Finite Group Databasebeta

← #29597379

Subgroups — #29597379

Order 15616.

526 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C16×D61C211yesmaximal, characteristic
2C2×#16840004C211yesmaximal, characteristic
3C2×#16839567C211yesmaximal, characteristic, complemented
4C2×#16839565C211yesmaximal, characteristic, complemented
5#21445845C211yesmaximal, characteristic
6#21443910C211yesmaximal, characteristic, complemented
7C61×#1380C211yesmaximal, characteristic, complemented
8C22×Dic244C2211yescharacteristic
9C22×C8×D61C2211yescharacteristic
10C2×#14519757—21no—
11C22×#11758185C2211yescharacteristic
12C2×C16×D61—22no—
13C2×C16×D61—22no—
14C2×C16×D61—22no—
15#16840004—22no—
16#16840004C414yescomplemented
17C2×#14519711C2211yescharacteristic
18C2×#14519700—21no—
19#16839619—21no—
20#16839567—22no—
21#16839565—22no—
22#16839524—21no—
23C22×C16×C61C2211yescharacteristic
24C2×#107×C61C2211yescharacteristic
25C2×#107×C61C2211yescharacteristic
26C22×Dic122—21no—
27C22×C4×D61D411yescharacteristic
28C2×Dic244—21no—
29C2×Dic244—21no—
30C2×Dic244—21no—
31C2×Dic244—21no—
32C2×Dic244—21no—
33C2×Dic244C2×C412yes—
34C22×#92418D411yescharacteristic
35C2×C8×D61—21no—
36C2×C8×D61—21no—
37C2×C8×D61—21no—
38C2×C8×D61—21no—
39C2×C8×D61C2×C412yes—
40C2×C8×D61D412yes—
41#14519757—24no—
42#14519757—42no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61—41no—
45C2×#11758198—41no—
46C2×#11758198—41no—
47C2×#11758185—21no—
48C2×#11758185—21no—
49C2×#11758185D412yes—
50C16×D61—28no—
51#14519711—21no—
52#14519711—21no—
53#14519700—24no—
54#14519684—41no—
55C22×C8×C61C2311yescharacteristic
56C2×C4⋊C4×C61—21no—
57C2×C4⋊C4×C61—21no—
58C2×C16×C61—21no—
59C2×C16×C61—21no—
60C2×C16×C61D412yes—
61#107×C61—21no—
62#107×C61—21no—
63#107×C61—21no—
64#107×C61C2×C412yes—
65C23×D61D811yescharacteristic
66C22×C4×C61—41no—
67C22×C4×C61—41no—
68C22×C4×C61C2×D411yescharacteristic
69C22×Dic61—41no—
70C22×Dic61—41no—
71C22×Dic61D811yescharacteristic
72C2×C4×D61—21no—
73C2×C4×D61—21no—
74C2×C4×D61—21no—
75C2×C4×D61—21no—
76C2×C4×D61C22⋊C412yes—
77C2×C4×D61SD1612yes—
78C2×Dic122—22no—
79C2×Dic122—41no—
80C2×Dic122—41no—
81C2×Dic122—41no—
82C2×Dic122—41no—
83C2×Dic122—41no—
84C2×C8×C61C22×C411yescharacteristic
85C2×C8×C61C2×D411yescharacteristic
86C2×C8×C61C2×D411yescharacteristic
87C2×C8×C61C2×D411yescharacteristic
88C2×C8×C61#4111yescharacteristic
89C2×C8×C61#4111yescharacteristic
90C4⋊C4×C61—22no—
91C4⋊C4×C61—41no—
92C4⋊C4×C61—41no—
93C4⋊C4×C61—41no—
94#11758198—44no—
95#11758198—82no—
96C4⋊Dic61—44no—
97C4⋊Dic61—41no—
98C4⋊Dic61—41no—
99C2×#92418—21no—
100C2×#92418—21no—
101C2×#92418C22⋊C412yes—
102Dic244—22no—
103Dic244—22no—
104Dic244—41no—
105Dic244—41no—
106C8×D61—22no—
107C8×D61—22no—
108C8×D61—22no—
109C8×D61—22no—
110C16×C61—22no—
111#11758185—22no—
112C23×C61C2×D811yescharacteristic
113C22×D61—21no—
114C22×D61—21no—
115C22×D61—21no—
116C22×D61—21no—
117C22×D61#10212yes—
118C22×D61SD3212yes—
119C22×D61Q3212yes—
120C2×C4×C61—42no—
121C2×C4×C61—81no—
122C2×C4×C61—81no—
123C2×C4×C61—81no—
124C2×C4×C61—81no—
