Finite Group Databasebeta

← #29707780

Subgroups — #29707780

Order 15616.

1530 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459152C211yesmaximal, characteristic
2C2×#16839735C212yesmaximal, complemented
3#21448307C212yesmaximal, complemented
4#21448301C211yesmaximal, characteristic, complemented
5#21447071C218yesmaximal, complemented
6C61×#1740C211yesmaximal, characteristic, complemented
7C22×#11758222C2212yes—
8#16840343—22no—
9#16840338—22no—
10#16840331—22no—
11#16840324C2214yes—
12#16840312—22no—
13#16840274—24no—
14C2×#14519736C2211yescharacteristic
15C22×C4×Dic61C2211yescharacteristic
16#16839735—28no—
17C2×#14519691C2218yes—
18#16839667—22no—
19#16839663—22no—
20#16839658—22no—
21#16839639C2218yes—
22#16839634C2214yes—
23C4×#11758196—22no—
24C4×#11758198—24no—
25C2×#117×C61C2212yes—
26C2×#117×C61C2211yescharacteristic
27C61×#342C2214yes—
28C2×#11758222—24no—
29C2×#11758222—22no—
30C2×#11758222—22no—
31C2×#11758222—22no—
32C2×#11758222—22no—
33C2×#11758222—24no—
34C2×#11758222—24no—
35C2×#11758222—24no—
36C22×Dic122D412yes—
37C22×D244D412yes—
38C22×C22⋊D61C2314yes—
39C23×Dic61C2312yes—
40C22×C4×D61D414yes—
41#14519777—22no—
42#14519774—22no—
43#14519765—24no—
44#14519761—28no—
45#14519757—28no—
46#14519752—28no—
47#14519751—28no—
48#14519747—28no—
49C4×D244—28no—
50#14519736—22no—
51#14519736D414yescomplemented
52C2×C4⋊Dic61—21no—
53C2×#11758196—24no—
54C2×C22⋊Dic61C2312yes—
55C2×#11758198—28no—
56C2×C4×Dic61—21no—
57C2×C4×Dic61—22no—
58C2×C4×Dic61—21no—
59C2×C4×Dic61—21no—
60C2×C4×Dic61—22no—
61#14519691—216no—
62#14519684—24no—
63#14519684—28no—
64C23×C4×C61C2311yescharacteristic
65#117×C61—22no—
66#117×C61—22no—
67#117×C61—24no—
68#117×C61—24no—
69C2×C4⋊C4×C61—22no—
70C2×C4⋊C4×C61D412yes—
71C2×C22⋊C4×C61C2314yes—
72C2×C22⋊C4×C61C2312yes—
73C2×C42×C61—22no—
74C2×C42×C61D412yes—
75C24×C61C2411yescharacteristic
76C23×D61C2×D414yes—
77C22×C4×C61—24no—
78C22×C4×C61C2×D414yes—
79C22×C4×C61C2×D412yes—
80C22×C4×C61C2×D412yes—
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D412yes—
84C22×C4×C61#4112yes—
85C22×C4×C61#4114yes—
86C2×C22⋊D61—24no—
87C2×C22⋊D61—24no—
88C2×C22⋊D61—24no—
89C2×C22⋊D61—24no—
90C2×C22⋊D61—24no—
91C2×C22⋊D61—24no—
92C22×Dic61—24no—
93C22×Dic61—22no—
94C22×Dic61—22no—
95C22×Dic61—44no—
96C22×Dic61C2×D414yes—
97#11758222—44no—
98#11758222—44no—
99#11758222—44no—
100#11758222—44no—
101#11758222—44no—
102#11758222—44no—
103#11758222—44no—
104#11758222—44no—
105C2×D244—22no—
106C2×D244—22no—
107C2×D244—24no—
108C2×D244—24no—
109C2×D244—28no—
110C2×C4×D61—24no—
111C2×C4×D61—24no—
112C2×C4×D61—28no—
113C2×C4×D61—28no—
114C2×Dic122—22no—
115C2×Dic122—22no—
116C2×Dic122—24no—
117C2×Dic122—24no—
118C4⋊C4×C61—24no—
119C4⋊C4×C61—44no—
120C22⋊C4×C61—24no—
121C22⋊C4×C61—24no—
122C22⋊C4×C61—24no—
123C42×C61—22no—
124C42×C61—22no—
125C42×C61—44no—
126C22⋊Dic61—22no—
127C22⋊Dic61C2×D414yes—
128#11758198—216no—
129#11758198—416no—
130C4⋊Dic61—28no—
131C4⋊Dic61—24no—
132#11758196—48no—
133C4×Dic61—24no—
134C4×Dic61—41no—
135C4×Dic61—42no—
136C4×Dic61—41no—
137C23×C61—42no—
138C23×C61C22×D412yes—
