Finite Group Databasebeta

← #29590815

Subgroups — #29590815

Order 15616.

612 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21449034C211yesmaximal, characteristic
2#21445811C211yesmaximal, characteristic, complemented
3#21443850C212yesmaximal, complemented
4#21443823C212yesmaximal, complemented
5C61×#1465C211yesmaximal, characteristic, complemented
6C22×#11758222C411yescharacteristic
7C2×C4×C22⋊D61—21no—
8#16839786—21no—
9C2×#14519694—21no—
10#16839682C2211yescharacteristic
11#16839679C2211yescharacteristic
12#16839677—21no—
13#16839674C411yescharacteristic
14#16839671C411yescharacteristic
15#16839670—21no—
16C4×#11758198—21no—
17C4×C22⋊Dic61C411yescharacteristic
18#16839531C2212yes—
19C2×C4×D4×C61C2211yescharacteristic
20C61×#407—21no—
21C61×#324C2212yes—
22C61×#323—21no—
23C2×#11758222—44no—
24C2×#11758222—41no—
25C2×#11758222—41no—
26C22×Dic122C2×C411yescharacteristic
27C22×D244C2×C411yescharacteristic
28C22×C22⋊D61—21no—
29C22×C22⋊D61—21no—
30C22×C4×D61—21no—
31C2×#11758194—22no—
32C4×C22⋊D61—44no—
33C2×#11758196—21no—
34C2×C22⋊Dic61—21no—
35C2×C22⋊Dic61—21no—
36C2×#11758198—21no—
37C2×#11758198—21no—
38C2×#11758198—21no—
39C2×C4×Dic61—21no—
40C4⋊#92418—41no—
41C4⋊#92418—41no—
42#14519694—42no—
43#14519684—21no—
44#14519684—21no—
45#14519684C2×C411yescharacteristic
46#14519684C2×C411yescharacteristic
47C22×D4×C61D411yescharacteristic
48C23×C4×C61C2×C411yescharacteristic
49C23×C4×C61D411yescharacteristic
50C2×M16×C61—41no—
51C4×D4×C61—41no—
52C4×D4×C61—41no—
53C2×C4⋊C4×C61C2311yescharacteristic
54C2×C22⋊C4×C61C2×C411yescharacteristic
55C2×C22⋊C4×C61D411yescharacteristic
56C2×C42×C61D411yescharacteristic
57C22⋊C8×C61—22no—
58C22⋊C8×C61—41no—
59C24×C61C22⋊C411yescharacteristic
60C24×C61C22⋊C411yescharacteristic
61C23×D61—21no—
62C2×D4×C61—21no—
63C2×D4×C61—21no—
64C2×D4×C61—41no—
65C2×D4×C61—41no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—42no—
71C22×C4×C61—42no—
72C22×C4×C61C22⋊C411yescharacteristic
73C22×C4×C61C22⋊C411yescharacteristic
74C22×C4×C61C22×C411yescharacteristic
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C2×C22⋊D61—42no—
78C2×C22⋊D61—42no—
79C2×C22⋊D61—42no—
80C2×C22⋊D61—42no—
81C2×C22⋊D61—42no—
82C2×C22⋊D61—42no—
83C22×Dic61—21no—
84C22×Dic61—21no—
85C22×Dic61—21no—
86#11758222—44no—
87#11758222—44no—
88#11758222—44no—
89#11758222—44no—
90C2×D244—41no—
91C2×D244—42no—
92C2×C4×D61—24no—
93C2×C4×D61—42no—
94C2×C4×D61—41no—
95C2×C4×D61—41no—
96C2×Dic122—41no—
97C2×Dic122—42no—
98M16×C61—82no—
99C2×C8×C61—42no—
100C4⋊C4×C61—41no—
101C22⋊C4×C61—41no—
102C22⋊C4×C61—41no—
103C42×C61—21no—
104C42×C61—21no—
105C22⋊Dic61—42no—
106C22⋊Dic61—42no—
107#11758198—42no—
108#11758198—42no—
109#11758198—41no—
110#11758198—41no—
111#11758196—42no—
112C4×Dic61—42no—
113#11758194—42no—
114#11758194—42no—
115C2×#92418—42no—
116C23×C61—21no—
117C23×C61—21no—
118C23×C61—21no—
