Finite Group Databasebeta

← #29590681

Subgroups — #29590681

Order 15616.

574 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 79 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#21451553C211yesmaximal, characteristic
2#21444146C211yesmaximal, characteristic, complemented
3#21443818C212yesmaximal, complemented
4#21443815C212yesmaximal, complemented
5C61×#1819C211yesmaximal, characteristic, complemented
6C2×#14519751C411yescharacteristic, complemented
7C2×#14519746—21no—
8#16839726—21no—
9#16839687C411yescharacteristic, complemented
10#16839687C2211yescharacteristic
11#16839678—21no—
12#16839678—21no—
13#16839671C411yescharacteristic, complemented
14#16839671C2211yescharacteristic
15#16839641—21no—
16#16839622C411yescharacteristic, complemented
17#16839486—21no—
18#16839486C2212yes—
19C22×C22⋊C4×C61C2211yescharacteristic
20C61×#379—21no—
21C61×#342—21no—
22C61×#342C2212yes—
23C22×D244C2×C411yescharacteristic
24C22×C22⋊D61—21no—
25C22×C22⋊D61—21no—
26C22×C22⋊D61—21no—
27C2×#11758194—41no—
28#14519751—41no—
29#14519751—41no—
30#14519746—44no—
31C2×C4⋊Dic61—21no—
32C2×C4⋊Dic61C2×C411yescharacteristic
33C2×C22⋊Dic61—21no—
34C2×C22⋊Dic61—21no—
35C2×C22⋊Dic61—21no—
36C2×#11758198—21no—
37C2×#11758198—21no—
38C2×#11758198—21no—
39C22⋊#92418—22no—
40C22⋊#92418—41no—
41#14519684—21no—
42#14519684—21no—
43#14519684C2×C411yescharacteristic
44#14519684C2×C411yescharacteristic
45C25×C61D411yescharacteristic
46C23×C4×C61C2×C411yescharacteristic
47C23×C4×C61D411yescharacteristic
48C2×C22⋊C4×C61—22no—
49C2×C22⋊C4×C61—41no—
50C2×C22⋊C4×C61—41no—
51C2×C22⋊C4×C61—41no—
52C2×C22⋊C4×C61—41no—
53C2×C22⋊C4×C61C2×C411yescharacteristic
54C2×C22⋊C4×C61D411yescharacteristic
55C2×C22⋊C4×C61D411yescharacteristic
56C2×C22⋊C4×C61C2311yescharacteristic
57C24×C61—21no—
58C24×C61—21no—
59C24×C61—22no—
60C24×C61—41no—
61C24×C61—81no—
62C24×C61—81no—
63C24×C61C22⋊C411yescharacteristic
64C24×C61C22⋊C411yescharacteristic
65C24×C61C2×D411yescharacteristic
66C23×D61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—42no—
72C22×C4×C61—81no—
73C22×C4×C61—81no—
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22⋊C411yescharacteristic
76C22×C4×C61C22×C411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C2×C22⋊D61—42no—
79C2×C22⋊D61—42no—
80C2×C22⋊D61—42no—
81C2×C22⋊D61—41no—
82C2×C22⋊D61—41no—
83C2×C22⋊D61—42no—
84C2×C22⋊D61—81no—
85C2×C22⋊D61—81no—
86C2×C22⋊D61—81no—
87C2×C22⋊D61—81no—
88C22×Dic61—21no—
89C22×Dic61—21no—
90C22×Dic61—21no—
91C2×D244—41no—
92C2×D244—81no—
93C22⋊C4×C61—41no—
94C22⋊C4×C61—41no—
95C22⋊C4×C61—41no—
96C22⋊C4×C61—41no—
97C22⋊C4×C61—42no—
98C22⋊C4×C61—42no—
99C22⋊C4×C61—82no—
100C22⋊C4×C61—82no—
101C22⋊Dic61—42no—
102C22⋊Dic61—42no—
103C22⋊Dic61—42no—
104#11758198—42no—
105#11758198—81no—
106#11758198—81no—
107C4⋊Dic61—41no—
108C4⋊Dic61—81no—
109#11758194—82no—
110C2×#92418—42no—
111C23×C61—21no—
112C23×C61—21no—
113C23×C61—21no—
114C23×C61—21no—
115C23×C61—21no—
116C23×C61—21no—
117C23×C61—21no—
118C23×C61—41no—
119C23×C61—41no—
120C23×C61—42no—