139C23×C61C22×D411yescharacteristic
140C23×C61C22×D411yescharacteristic
141C23×C61C2×#4111yescharacteristic
142C23×C61C2×#4112yes—
143C22×D61—24no—
144C22×D61—24no—
145C22×D61—24no—
146C22×D61—48no—
147C2×C4×C61—24no—
148C2×C4×C61—44no—
149C2×C4×C61—48no—
150C2×C4×C61—44no—
151C2×C4×C61#12011yescharacteristic
152C2×C4×C61#12012yes—
153C2×C4×C61#12011yescharacteristic
154C2×C4×C61#12114yes—
155C2×C4×C61#12114yes—
156C2×C4×C61#12112yes—
157C2×C4×C61#12112yes—
158C2×C4×C61#12314yes—
159C2×C4×C61#12314yes—
160C2×C4×C61#12412yes—
161C2×C4×C61#12412yes—
162C22⋊D61—44no—
163C22⋊D61—44no—
164C22⋊D61—44no—
165C22⋊D61—44no—
166C2×Dic61—24no—
167C2×Dic61—28no—
168C2×Dic61—24no—
169C2×Dic61—24no—
170C2×Dic61—24no—
171C2×Dic61—24no—
172C2×Dic61—44no—
173C2×Dic61—42no—
174C2×Dic61—44no—
175C2×Dic61—42no—
176D244—44no—
177D244—44no—
178D244—88no—
179C4×D61—48no—
180C4×D61—48no—
181Dic122—44no—
182Dic122—44no—
183#23514—611nomaximal, Sylow
184C22×C61—22no—
185C22×C61—24no—
186C22×C61—41no—
187C22×C61—42no—
188C22×C61—41no—
189C22×C61C2×#12011yescharacteristic
190C22×C61C2×#12112yes—
191C22×C61C2×#12111yescharacteristic
192C22×C61C2×#12312yes—
193C22×C61C2×#12411yescharacteristic
194C2×D61—48no—
195C2×D61—44no—
196C2×D61—48no—
197C2×D61—84no—
198C4×C61—44no—
199C4×C61—84no—
200C4×C61#39412yes—
201C4×C61#39414yes—
202C4×C61#39412yes—
203Dic61—44no—
204Dic61—44no—
205Dic61—44no—
206#3449—611no—
207C2×#410—612no—
208#2297—612no—
209#2296—611no—
210#2044—618no—
211#1740D6111yescharacteristic, complemented
212C2×C61—21no—
213C2×C61—22no—
214C2×C61—22no—
215C2×C61—21no—
216C2×C61C2×#39411yescharacteristic
217C2×C61#302512yes—
218D61—84no—
219C22×#41—612no—
220#537—1222no—
221#536—1222no—
222#535—1222no—
223#534—614no—
224#532—1222no—
225#525—1224no—
226C2×#117—611no—
227C2×#117C2×D6112yes—
228C2×#117C2×D6111yescharacteristic
229C22×C42—611no—
230#410—1228no—
231C2×#99—618no—
232#391—1222no—
233#390—1222no—
234#389—1222no—
235#386—618no—
236#385—614no—
237C4×C4⋊C4—1222no—
238C4×C22⋊C4—1224no—
239#342C2×D6114yes—
240C61#2351411yescharacteristic, complemented, Sylow
241C2×#41—1222no—
242C2×#41—1222no—
243C2×#41—1224no—
244C2×#41—1224no—
245C2×#41—1222no—
246C2×#41—1222no—
247C2×#41—1224no—
248C2×#41—1224no—
249C22×Q8—612no—
250C22×D4—614no—
251C22×D4—612no—
252C23×C4—614no—
253C23×C4—612no—
254C23×C4C22×D6111yescharacteristic
255C4⋊Q8—1222no—
256#127—1222no—
257#124—1224no—
258#123—1228no—
259C22⋊Q8—1228no—
260#121—1228no—
261#121—1228no—
262#120—1228no—
263C4×D4—1228no—
264#117—22no—
265#117—22no—
266#117—24no—
267#117—24no—
268#117—614no—
269#117—1222no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—1221no—
272C2×C4⋊C4—1224no—
273C2×C4⋊C4D24412yes—
274C2×C22⋊C4—612no—
275C2×C22⋊C4—1228no—
276C2×C22⋊C4C22×D6114yes—
277C2×C22⋊C4C22×D6112yes—
278C2×C42—22no—
279C2×C42—1221no—
280C2×C42—1221no—
281C2×C42—1221no—
282C2×C42—1222no—
283C2×C42—1222no—
284C2×C42D24412yes—
285#99—12216no—
286#95—1228no—
287#95—1224no—
288C24—614no—
289C24C23×D6111yescharacteristic
290#41—2444no—
291#41—2444no—
292#41—2444no—
293#41—2444no—
294#41—2444no—
295#41—2444no—
296#41—2444no—
