Finite Group Databasebeta

← #29576965

Subgroups — #29576965

Order 15616.

438 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 85 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#21456651C211yesmaximal, characteristic
2#21454879C211yesmaximal, characteristic
3#21444376C211yesmaximal, characteristic
4#21444353C211yesmaximal, characteristic, complemented
5#21444182C211yesmaximal, characteristic, complemented
6#21443728C211yesmaximal, characteristic, complemented
7C61×#1293C211yesmaximal, characteristic, complemented
8#16840273C2211yescharacteristic
9C2×#14519756C2211yescharacteristic
10#16840104—21no—
11#16840098C411yescharacteristic, complemented
12#16840081C411yescharacteristic, complemented
13#16840020—21no—
14#16840008—21no—
15#16839926—21no—
16#16839920C411yescharacteristic, complemented
17#16839903C411yescharacteristic, complemented
18#16839838—21no—
19#16839837—21no—
20#16839836—21no—
21#16839831—21no—
22#16839820—21no—
23#16839727—21no—
24#16839726C2211yescharacteristic
25#16839625C2211yescharacteristic
26#16839500—21no—
27#16839500—21no—
28#16839500—21no—
29#16839497—21no—
30#16839487—21no—
31#16839487—21no—
32C61×#407C2211yescharacteristic
33C61×#407C2211yescharacteristic
34C2×C22⋊C8×C61C2211yescharacteristic
35C2×#11758222D411yescharacteristic
36C22×Dic122C2×C411yescharacteristic
37C2×Dic244—21no—
38C2×#11758190—21no—
39C2×#11758194—21no—
40C2×#11758189—21no—
41C2×C8×D61—21no—
42#14519760—21no—
43#14519756—21no—
44#14519756—21no—
45#14519756—21no—
46#14519756—21no—
47#14519756C2×C411yescharacteristic
48#14519756D411yescharacteristic
49#14519751C2×C411yescharacteristic
50#14519751D411yescharacteristic
51#14519746—21no—
52C4×C22⋊D61—21no—
53C4×C22⋊D61—21no—
54#14519736D411yescharacteristic
55C2×C4⋊Dic61C2×C411yescharacteristic
56C2×#11758196—21no—
57C2×C22⋊Dic61—21no—
58C2×C22⋊Dic61—21no—
59#14519711—21no—
60#14519709—21no—
61#14519706—21no—
62#14519706—21no—
63#14519702—21no—
64#14519702—21no—
65#14519702—21no—
66#14519699—21no—
67C22⋊#92418—21no—
68C22⋊#92418D411yescharacteristic
69C22⋊#92418D411yescharacteristic
70#14519689—21no—
71#14519689—21no—
72#14519687—21no—
73C8×Dic61—21no—
74#14519684—21no—
75#14519684—21no—
76C23×C4×C61C2311yescharacteristic
77C2×M16×C61—21no—
78C2×M16×C61—21no—
79C22×C8×C61—21no—
80C22⋊C8×C61—21no—
81C22⋊C8×C61—21no—
82C22⋊C8×C61—21no—
83C22⋊C8×C61—21no—
84C22⋊C8×C61C2×C411yescharacteristic
85C22⋊C8×C61C2×C411yescharacteristic
86C22⋊C8×C61D411yescharacteristic
87C22⋊C8×C61D411yescharacteristic
88C24×C61#4111yescharacteristic
89C22×C4×C61—81no—
90C22×C4×C61C22×C411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61#4111yescharacteristic
96C2×C22⋊D61—21no—
97C2×C22⋊D61—21no—
98C2×C22⋊D61—21no—
99C22×Dic61—21no—
100C22×Dic61—21no—
101C22×Dic61—41no—
102#11758222—81no—
103C2×D244C22⋊C411yescharacteristic
104C2×C4×D61—21no—
105C2×Dic122—21no—
106C2×Dic122—21no—
107C2×Dic122—81no—
108C2×Dic122C22⋊C411yescharacteristic
109M16×C61—81no—
110M16×C61—81no—
111C2×C8×C61—21no—
112C2×C8×C61—21no—
113C2×C8×C61—21no—
114C2×C8×C61—21no—
115C2×C8×C61—41no—
116C2×C8×C61—81no—
117C22⋊Dic61—21no—
118C22⋊Dic61—21no—
119C22⋊Dic61—21no—
120C22⋊Dic61—41no—
