—41no—
77C2×D244—41no—
78C2×D244—41no—
79C2×D244—82no—
80C2×Dic122—22no—
81C2×Dic122—41no—
82C2×Dic122—82no—
83C2×C8×C61—21no—
84C2×C8×C61—21no—
85C2×C8×C61—21no—
86C2×C8×C61—21no—
87C2×C8×C61—41no—
88C2×C8×C61—41no—
89C2×C8×C61—42no—
90C2×C8×C61—42no—
91C2×C8×C61C22⋊C412yes—
92C2×C8×C61C22⋊C412yes—
93C42×C61C22×C411yescharacteristic
94C42×C61C2×D411yescharacteristic
95C42×C61C2×D411yescharacteristic
96C42×C61#4111yescharacteristic
97C4⋊Dic61—22no—
98C4⋊Dic61—22no—
99C4⋊Dic61—22no—
100C4⋊Dic61—22no—
101C4⋊Dic61—41no—
102C4⋊Dic61—41no—
103C4×Dic61—41no—
104C4×Dic61—41no—
105C2×#92418—22no—
106C2×#92418—41no—
107C2×#92418—41no—
108C23×C61#12111yescharacteristic
109C22×D61—42no—
110C22×D61—81no—
111C22×D61—81no—
112C22×D61—82no—
113C22×D61—82no—
114C2×C4×C61—22no—
115C2×C4×C61C2×C22⋊C411yescharacteristic
116C2×C4×C61C4×D411yescharacteristic
117C2×C4×C61C4×D411yescharacteristic
118C2×C4×C61#12111yescharacteristic
119C2×C4×C61#12411yescharacteristic
120C2×C4×C61#12711yescharacteristic
121C2×C4×C61#13511yescharacteristic
122C2×C4×C61#13511yescharacteristic
123C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C61AGL(1,ℤ/8)12yes—
125C2×C4×C61#13711yescharacteristic
126C2×C4×C61#13712yes—
127C2×Dic61—42no—
128C2×Dic61—42no—
129C2×Dic61—42no—
130C2×Dic61—42no—
131C2×Dic61—82no—
132D244—42no—
133D244—42no—
134D244—44no—
135D244—82no—
136D244—82no—
137D244—81no—
138D244—81no—
139D244—81no—
140D244—81no—
141Dic122—82no—
142Dic122—82no—
143C8×C61—22no—
144C8×C61—22no—
145C8×C61—81no—
146C8×C61—81no—
147#92418—42no—
148#9747—611nomaximal, Sylow
149C22×C61#39011yescharacteristic
150C22×C61#46311yescharacteristic
151C22×C61#46411yescharacteristic
152C22×C61#46812yes—
153C22×C61#47212yes—
154C2×D61—82no—
155C2×D61—82no—
156C2×D61—82no—
157C2×D61—162no—
158C2×D61—162no—
159C4×C61—21no—
160C4×C61—21no—
161C4×C61#41611yescharacteristic
162C4×C61#41711yescharacteristic
163C4×C61#44012yes—
164C4×C61#46311yescharacteristic
165C4×C61#46411yescharacteristic
166C4×C61#49611yescharacteristic
167C4×C61#49711yescharacteristic
168Dic61—82no—
169Dic61—82no—
170C2×#486—611no—
171#1987—611no—
172#1982—611no—
173#1770—611no—
174C2×#335D6111yescharacteristic, complemented
175C2×#333—611no—
176C2×#331—611no—
177C2×C61#169712yes—
178C2×C61#169812yes—
179C2×C61#197311yescharacteristic
180C2×C61#197811yescharacteristic
181C2×C61#198311yescharacteristic
182D61—162no—
183C2×C4⋊Q8—611no—
184C2×#127—611no—
185#486—614no—
186#486—1222no—
187C2×C4⋊C8—611no—
188C2×C4⋊C8C2×D6111yescharacteristic
189C2×C4⋊C8C2×D6111yescharacteristic
190C2×#102—1221no—
191C2×#102—1221no—
192C2×#102—1222no—
193C2×#97—1221no—
194C2×C4×C8C2×D6111yescharacteristic
195C4×C4⋊C4—611no—
196#340—1222no—
197#336—1221no—
198#335—22no—
199#333—1222no—
200#331—1222no—
201C61#974711yescharacteristic, complemented, Sylow
202C22×Q8—1221no—
203C22×D4—1221no—
204C22×D4—1221no—
205C22×D4—1221no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—1221no—
209C22×C8D24411yescharacteristic
210C22×C8D24411yescharacteristic
211C4⋊Q8—612no—
212C4⋊Q8—1221no—
213C4⋊Q8—1221no—
214C4⋊Q8—1221no—
215#127—612no—
216#127—1221no—
217#127—1221no—
218#127—1221no—
219C2×C4⋊C4—612no—
220C2×C4⋊C4—1221no—
221C2×C4⋊C4—1221no—
222C2×C42—1221no—
223C2×C42C22×D6111yescharacteristic