297#41—2444no—
298C2×Q8—1222no—
299C2×Q8—1222no—
300C2×Q8—1224no—
301C2×Q8—1224no—
302C2×D4—1228no—
303C2×D4—1224no—
304C2×D4—1224no—
305C2×D4—1224no—
306C2×D4—1224no—
307C2×D4—1224no—
308C2×D4—1224no—
309C2×D4—1222no—
310C2×D4—1222no—
311C2×D4—1224no—
312C2×D4—1224no—
313C22×C4—24no—
314C22×C4—614no—
315C22×C4—1224no—
316C22×C4—1222no—
317C22×C4—1222no—
318C22×C4—1224no—
319C22×C4—1224no—
320C22×C4—1228no—
321C22×C4—1228no—
322C22×C4—2444no—
323C22×C4C2×D24414yes—
324C22×C4C2×D24411yescharacteristic
325C22×C4C2×D24411yescharacteristic
326C22×C4#1175822214yes—
327C22×C4D4×D6112yes—
328C22×C4D4×D6112yes—
329C22×C4D4×D6112yes—
330C22×C4#1175822612yes—
331C4⋊C4—24no—
332C4⋊C4—44no—
333C4⋊C4—1228no—
334C4⋊C4—1224no—
335C4⋊C4—2448no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—614no—
340C22⋊C4—12216no—
341C22⋊C4—1222no—
342C22⋊C4—24416no—
343C42—22no—
344C42—22no—
345C42—44no—
346C42—1224no—
347C42—2441no—
348C42—2442no—
349C42—2441no—
350C23—42no—
351C23—1224no—
352C23—1224no—
353C23—1224no—
354C23—2448no—
355C23C2×D4×D6112yes—
356C23C2×D4×D6111yescharacteristic
357C23C22×D24411yescharacteristic
358C23C2×#1175822212yes—
359C23C2×#1175822611yescharacteristic
360Q8—2444no—
361Q8—2444no—
362D4—4888no—
363D4—2444no—
364D4—2444no—
365D4—2444no—
366D4—2444no—
367D4—2444no—
368D4—2444no—
369C2×C4—24no—
370C2×C4—44no—
371C2×C4—48no—
372C2×C4—44no—
373C2×C4—1228no—
374C2×C4—1224no—
375C2×C4—1224no—
376C2×C4—1224no—
377C2×C4—1224no—
378C2×C4—1224no—
379C2×C4—2444no—
380C2×C4—2442no—
381C2×C4—2444no—
382C2×C4—2442no—
383C2×C4—2448no—
384C2×C4—2448no—
385C2×C4#1451974711yescharacteristic
386C2×C4#1451974712yes—
387C2×C4#1451974711yescharacteristic
388C2×C4#1451975014yes—
389C2×C4#1451975014yes—
390C2×C4#1451975212yes—
391C2×C4#1451975212yes—
392C2×C4#1451976214yes—
393C2×C4#1451976214yes—
394C2×C4#1451976512yes—
395C2×C4#1451976512yes—
396C22—22no—
397C22—24no—
398C22—41no—
399C22—42no—
400C22—41no—
401C22—4884no—
402C22—2448no—
403C22—2444no—
404C22—2448no—
405C22C2×#1451974711yescharacteristic
406C22C2×#1451975012yes—
407C22C2×#1451975211yescharacteristic
408C22C2×#1451976212yes—
409C22C2×#1451976511yescharacteristic
410C4—44no—
411C4—84no—
412C4—2444no—
413C4—2444no—
414C4—2444no—
415C4#1683968012yes—
416C4#1683968014yes—
417C4#1683968012yes—
418C2—22no—
419C2—21no—
420C2—22no—
421C2—21no—
422C2—4884no—
423C2C2×#1683968011yescharacteristic
424C2#2145521312yes—
subgroupG/H
centerC2×C4#14519747
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC61×#1740C2
socleC22×C61C2×#120
prime radical#21459152C2
hypercenter#1740D61
hypocenterC61#23514
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1740

Complements

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

NH ≅ G/Ntype
2. C2×#16839735C2semidirect
3. #21448307C2semidirect
4. #21448301C2semidirect
5. #21447071C2semidirect
6. C61×#1740C2semidirect
51. #14519736D4semidirect
211. #1740D61semidirect
240. C61#23514semidirect

Maximal subgroups

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

subgroupindex
#214591522
C2×#168397352
#214483072
#214483012
#214470712
C61×#17402
#2351461