119C23×C61—21no—
120C23×C61—21no—
121C23×C61—21no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61—41no—
126C23×C61C2×C22⋊C411yescharacteristic
127C22×D61—22no—
128C22×D61—41no—
129C22×D61—41no—
130C22×D61—41no—
131C22×D61—41no—
132C22×D61—41no—
133C22×D61—41no—
134D4×C61—42no—
135D4×C61—42no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—22no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—42no—
143C2×C4×C61—42no—
144C2×C4×C61—42no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—81no—
149C2×C4×C61#9911yescharacteristic
150C2×C4×C61#9911yescharacteristic
151C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C61#13711yescharacteristic
153C22⋊D61—48no—
154C22⋊D61—48no—
155C2×Dic61—22no—
156C2×Dic61—22no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164D244—42no—
165D244—42no—
166C4×D61—44no—
167C4×D61—44no—
168Dic122—42no—
169Dic122—42no—
170C8×C61—82no—
171#92418—82no—
172#11828—611nomaximal, Sylow
173C22×C61—21no—
174C22×C61—21no—
175C22×C61—42no—
176C22×C61—41no—
177C22×C61—41no—
178C22×C61—41no—
179C22×C61—41no—
180C22×C61—41no—
181C22×C61—41no—
182C22×C61—81no—
183C22×C61—81no—
184C22×C61—81no—
185C22×C61C2×#9911yescharacteristic
186C22×C61#41711yescharacteristic
187C22×C61#42111yescharacteristic
188C2×D61—42no—
189C2×D61—42no—
190C2×D61—42no—
191C2×D61—81no—
192C2×D61—81no—
193C2×D61—81no—
194C2×D61—81no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—41no—
199C4×C61—41no—
200C4×C61—41no—
201C4×C61AΣL(1,𝔽16)12yes—
202Dic61—42no—
203Dic61—42no—
204Dic61—81no—
205#2388—611no—
206#1800—611no—
207#1465D6111yescharacteristic, complemented
208#1354—612no—
209#1345—612no—
210C2×C61—21no—
211C2×C61—21no—
212C2×C61—81no—
213C2×C61—81no—
214C2×C61#151311yescharacteristic
215C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
216C2×C61#213211yescharacteristic
217D61—82no—
218C22×#41—611no—
219C2×C4×D4—1221no—
220C2×C4×D4C2×D6111yescharacteristic
221#423—1221no—
222C2×#100—1221no—
223#407—21no—
224#395—611no—
225#394—611no—
226#394—1221no—
227#393—611no—
228#392—611no—
229#392—1221no—
230C4×C22⋊C4—611no—
231C4×C22⋊C4—1221no—
232#343—612no—
233#324C2×D6112yes—
234#323—21no—
235C61#1182811yescharacteristic, complemented, Sylow
236C2×#41—2441no—
237C2×#41—2441no—
238C2×#41—2444no—
239C22×Q8—611no—
240C22×D4—611no—
241C22×D4—1221no—
242C22×D4—1221no—
243C22×D4D24411yescharacteristic
244C23×C4—1221no—
245C23×C4C4×D6111yescharacteristic
246C23×C4C22⋊D6111yescharacteristic
247C2×M16—41no—
248C2×M16—1222no—
249C4×D4—41no—
250C4×D4—41no—
251C4×D4—2444no—
252C2×C4⋊C4—1221no—
253C2×C4⋊C4C22×D6111yescharacteristic
254C2×C22⋊C4—1221no—
255C2×C22⋊C4—1221no—
256C2×C22⋊C4—1221no—
257C2×C22⋊C4—1221no—
258C2×C22⋊C4—1221no—
259C2×C22⋊C4C4×D6111yescharacteristic
260C2×C22⋊C4C22⋊D6111yescharacteristic
261C2×C42—1221no—
262C2×C42D24411yescharacteristic
263C4⋊C8—2441no—
264C4⋊C8—2441no—
