Finite Group Databasebeta

← #29576489

Subgroups — #29576489

Order 15616.

478 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 93 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#21456602C211yesmaximal, characteristic
2#21454873C211yesmaximal, characteristic
3#21444324C211yesmaximal, characteristic
4C2×#16839500C211yesmaximal, characteristic, complemented
5#21444184C211yesmaximal, characteristic, complemented
6#21443661C211yesmaximal, characteristic, complemented
7C61×#1298C211yesmaximal, characteristic, complemented
8#16840273C2211yescharacteristic
9C2×#14519756C2211yescharacteristic
10#16840104—21no—
11#16840093C411yescharacteristic, complemented
12#16840076C411yescharacteristic, complemented
13#16840003—21no—
14#16839991—21no—
15#16839926—21no—
16#16839920C411yescharacteristic, complemented
17#16839903C411yescharacteristic, complemented
18#16839833—21no—
19#16839830—21no—
20C8×C22⋊D61—21no—
21#16839825—21no—
22#16839820—21no—
23#16839727—21no—
24C2×C22⋊#92418C2211yescharacteristic
25C2×#14519684C2211yescharacteristic
26#16839500—21no—
27#16839500—21no—
28#16839500—21no—
29#16839497—21no—
30#16839487—21no—
31#16839487—21no—
32C61×#407C2211yescharacteristic
33C2×C22⋊C8×C61C2211yescharacteristic
34C2×C22⋊C8×C61C2211yescharacteristic
35C2×#11758222D411yescharacteristic
36C22×Dic122C2×C411yescharacteristic
37C23×Dic61—21no—
38C2×Dic244—21no—
39C2×#11758190—21no—
40C2×#11758189—21no—
41C22×#92418—21no—
42C2×C8×D61—21no—
43#14519760—21no—
44#14519756—21no—
45#14519756—21no—
46#14519756—21no—
47#14519756—21no—
48#14519756C2×C411yescharacteristic
49#14519756D411yescharacteristic
50#14519751C2×C411yescharacteristic
51#14519751D411yescharacteristic
52#14519746—21no—
53C4×C22⋊D61—21no—
54C4×C22⋊D61—21no—
55#14519736D411yescharacteristic
56C2×C4⋊Dic61C2×C411yescharacteristic
57C2×#11758196—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—
78C22×C8×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—22no—
101C22×Dic61—21no—
102C22×Dic61—41no—
103C22×Dic61—41no—
104C22×Dic61—81no—
105C22×Dic61—81no—
106#11758222—81no—
107C2×D244C22⋊C411yescharacteristic
108C2×C4×D61—21no—
109C2×Dic122—21no—
110C2×Dic122—21no—
111C2×Dic122—81no—
112C2×Dic122C22⋊C411yescharacteristic
113M16×C61—81no—
114C2×C8×C61—21no—
115C2×C8×C61—21no—
116C2×C8×C61—21no—
117C2×C8×C61—21no—
118C2×C8×C61—41no—
119C2×C8×C61—81no—
120C2×C8×C61—81no—
121C22⋊Dic61—21no—
122C22⋊Dic61—21no—
123C22⋊Dic61—21no—
124C22⋊Dic61—41no—
125C22⋊Dic61—41no—
126#11758198—21no—
127#11758198—21no—
128C4⋊Dic61—21no—
129C4⋊Dic61—21no—
130C4⋊Dic61C22⋊C411yescharacteristic
131C4⋊Dic61C22⋊C411yescharacteristic
132#11758196—21no—
133#11758196—21no—
134#11758196—41no—
135#11758196—41no—
136C4×Dic61—21no—
137C2×#92418—21no—
138C2×#92418—21no—
139C2×#92418—81no—
140Dic244—81no—
141#11758190—81no—
142#11758189—81no—
143C8×D61—81no—
144C23×C61—21no—
145C23×C61—21no—
146C23×C61—21no—
147C23×C61—21no—
148C23×C61—41no—
149C23×C61C4×D411yescharacteristic
150C23×C61#12111yescharacteristic
151C23×C61#12311yescharacteristic
152C22×D61—21no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61C2×C22⋊C411yescharacteristic
160C2×C4×C61C4×D411yescharacteristic
161C2×C4×C61#12011yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C22⋊D61—41no—
164C22⋊D61—41no—
165C22⋊D61—41no—
166C22⋊D61—41no—
167C22⋊D61—81no—
168C2×Dic61—21no—
169C2×Dic61—21no—
170C2×Dic61—21no—
171C2×Dic61—42no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176C2×Dic61—81no—
177C2×Dic61—82no—
178C2×Dic61—81no—
179C2×Dic61—81no—
180C2×Dic61—82no—
181D244—41no—
182C4×D61—41no—
183C4×D61—41no—
184Dic122—41no—
185Dic122—41no—
186Dic122—41no—
187Dic122—81no—
188C8×C61—41no—
189C8×C61—41no—
190C8×C61—41no—
191C8×C61—41no—
