Finite Group Databasebeta

← #29590296

Subgroups — #29590296

Order 15616.

592 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 107 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#21451624C211yesmaximal, characteristic
2C2×#16839486C211yesmaximal, characteristic
3#21443842C212yesmaximal, complemented
4#21443839C212yesmaximal, complemented
5C2×C61×#342C211yesmaximal, characteristic
6C2×#14519756C411yescharacteristic, complemented
7C2×C22⋊#92418—21no—
8#16839698C411yescharacteristic, complemented
9#16839698C2211yescharacteristic
10#16839685—21no—
11#16839685—21no—
12#16839674C411yescharacteristic, complemented
13#16839674C2211yescharacteristic
14#16839635—21no—
15#16839622C411yescharacteristic, complemented
16#16839618—21no—
17#16839486—21no—
18#16839486C2212yes—
19C22×C22⋊C4×C61—21no—
20C22×C22⋊C4×C61C2211yescharacteristic
21C61×#342—21no—
22C61×#342C2212yes—
23C22×Dic122C2×C411yescharacteristic, complemented
24C22×#92418—41no—
25#14519756—41no—
26#14519756—41no—
27C2×C4⋊Dic61—21no—
28C2×C4⋊Dic61C2×C411yescharacteristic, complemented
29C2×#11758196—21no—
30C2×#11758196—21no—
31C2×#11758196—21no—
32C2×C22⋊Dic61—21no—
33C2×C22⋊Dic61—21no—
34C2×C22⋊Dic61—21no—
35C2×C22⋊Dic61—21no—
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61—21no—
38C22⋊#92418—22no—
39C22⋊#92418—41no—
40#14519684—21no—
41#14519684—21no—
42#14519684C2×C411yescharacteristic, complemented
43#14519684C2×C411yescharacteristic, complemented
44C25×C61D411yescharacteristic
45C23×C4×C61—41no—
46C23×C4×C61C2×C411yescharacteristic
47C23×C4×C61D411yescharacteristic
48C2×C22⋊C4×C61—24no—
49C2×C22⋊C4×C61—22no—
50C2×C22⋊C4×C61—41no—
51C2×C22⋊C4×C61—41no—
52C2×C22⋊C4×C61—41no—
53C2×C22⋊C4×C61—42no—
54C2×C22⋊C4×C61C2×C411yescharacteristic
55C2×C22⋊C4×C61D411yescharacteristic
56C2×C22⋊C4×C61D411yescharacteristic
57C2×C22⋊C4×C61C2311yescharacteristic
58C24×C61—21no—
59C24×C61—21no—
60C24×C61—22no—
61C24×C61—81no—
62C24×C61—81no—
63C24×C61C22⋊C411yescharacteristic
64C24×C61C22⋊C411yescharacteristic
65C24×C61SD1612yes—
66C24×C61Q1612yes—
67C24×C61C2×D411yescharacteristic
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—44no—
73C22×C4×C61—42no—
74C22×C4×C61—81no—
75C22×C4×C61—81no—
76C22×C4×C61—82no—
77C22×C4×C61—82no—
78C22×C4×C61C22⋊C411yescharacteristic
79C22×C4×C61C22⋊C411yescharacteristic
80C22×C4×C61C22×C411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C22×Dic61—21no—
83C22×Dic61—21no—
84C22×Dic61—21no—
85C22×Dic61—21no—
86C2×Dic122—41no—
87C2×Dic122—81no—
88C22⋊C4×C61—24no—
89C22⋊C4×C61—41no—
90C22⋊C4×C61—41no—
91C22⋊C4×C61—41no—
92C22⋊C4×C61—41no—
93C22⋊C4×C61—44no—
94C22⋊C4×C61—42no—
95C22⋊Dic61—41no—
96C22⋊Dic61—41no—
97C22⋊Dic61—42no—
98C22⋊Dic61—81no—
99C22⋊Dic61—81no—
100C22⋊Dic61—81no—
101C22⋊Dic61—81no—
102C4⋊Dic61—41no—
103C4⋊Dic61—81no—
104#11758196—42no—
105#11758196—81no—
106#11758196—81no—
107C2×#92418—42no—
108C2×#92418—82no—
109C23×C61—22no—
110C23×C61—22no—
111C23×C61—21no—
112C23×C61—21no—
113C23×C61—22no—
114C23×C61—22no—
115C23×C61—21no—
116C23×C61—21no—