265#100—2442no—
266C22⋊C8—22no—
267C22⋊C8—41no—
268#95—611no—
269#95—611no—
270#95—1221no—
271#95—1221no—
272C24—1221no—
273C24#1175819811yescharacteristic
274C24#1175819811yescharacteristic
275#41—2444no—
276#41—2444no—
277#41—2444no—
278#41—2444no—
279C2×Q8—2441no—
280C2×Q8—2442no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—2441no—
286C2×D4—2442no—
287C2×D4—2442no—
288C2×D4—2442no—
289C2×D4—2442no—
290C2×D4—2442no—
291C2×D4—2442no—
292C2×D4—2442no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—1224no—
300C22×C4—1221no—
301C22×C4—1221no—
302C22×C4—1221no—
303C22×C4—2442no—
304C22×C4—2441no—
305C22×C4—2441no—
306C22×C4#1175819811yescharacteristic
307C22×C4#1175819811yescharacteristic
308C22×C4C2×C4×D6111yescharacteristic
309C22×C4C2×D24411yescharacteristic
310C22×C4C2×C22⋊D6111yescharacteristic
311M16—82no—
312M16—2442no—
313M16—2442no—
314C2×C8—42no—
315C2×C8—2442no—
316C4⋊C4—41no—
317C4⋊C4—2442no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—2442no—
321C22⋊C4—2442no—
322C22⋊C4—2442no—
323C22⋊C4—2442no—
324C22⋊C4—2441no—
325C22⋊C4—2441no—
326C42—21no—
327C42—21no—
328C42—2442no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—1222no—
340C23—2441no—
341C23—2441no—
342C23—2441no—
343C23—2441no—
344C23—2441no—
345C23—2441no—
346C23C2×#1175819811yescharacteristic
347Q8—2442no—
348Q8—2442no—
349D4—42no—
350D4—42no—
351D4—2442no—
352D4—2442no—
353D4—2448no—
354D4—2448no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—81no—
368C2×C4—1222no—
369C2×C4—1222no—
370C2×C4—2444no—
371C2×C4—2441no—
372C2×C4—2441no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4—2444no—
379C2×C4#1451969111yescharacteristic
380C2×C4#1451969111yescharacteristic
381C2×C4#1451980511yescharacteristic
382C2×C4#1451981011yescharacteristic
383C8—82no—
384C8—4882no—
385C22—21no—
386C22—21no—
387C22—42no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—4881no—
398C22—4881no—
399C22—4881no—
400C22—4881no—
401C22—2442no—
402C22—2442no—
403C22—2442no—
404C22C2×#1451969111yescharacteristic
405C22#1683976411yescharacteristic
406C22#1683977811yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—4881no—
414C4—2442no—
415C4—2442no—
416C4#1683954812yes—
417C2—21no—
418C2—21no—
419C2—81no—
420C2—81no—
421C2—4882no—
422C2#2144436311yescharacteristic
423C2C2×#1683954811yescharacteristic
424C2#2144750211yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1465C2
socleC22×C61C2×#99
prime radical#21449034C2
cosocleC2×C4⋊C4×C61C23
hypercenter#1465D61
hypocenterC61#11828
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
2. #21445811C2semidirect
3. #21443850C2semidirect
4. #21443823C2semidirect
5. C61×#1465C2semidirect
207. #1465D61semidirect
235. C61#11828semidirect

Maximal subgroups

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

subgroupindex
#214490342
#214458112
#214438502
#214438232
C61×#14652
#1182861