121C23×C61—42no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61—41no—
126C23×C61—42no—
127C23×C61—42no—
128C23×C61—81no—
129C23×C61—81no—
130C23×C61—84no—
131C23×C61—84no—
132C23×C61#9911yescharacteristic
133C23×C61#9911yescharacteristic
134C23×C61C2×C22⋊C411yescharacteristic
135C23×C61AGL(1,ℤ/8)12yes—
136C22×D61—41no—
137C22×D61—41no—
138C22×D61—41no—
139C22×D61—41no—
140C22×D61—41no—
141C22×D61—41no—
142C22×D61—41no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—42no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—81no—
153C2×C4×C61—81no—
154C2×C4×C61—81no—
155C2×C4×C61—82no—
156C2×C4×C61—82no—
157C2×C4×C61—82no—
158C22⋊D61—82no—
159C22⋊D61—82no—
160C22⋊D61—84no—
161C22⋊D61—84no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171D244—82no—
172#92418—82no—
173#11799—611nomaximal, Sylow
174C22×C61—21no—
175C22×C61—21no—
176C22×C61—21no—
177C22×C61—21no—
178C22×C61—21no—
179C22×C61—21no—
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—22no—
183C22×C61—42no—
184C22×C61—42no—
185C22×C61—42no—
186C22×C61—42no—
187C22×C61—41no—
188C22×C61—41no—
189C22×C61—41no—
190C22×C61—41no—
191C22×C61—41no—
192C22×C61—81no—
193C22×C61—81no—
194C22×C61—84no—
195C22×C61—84no—
196C22×C61C2×#9911yescharacteristic
197C22×C61#41811yescharacteristic
198C22×C61#42111yescharacteristic
199C2×D61—81no—
200C2×D61—81no—
201C2×D61—81no—
202C2×D61—81no—
203C2×D61—81no—
204C2×D61—81no—
205C2×D61—81no—
206C4×C61—41no—
207C4×C61—41no—
208C4×C61—41no—
209C4×C61—41no—
210C4×C61—162no—
211Dic61—81no—
212Dic61—81no—
213Dic61—81no—
214#2617—611no—
215#1819D6111yescharacteristic, complemented
216#1462—611no—
217#1344—612no—
218#1343—612no—
219C2×C61—21no—
220C2×C61—21no—
221C2×C61—42no—
222C2×C61—41no—
223C2×C61—41no—
224C2×C61—81no—
225C2×C61#151011yescharacteristic
226C2×C61#212111yescharacteristic
227C2×C61#212711yescharacteristic
228D61—161no—
229C2×#121—611no—
230C2×#120—1221no—
231C22×C22⋊C4C2×D6111yescharacteristic
232#407—1221no—
233#397—611no—
234#397—611no—
235#394—1221no—
236#394—1221no—
237#392—611no—
238#392—611no—
239#386—1221no—
240#380—611no—
241#379—21no—
242#342—21no—
243#342C2×D6112yes—
244#323—612no—
245#323—1221no—
246C61#1179911yescharacteristic, complemented, Sylow
247C25C22⋊D6111yescharacteristic
248C22×D4—611no—
249C22×D4—1221no—
250C22×D4—1221no—
251C22×D4—1221no—
252C23×C4C4×D6111yescharacteristic
253C23×C4D24411yescharacteristic
254C2×M16—2441no—
255#121—2441no—
256#121—2441no—
257#120—2444no—
258C2×C4⋊C4—611no—
259C2×C4⋊C4—1221no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—1221no—
266C2×C22⋊C4—1221no—
267C2×C22⋊C4—1221no—
268C2×C22⋊C4—1221no—
269C2×C22⋊C4—1221no—
270C2×C22⋊C4—1221no—
271C2×C22⋊C4C4×D6111yescharacteristic
272C2×C22⋊C4D24411yescharacteristic
273C2×C22⋊C4C22⋊D6111yescharacteristic
274C2×C22⋊C4C22×D6111yescharacteristic
275C22⋊C8—1222no—
276C22⋊C8—2441no—
277#95—611no—
278#95—611no—
279#95—1221no—
280#95—1221no—
281C24—21no—
282C24—21no—
283C24—22no—
284C24—41no—
285C24—81no—
286C24—81no—
287C24—1221no—