117C23×C61—41no—
118C23×C61—41no—
119C23×C61—42no—
120C23×C61—42no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61—81no—
126C23×C61—81no—
127C23×C61—82no—
128C23×C61—82no—
129C23×C61—84no—
130C23×C61#9911yescharacteristic
131C23×C61#9911yescharacteristic
132C23×C61#10312yes—
133C23×C61#10312yes—
134C23×C61#10412yes—
135C23×C61C2×C22⋊C411yescharacteristic
136C23×C61C2×SD1611yescharacteristic
137C23×C61C2×Q1611yescharacteristic
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—42no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—44no—
148C2×C4×C61—81no—
149C2×C4×C61—81no—
150C2×C4×C61—81no—
151C2×C4×C61—88no—
152C2×C4×C61—82no—
153C2×C4×C61—82no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166Dic122—82no—
167#92418—82no—
168#11716—611nomaximal, Sylow
169C22×C61—21no—
170C22×C61—21no—
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—21no—
174C22×C61—21no—
175C22×C61—21no—
176C22×C61—22no—
177C22×C61—21no—
178C22×C61—42no—
179C22×C61—42no—
180C22×C61—42no—
181C22×C61—41no—
182C22×C61—41no—
183C22×C61—44no—
184C22×C61—44no—
185C22×C61—41no—
186C22×C61—41no—
187C22×C61—41no—
188C22×C61—41no—
189C22×C61—81no—
190C22×C61—81no—
191C22×C61—82no—
192C22×C61—82no—
193C22×C61#32812yes—
194C22×C61#32812yes—
195C22×C61C2×#9911yescharacteristic
196C22×C61C2×#10311yescharacteristic
197C22×C61C2×#10411yescharacteristic
198C4×C61—41no—
199C4×C61—41no—
200C4×C61—41no—
201C4×C61—41no—
202C4×C61—84no—
203Dic61—81no—
204Dic61—81no—
205Dic61—81no—
206Dic61—81no—
207#2626—611no—
208C2×#342D6111yescharacteristic
209C2×#323—611no—
210#1352—612no—
211#1351—612no—
212C2×C61—21no—
213C2×C61—22no—
214C2×C61—21no—
215C2×C61—22no—
216C2×C61—41no—
217C2×C61—42no—
218C2×C61—41no—
219C2×C61C2×#32811yescharacteristic
220C2×C61#212111yescharacteristic
221C2×C61#212711yescharacteristic
222C2×C22⋊Q8—611no—
223C22×C22⋊C4—21no—
224C22×C22⋊C4C2×D6111yescharacteristic
225C2×C22⋊C8—1221no—
226#399—611no—
227#399—611no—
228#396—1221no—
229#396—1221no—
230#393—611no—
231#393—611no—
232#385—1221no—
233#380—611no—
234#379—1221no—
235#342—21no—
236#342C2×D6112yes—
237#323—612no—
238#323—1221no—
239C61#1171611yescharacteristic, complemented, Sylow
240C25C22⋊D6111yescharacteristic
241C22×Q8—611no—
242C23×C4—41no—
243C23×C4C4×D6111yescharacteristic
244C23×C4D24411yescharacteristic
245C22×C8—2441no—
246C22⋊Q8—2441no—
247C22⋊Q8—2441no—
248C2×C4⋊C4—611no—
249C2×C4⋊C4—1221no—
250C2×C4⋊C4—1221no—
251C2×C4⋊C4—1221no—
252C2×C4⋊C4—1221no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—1221no—
260C2×C22⋊C4—1221no—
261C2×C22⋊C4—1221no—
262C2×C22⋊C4—1221no—
263C2×C22⋊C4—1221no—
264C2×C22⋊C4—1221no—
265C2×C22⋊C4C4×D6111yescharacteristic
266C2×C22⋊C4D24411yescharacteristic
267C2×C22⋊C4C22⋊D6111yescharacteristic
268C2×C22⋊C4C22×D6111yescharacteristic
269C22⋊C8—1222no—
270C22⋊C8—2441no—
271#95—611no—
272#95—611no—
273#95—1221no—
274#95—1221no—
275C24—21no—
276C24—21no—
277C24—22no—
278C24—81no—
279C24—81no—
280C24#1175819811yescharacteristic
281C24#1175819811yescharacteristic
282C24#1175820012yes—