125C2×C4×C61—81no—
126C2×C4×C61C2×C22⋊C411yescharacteristic
127C2×C4×C61#12111yescharacteristic
128C2×C4×C61#12111yescharacteristic
129C2×C4×C61C2×SD1611yescharacteristic
130C2×C4×C61#13511yescharacteristic
131C2×C4×C61#13511yescharacteristic
132C2×Dic61—21no—
133C2×Dic61—21no—
134C2×Dic61—42no—
135C2×Dic61—81no—
136C2×Dic61—81no—
137C2×Dic61—81no—
138C2×Dic61—81no—
139C2×Dic61—81no—
140C2×Dic61#10212yes—
141C4×D61—22no—
142C4×D61—22no—
143C4×D61—22no—
144C4×D61—22no—
145Dic122—42no—
146Dic122—42no—
147Dic122—81no—
148Dic122—81no—
149C8×C61C4×D411yescharacteristic
150C8×C61C4×D411yescharacteristic
151C8×C61#12011yescharacteristic
152C8×C61#12311yescharacteristic
153#92418—22no—
154#13060—611nomaximal, Sylow
155C22×C61C2×#10211yescharacteristic
156C22×C61#46611yescharacteristic
157C22×C61#46611yescharacteristic
158C22×C61C2×SD3211yescharacteristic
159C22×C61C2×Q3211yescharacteristic
160C22×C61#50811yescharacteristic
161C22×C61#50811yescharacteristic
162C2×D61—22no—
163C2×D61—22no—
164C2×D61—22no—
165C2×D61—22no—
166C2×D61—22no—
167C2×D61—22no—
168C2×D61#35814yescomplemented
169C4×C61—82no—
170C4×C61—161no—
171C4×C61#38611yescharacteristic
172C4×C61#41611yescharacteristic
173C4×C61#46511yescharacteristic
174C4×C61#46511yescharacteristic
175Dic61—22no—
176Dic61—82no—
177Dic61—161no—
178C23×C16—611no—
179C2×#467—611no—
180C2×#358—611no—
181C2×#358—611no—
182#1808—611no—
183#1380—611no—
184#1380D6111yescharacteristic, complemented
185C2×C61#189311yescharacteristic
186C2×C61C2×#35811yescharacteristic
187C2×C61#213811yescharacteristic
188C2×C61#221611yescharacteristic
189C2×C61#221611yescharacteristic
190C2×C61#221711yescharacteristic
191C2×C61#221711yescharacteristic
192D61—24no—
193C22×Q16—611no—
194C23×C8—611no—
195C2×C22⋊Q8—1221no—
196C22×C16—611no—
197C22×C16—1222no—
198C22×C16—1222no—
199C22×C16—1222no—
200C22×C16C2×D6111yescharacteristic
201#467—614no—
202#467—1222no—
203C2×#107—611no—
204C2×#107C2×D6111yescharacteristic
205C2×#107C2×D6111yescharacteristic
206C2×#103—1221no—
207#380—1221no—
208#358—1222no—
209#358—1222no—
210#340—1221no—
211C61#1306011yescharacteristic, complemented, Sylow
212C22×Q8—1221no—
213C23×C4—611no—
214C2×Q16—612no—
215C2×Q16—1221no—
216C2×Q16—1221no—
217C2×Q16—1221no—
218C2×Q16—1221no—
219C2×Q16—1221no—
220C22×C8—611no—
221C22×C8—612no—
222C22×C8—612no—
223C22×C8—1221no—
224C22×C8—1221no—
225C22×C8—1221no—
226C22×C8—1221no—
227C22×C8C22×D6111yescharacteristic
228C22⋊Q8—1224no—
229C22⋊Q8—2442no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—1221no—
233C2×C4⋊C4—2441no—
234C2×C22⋊C4—2441no—
235C2×C22⋊C4—2441no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—612no—
239C2×C16—1228no—
240C2×C16—1221no—
241C2×C16—1221no—
242C2×C16D24412yes—
243#107—21no—
244#107—21no—
245#107—21no—
246#107—1221no—
247#107—1221no—
248#107C4×D6112yes—
249#103—1224no—
250#95—2441no—
251C24—611no—
252C2×Q8—1222no—
253C2×Q8—2441no—
254C2×Q8—2441no—
255C2×Q8—2441no—
256C2×Q8—2441no—
257C2×Q8—2441no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—611no—