121C22⋊Dic61—41no—
122C22⋊Dic61—81no—
123#11758198—21no—
124#11758198—21no—
125C4⋊Dic61—21no—
126C4⋊Dic61—21no—
127C4⋊Dic61C22⋊C411yescharacteristic
128C4⋊Dic61C22⋊C411yescharacteristic
129#11758196—21no—
130#11758196—21no—
131#11758196—41no—
132#11758196—41no—
133C4×Dic61—21no—
134#11758194—81no—
135C2×#92418—21no—
136C2×#92418—21no—
137Dic244—81no—
138#11758190—81no—
139#11758189—81no—
140C8×D61—81no—
141C23×C61—41no—
142C23×C61—41no—
143C23×C61—41no—
144C23×C61C4×D411yescharacteristic
145C23×C61#12111yescharacteristic
146C23×C61#12311yescharacteristic
147C22×D61—21no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61C2×C22⋊C411yescharacteristic
155C2×C4×C61C4×D411yescharacteristic
156C2×C4×C61#12011yescharacteristic
157C2×C4×C61#12111yescharacteristic
158C22⋊D61—81no—
159C22⋊D61—81no—
160C22⋊D61—81no—
161C2×Dic61—21no—
162C2×Dic61—21no—
163C2×Dic61—21no—
164C2×Dic61—42no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—81no—
168C2×Dic61—81no—
169C2×Dic61—81no—
170C2×Dic61—81no—
171D244—41no—
172C4×D61—41no—
173C4×D61—41no—
174Dic122—41no—
175Dic122—41no—
176Dic122—41no—
177Dic122—81no—
178C8×C61—41no—
179C8×C61—41no—
180C8×C61—41no—
181C8×C61—41no—
182C8×C61—81no—
183#92418—41no—
184#92418—41no—
185#9331—611nomaximal, Sylow
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—82no—
196C22×C61#38611yescharacteristic
197C2×D61—41no—
198C2×D61—41no—
199C2×D61—41no—
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61—21no—
204Dic61—41no—
205Dic61—41no—
206Dic61—41no—
207Dic61—81no—
208Dic61—81no—
209Dic61—161no—
210#3196—611no—
211#2978—611no—
212#1516—611no—
213#1511—611no—
214#1472—611no—
215#1306—611no—
216#1293D6111yescharacteristic, complemented
217C2×C61—41no—
218C2×C61—41no—
219C2×C61—41no—
220C2×C61#188111yescharacteristic
221C2×C61#190211yescharacteristic
222C2×C61#191311yescharacteristic
223D61—81no—
224#525—611no—
225C2×C22⋊Q8—611no—
226#485—1221no—
227#484—611no—
228#481—611no—
229#470—1221no—
230#468—1221no—
231#452—1221no—
232#451—611no—
233#448—611no—
234#436—1221no—
235#436—1221no—
236#435—1221no—
237#435—1221no—
238#433—1221no—
239#408—1221no—
240#407—611no—
241#407C2×D6111yescharacteristic
242#407C2×D6111yescharacteristic
243C2×C22⋊C8C2×D6111yescharacteristic
244#381—611no—
245#328—1221no—
246#328—1221no—
247#328—1221no—
248#327—1221no—
249#324—1221no—
250#324—1221no—
251C61#933111yescharacteristic, complemented, Sylow
252C2×#41—611no—
253C22×Q8—611no—
254C23×C4C22×D6111yescharacteristic
255C2×Q16—1221no—
256C2×SD16—1221no—
257C2×M16—21no—
258C2×M16—21no—
259C2×M16—1221no—
260C2×M16—1221no—
261C22×C8—21no—
262C22×C8—1221no—
263#123—1221no—
264C22⋊Q8—611no—
265C22⋊Q8—611no—
266C22⋊Q8—1221no—
267C22⋊Q8—1221no—
268C22⋊Q8—1221no—
269C22⋊Q8—1221no—
270#121—611no—
271#121—611no—
272#120—1221no—
273C4×D4—1221no—
274C4×D4—1221no—
275#117—611no—
276C2×C4⋊C4—611no—
277C2×C4⋊C4—1221no—
278C2×C22⋊C4—1221no—
279C2×C22⋊C4—1221no—
280#107—1221no—
281#106—1221no—
282C4⋊C8—1221no—
283C4⋊C8—1221no—
284#103—1221no—
285#103—1221no—
286#103—1221no—
287#102—1221no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292C22⋊C8—611no—
293C22⋊C8—611no—
294C22⋊C8—1221no—