283C24#1175820212yes—
284C24C2×C22⋊D6111yescharacteristic
285C2×Q8—4881no—
286C2×Q8—2441no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—81no—
294C22×C4—81no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4—1221no—
298C22×C4—1221no—
299C22×C4—1221no—
300C22×C4—1221no—
301C22×C4#1175819811yescharacteristic
302C22×C4#1175819811yescharacteristic
303C22×C4C2×C4×D6111yescharacteristic
304C22×C4C2×D24411yescharacteristic
305C2×C8—4882no—
306C2×C8—2442no—
307C4⋊C4—4881no—
308C4⋊C4—4881no—
309C4⋊C4—4881no—
310C4⋊C4—2441no—
311C4⋊C4—2442no—
312C22⋊C4—24no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—44no—
318C22⋊C4—42no—
319C22⋊C4—4881no—
320C22⋊C4—4881no—
321C22⋊C4—4881no—
322C22⋊C4—4881no—
323C22⋊C4—2441no—
324C22⋊C4—2441no—
325C22⋊C4—2442no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—21no—
333C23—21no—
334C23—41no—
335C23—41no—
336C23—42no—
337C23—42no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—81no—
343C23—81no—
344C23—82no—
345C23—82no—
346C23—84no—
347C23#1451969111yescharacteristic
348C23#1451969111yescharacteristic
349C23#1451970012yes—
350C23#1451970012yes—
351C23#1451970512yes—
352C23C2×#1175819811yescharacteristic
353C23C2×#1175820011yescharacteristic
354C23C2×#1175820211yescharacteristic
355Q8—4882no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—44no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—88no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—2441no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4—2441no—
379C2×C4—2441no—
380C2×C4—2441no—
381C2×C4—2441no—
382C2×C4—2441no—
383C2×C4—2441no—
384C8—4882no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—41no—
398C22—41no—
399C22—44no—
400C22—44no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—81no—
406C22—81no—
407C22—82no—
408C22—82no—
409C22#1683949812yes—
410C22#1683949812yes—
411C22C2×#1451969111yescharacteristic
412C22C2×#1451970011yescharacteristic
413C22C2×#1451970511yescharacteristic
414C4—41no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—84no—
419C4—4881no—
420C4—4881no—
421C4—4881no—
422C4—4881no—
423C2—21no—
424C2—21no—
425C2—22no—
426C2—22no—
427C2—41no—
428C2—42no—
429C2—41no—
430C2C2×#1683949811yescharacteristic
431C2#2144746311yescharacteristic
432C2#2144748611yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC2×C61×#342C2
socleC22×C61C2×#99
prime radicalC25×C61D4
cosocleC2×C22⋊C4×C61C23
hypercenterC2×#342D61
hypocenterC61#11716
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#342

Complements

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

NH ≅ G/Ntype
3. #21443842C2semidirect
4. #21443839C2semidirect
6. C2×#14519756C4semidirect
8. #16839698C4semidirect
12. #16839674C4semidirect
15. #16839622C4semidirect
23. C22×Dic122C2×C4semidirect
28. C2×C4⋊Dic61C2×C4semidirect
42. #14519684C2×C4semidirect
43. #14519684C2×C4semidirect
239. C61#11716semidirect

Maximal subgroups

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

subgroupindex
#214516242
C2×#168394862
#214438422
#214438392
C2×C61×#3422
#1171661