288C24#1175819811yescharacteristic
289C24#1175819811yescharacteristic
290C24C2×C22⋊D6111yescharacteristic
291C2×D4—4881no—
292C2×D4—4881no—
293C2×D4—4881no—
294C2×D4—4881no—
295C2×D4—4881no—
296C2×D4—2441no—
297C2×D4—2442no—
298C2×D4—2442no—
299C2×D4—2442no—
300C2×D4—2441no—
301C2×D4—2441no—
302C2×D4—2442no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—42no—
308C22×C4—81no—
309C22×C4—81no—
310C22×C4—1221no—
311C22×C4—1221no—
312C22×C4—1221no—
313C22×C4#1175819811yescharacteristic
314C22×C4#1175819811yescharacteristic
315C22×C4C2×C4×D6111yescharacteristic
316C22×C4C2×D24411yescharacteristic
317M16—4882no—
318C2×C8—2442no—
319C4⋊C4—4881no—
320C4⋊C4—2441no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C22⋊C4—4881no—
330C22⋊C4—4881no—
331C22⋊C4—2442no—
332C22⋊C4—2442no—
333C22⋊C4—2442no—
334C22⋊C4—2442no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—41no—
343C23—41no—
344C23—42no—
345C23—42no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—42no—
351C23—42no—
352C23—81no—
353C23—81no—
354C23—84no—
355C23—84no—
356C23—2441no—
357C23—2441no—
358C23—2441no—
359C23—2441no—
360C23—2441no—
361C23—2441no—
362C23—2441no—
363C23#1451969111yescharacteristic
364C23#1451969111yescharacteristic
365C23C2×#1175819811yescharacteristic
366C23#1451980212yes—
367D4—4882no—
368D4—4882no—
369D4—4882no—
370D4—4884no—
371D4—4884no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—2441no—
388C2×C4—2441no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4—2441no—
392C2×C4—2441no—
393C2×C4—2441no—
394C2×C4—2441no—
395C2×C4—2441no—
396C8—4882no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—22no—
404C22—21no—
405C22—21no—
406C22—42no—
407C22—42no—
408C22—42no—
409C22—42no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—81no—
416C22—81no—
417C22—84no—
418C22—84no—
419C22—4881no—
420C22—4881no—
421C22—4881no—
422C22—4881no—
423C22—4881no—
424C22—4881no—
425C22—4881no—
426C22C2×#1451969111yescharacteristic
427C22#1683976611yescharacteristic
428C22#1683978011yescharacteristic
429C4—41no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—162no—
434C4—4881no—
435C4—4881no—
436C4—4881no—
437C2—21no—
438C2—21no—
439C2—42no—
440C2—41no—
441C2—41no—
442C2—81no—
443C2—9761no—
444C2#2144434711yescharacteristic
445C2#2144746311yescharacteristic
446C2#2144748611yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC61×#1819C2
socleC22×C61C2×#99
prime radical#21451553C2
cosocleC2×C22⋊C4×C61C23
hypercenter#1819D61
hypocenterC61#11799
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #1819

Complements

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

NH ≅ G/Ntype
2. #21444146C2semidirect
3. #21443818C2semidirect
4. #21443815C2semidirect
5. C61×#1819C2semidirect
6. C2×#14519751C4semidirect
9. #16839687C4semidirect
13. #16839671C4semidirect
16. #16839622C4semidirect
215. #1819D61semidirect
246. C61#11799semidirect

Maximal subgroups

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

subgroupindex
#214515532
#214441462
#214438182
#214438152
C61×#18192
#1179961