295C22⋊C8—1221no—
296C22⋊C8—1221no—
297C22⋊C8C4×D6111yescharacteristic
298C22⋊C8C4×D6111yescharacteristic
299C22⋊C8D24411yescharacteristic
300C22⋊C8D24411yescharacteristic
301#97—1221no—
302C4×C8—1221no—
303#95—1221no—
304#95—1221no—
305C24#1175822611yescharacteristic
306#41—4881no—
307C2×Q8—611no—
308C2×Q8—1221no—
309C2×Q8—1221no—
310C2×Q8—4881no—
311C2×D4—611no—
312C2×D4—1221no—
313C2×D4—1221no—
314C2×D4—1221no—
315C22×C4—81no—
316C22×C4—1221no—
317C22×C4—1221no—
318C22×C4—1221no—
319C22×C4—2441no—
320C22×C4C2×C4×D6111yescharacteristic
321C22×C4C2×D24411yescharacteristic
322C22×C4#1175822211yescharacteristic
323C22×C4D4×D6111yescharacteristic
324C22×C4D4×D6111yescharacteristic
325C22×C4D4×D6111yescharacteristic
326Q16—4881no—
327SD16—4881no—
328M16—81no—
329M16—81no—
330M16—4881no—
331M16—4881no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—41no—
337C2×C8—81no—
338C2×C8—1221no—
339C2×C8—1221no—
340C2×C8—4881no—
341C4⋊C4—611no—
342C4⋊C4—611no—
343C4⋊C4—1221no—
344C4⋊C4—1221no—
345C4⋊C4—1221no—
346C4⋊C4—1221no—
347C4⋊C4—2441no—
348C4⋊C4—2441no—
349C22⋊C4—1221no—
350C22⋊C4—1221no—
351C22⋊C4—1221no—
352C22⋊C4—1221no—
353C22⋊C4—1221no—
354C22⋊C4—4881no—
355C22⋊C4—2441no—
356C22⋊C4—2441no—
357C42—1221no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—1221no—
362C23#1451973911yescharacteristic
363C23#1451975211yescharacteristic
364C23#1451976211yescharacteristic
365Q8—4881no—
366Q8—2441no—
367Q8—2441no—
368Q8—2441no—
369D4—4881no—
370D4—4881no—
371D4—4881no—
372D4—2441no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—1221no—
380C2×C4—1221no—
381C2×C4—1221no—
382C2×C4—4881no—
383C2×C4—4881no—
384C2×C4—4881no—
385C2×C4—4881no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4—2442no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4C22⋊C4×D6111yescharacteristic
392C2×C4C4×D24411yescharacteristic
393C2×C4#1451974711yescharacteristic
394C2×C4#1451975011yescharacteristic
395C8—41no—
396C8—41no—
397C8—41no—
398C8—41no—
399C8—81no—
400C8—2441no—
401C8—2441no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—82no—
412C22—2441no—
413C22—2441no—
414C22—2441no—
415C22#1683964011yescharacteristic
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—9761no—
421C4—4881no—
422C4—4881no—
423C4—2441no—
424C4—2441no—
425C4—2441no—
426C2—41no—
427C2—41no—
428C2—41no—
429C2—4881no—
430C2#2144618211yescharacteristic
431C2#2144629311yescharacteristic
432C2#2144636011yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1293C2
socleC22×C61#386
prime radical#16840273C22
cosocleC23×C4×C61C23
hypercenter#1293D61
hypocenterC61#9331
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1293

Complements

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

NH ≅ G/Ntype
4. #21444353C2semidirect
5. #21444182C2semidirect
6. #21443728C2semidirect
7. C61×#1293C2semidirect
11. #16840098C4semidirect
12. #16840081C4semidirect
16. #16839920C4semidirect
17. #16839903C4semidirect
216. #1293D61semidirect
251. C61#9331semidirect

Maximal subgroups

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

subgroupindex
#214566512
#214548792
#214443762
#214443532
#214441822
#214437282
C61×#12932
#933161