261C22×C4—612no—
262C22×C4—612no—
263C22×C4—1221no—
264C22×C4—1221no—
265C22×C4—1221no—
266C22×C4—1221no—
267C22×C4—2441no—
268C22×C4—2441no—
269C22×C4D4×D6111yescharacteristic
270Q16—1222no—
271Q16—1222no—
272Q16—2441no—
273Q16—2441no—
274C2×C8—612no—
275C2×C8—1221no—
276C2×C8—1221no—
277C2×C8—1222no—
278C2×C8—1222no—
279C2×C8—1222no—
280C2×C8—1222no—
281C2×C8C2×C4×D6111yescharacteristic
282C2×C8C2×D24411yescharacteristic
283C2×C8#1175822211yescharacteristic
284C2×C8D4×D6111yescharacteristic
285C2×C8D4×D6111yescharacteristic
286C2×C8#1175822611yescharacteristic
287C4⋊C4—22no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—2444no—
292C4⋊C4—2441no—
293C4⋊C4—2441no—
294C22⋊C4—4882no—
295C22⋊C4—2444no—
296C16—22no—
297C16—1222no—
298C23—612no—
299C23—612no—
300C23—612no—
301C23—1221no—
302C23—1221no—
303C23—1221no—
304C23—1221no—
305C23D8×D6111yescharacteristic
306Q8—4881no—
307Q8—4881no—
308Q8—2442no—
309Q8—2442no—
310C2×C4—42no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—612no—
317C2×C4—1221no—
318C2×C4—1221no—
319C2×C4—1222no—
320C2×C4—1222no—
321C2×C4—1222no—
322C2×C4—1222no—
323C2×C4—4881no—
324C2×C4—4881no—
325C2×C4—4881no—
326C2×C4—4881no—
327C2×C4—4881no—
328C2×C4—2442no—
329C2×C4C22⋊C4×D6111yescharacteristic
330C2×C4#1451975011yescharacteristic
331C2×C4#1451975211yescharacteristic
332C2×C4SD16×D6111yescharacteristic
333C2×C4#1451979811yescharacteristic
334C2×C4#1451980011yescharacteristic
335C8—1222no—
336C8#1451973911yescharacteristic
337C8C4×D24411yescharacteristic
338C8#1451974711yescharacteristic
339C8#1451976211yescharacteristic
340C22—614no—
341C22—1222no—
342C22—1222no—
343C22—1222no—
344C22—1222no—
345C22—1222no—
346C22—1222no—
347C22#102×D6111yescharacteristic
348C22#1683999711yescharacteristic
349C22#1683999911yescharacteristic
350C22SD32×D6111yescharacteristic
351C22Q32×D6111yescharacteristic
352C22#1684019111yescharacteristic
353C22#1684019311yescharacteristic
354C4—82no—
355C4—161no—
356C4—1222no—
357C4—9761no—
358C4—4882no—
359C4#1683964011yescharacteristic
360C4#1683975711yescharacteristic
361C4#1683999211yescharacteristic
362C4#1683999411yescharacteristic
363C2—1224no—
364C2#2144624611yescharacteristic
365C2#358×D6111yescharacteristic
366C2#2144752311yescharacteristic
367C2#2144790911yescharacteristic
368C2#2144791111yescharacteristic
369C2#2144791411yescharacteristic
370C2#2144791611yescharacteristic
subgroupG/H
centerC23D8×D61
commutatorC2×C8×C61C22×C4
FrattiniC22×C8C22×D61
FittingC61×#1380C2
socleC23×C61C2×D8
prime radicalC23×D61D8
cosocleC22×C8×C61C23
hypercenter#1380D61
hypocenterC61#13060
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
3. C2×#16839567C2semidirect
4. C2×#16839565C2semidirect
6. #21443910C2semidirect
7. C61×#1380C2semidirect
16. #16840004C4semidirect
168. C2×D61#358semidirect
184. #1380D61semidirect
211. C61#13060semidirect

Maximal subgroups

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

subgroupindex
C22×C16×D612
C2×#168400042
C2×#168395672
C2×#168395652
#214458452
#214439102
C61×#13802
#1306061