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—612no—
229C4⋊C8—1221no—
230#102—1224no—
231#102—1224no—
232#102—2444no—
233#97—2442no—
234C4×C8C4×D6112yes—
235C4×C8C22⋊D6112yes—
236#95—1221no—
237C24—2441no—
238C2×Q8—1222no—
239C2×Q8—4882no—
240C2×Q8—2441no—
241C2×D4—1222no—
242C2×D4—1222no—
243C2×D4—1222no—
244C2×D4—4882no—
245C2×D4—2442no—
246C2×D4—2441no—
247C2×D4—2441no—
248C2×D4—2441no—
249C2×D4—2441no—
250C2×D4—2441no—
251C2×D4—2441no—
252C2×D4—2442no—
253C2×D4—2441no—
254C22×C4—1222no—
255C22×C4—2441no—
256C22×C4C2×D24411yescharacteristic
257C22×C4D4×D6111yescharacteristic
258C22×C4#1175822611yescharacteristic
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—1222no—
268C2×C8—2441no—
269C2×C8—2441no—
270C2×C8#1175819812yes—
271C2×C8#1175819812yes—
272C4⋊C4—1222no—
273C4⋊C4—1222no—
274C4⋊C4—1222no—
275C4⋊C4—1222no—
276C4⋊C4—2441no—
277C4⋊C4—2441no—
278C42—2441no—
279C42—2441no—
280C42C2×C4×D6111yescharacteristic
281C42D4×D6111yescharacteristic
282C42#1175822611yescharacteristic
283C42C2×C22⋊D6111yescharacteristic
284C23—4881no—
285C23—4881no—
286C23—4882no—
287C23—4882no—
288C23—2442no—
289C23#1451975211yescharacteristic
290Q8—4882no—
291Q8—4882no—
292D4—4882no—
293D4—4882no—
294D4—4881no—
295D4—4881no—
296D4—4881no—
297D4—4881no—
298D4—2442no—
299D4—2442no—
300D4—2444no—
301C2×C4—22no—
302C2×C4—4882no—
303C2×C4—2442no—
304C2×C4—2442no—
305C2×C4—2442no—
306C2×C4—2442no—
307C2×C4C2×#1175819811yescharacteristic
308C2×C4#1451973911yescharacteristic
309C2×C4#1451973911yescharacteristic
310C2×C4#1451975211yescharacteristic
311C2×C4#1451976611yescharacteristic
312C2×C4#1451977311yescharacteristic
313C2×C4#1451979911yescharacteristic
314C2×C4#1451979911yescharacteristic
315C2×C4#1451980112yes—
316C2×C4#1451980511yescharacteristic
317C2×C4#1451980912yes—
318C2×C4#1451981011yescharacteristic
319C8—22no—
320C8—22no—
321C8—81no—
322C8—81no—
323C8—2442no—
324C22—9762no—
325C22—9762no—
326C22—4882no—
327C22—4882no—
328C22—4882no—
329C22#1683966311yescharacteristic
330C22#1683997811yescharacteristic
331C22#1683998511yescharacteristic
332C22#1684000912yes—
333C22#1684003312yes—
334C4—21no—
335C4—21no—
336C4—4882no—
337C4—4882no—
338C4#1683975911yescharacteristic
339C4#1683976411yescharacteristic
340C4#1683986112yes—
341C4#1683997811yescharacteristic
342C4#1683998511yescharacteristic
343C4#1684015011yescharacteristic
344C4#1684015511yescharacteristic
345C2—9762no—
346C2#2144535312yes—
347C2#2144536012yes—
348C2#2144669811yescharacteristic
349C2#2144672711yescharacteristic
350C2#2144674911yescharacteristic
subgroupG/H
centerC23#14519752
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC2×C61×#335C2
socleC23×C61#121
prime radicalC2×#14519774C22
cosocleC2×C42×C61C23
hypercenterC2×#335D61
hypocenterC61#9747
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#335

Complements

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

NH ≅ G/Ntype
2. #21446766C2semidirect
4. #21445686C2semidirect
5. C2×#16839510C2semidirect
7. C2×C61×#335C2semidirect
11. #16840109C4semidirect
174. C2×#335D61semidirect
201. C61#9747semidirect

Maximal subgroups

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

subgroupindex
C2×#168401092
#214467662
#214467452
#214456862
C2×#168395102
C2×#168395052
C2×C61×#3352
#974761