192C8×C61—81no—
193#92418—41no—
194#92418—41no—
195#9263—611nomaximal, Sylow
196C22×C61—21no—
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—41no—
204C22×C61—41no—
205C22×C61—41no—
206C22×C61—41no—
207C22×C61#38611yescharacteristic
208C22×C61#44311yescharacteristic
209C22×C61#44411yescharacteristic
210C22×C61#47111yescharacteristic
211C22×C61#47311yescharacteristic
212C2×D61—41no—
213C2×D61—41no—
214C2×D61—41no—
215C4×C61—21no—
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—21no—
219Dic61—41no—
220Dic61—41no—
221Dic61—41no—
222Dic61—82no—
223Dic61—81no—
224Dic61—81no—
225#3190—611no—
226#2977—611no—
227#1504—611no—
228C2×#328—611no—
229#1472—611no—
230#1298D6111yescharacteristic, complemented
231#1280—611no—
232C2×C61—21no—
233C2×C61—21no—
234C2×C61—21no—
235C2×C61—21no—
236C2×C61—41no—
237C2×C61#187611yescharacteristic
238C2×C61#190811yescharacteristic
239C2×C61#191311yescharacteristic
240D61—81no—
241#525—611no—
242C2×C22⋊Q8—611no—
243#485—1221no—
244#483—611no—
245#480—611no—
246#467—1221no—
247#465—1221no—
248#452—1221no—
249#451—611no—
250#448—611no—
251#435—1221no—
252#435—1221no—
253C8×D4—1221no—
254C8×D4—1221no—
255#433—1221no—
256#408—1221no—
257#407C2×D6111yescharacteristic
258C2×C22⋊C8—611no—
259C2×C22⋊C8C2×D6111yescharacteristic
260C2×C22⋊C8C2×D6111yescharacteristic
261C2×#95—611no—
262#328—1221no—
263#328—1221no—
264#328—1221no—
265#327—1221no—
266#324—1221no—
267#324—1221no—
268C61#926311yescharacteristic, complemented, Sylow
269C2×#41—611no—
270C22×Q8—611no—
271C23×C4—1221no—
272C23×C4C22×D6111yescharacteristic
273C2×Q16—1221no—
274C2×SD16—1221no—
275C2×M16—21no—
276C2×M16—1221no—
277C22×C8—21no—
278C22×C8—21no—
279C22×C8—1221no—
280C22×C8—1221no—
281#123—1221no—
282C22⋊Q8—611no—
283C22⋊Q8—611no—
284C22⋊Q8—1221no—
285C22⋊Q8—1221no—
286C22⋊Q8—1221no—
287C22⋊Q8—1221no—
288#121—611no—
289#121—611no—
290#120—1221no—
291C4×D4—1221no—
292C4×D4—1221no—
293#117—611no—
294C2×C4⋊C4—611no—
295C2×C4⋊C4—1221no—
296C2×C22⋊C4—1221no—
297#107—1221no—
298#106—1221no—
299C4⋊C8—1221no—
300C4⋊C8—1221no—
301#103—1221no—
302#103—1221no—
303#103—1221no—
304#102—1221no—
305C22⋊C8—21no—
306C22⋊C8—21no—
307C22⋊C8—21no—
308C22⋊C8—21no—
309C22⋊C8—611no—
310C22⋊C8—611no—
311C22⋊C8—1221no—
312C22⋊C8—1221no—
313C22⋊C8—1221no—
314C22⋊C8C4×D6111yescharacteristic
315C22⋊C8C4×D6111yescharacteristic
316C22⋊C8D24411yescharacteristic
317C22⋊C8D24411yescharacteristic
318#97—1221no—
319C4×C8—1221no—
320#95—1221no—
321#95—1221no—
322C24#1175822611yescharacteristic
323#41—4881no—
324C2×Q8—611no—
325C2×Q8—1221no—
326C2×Q8—1221no—
327C2×Q8—4881no—
328C2×D4—611no—
329C2×D4—1221no—
330C2×D4—1221no—
331C2×D4—1221no—
332C22×C4—81no—
333C22×C4—1221no—
334C22×C4—1222no—
335C22×C4—1221no—
336C22×C4—1221no—
337C22×C4—4881no—
338C22×C4—4881no—
339C22×C4—2441no—
340C22×C4—2441no—
341C22×C4C2×C4×D6111yescharacteristic
342C22×C4C2×D24411yescharacteristic
343C22×C4#1175822211yescharacteristic
344C22×C4D4×D6111yescharacteristic
345C22×C4D4×D6111yescharacteristic
346C22×C4D4×D6111yescharacteristic
347Q16—4881no—
348SD16—4881no—
349M16—81no—
350M16—4881no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—41no—
356C2×C8—81no—
357C2×C8—81no—
358C2×C8—1221no—
359C2×C8—1221no—
360C2×C8—4881no—
361C2×C8—4881no—
362C4⋊C4—611no—
363C4⋊C4—611no—
364C4⋊C4—1221no—
365C4⋊C4—1221no—
366C4⋊C4—1221no—
367C4⋊C4—1221no—
368C4⋊C4—2441no—
369C4⋊C4—2441no—
370C22⋊C4—1221no—
371C22⋊C4—1221no—
372C22⋊C4—1221no—
373C22⋊C4—1221no—
374C22⋊C4—1221no—
375C22⋊C4—2441no—
376C22⋊C4—2441no—
377C42—1221no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—41no—
383C23—1221no—
384C23#1451973911yescharacteristic
385C23#1451975211yescharacteristic
386C23#1451976211yescharacteristic
387Q8—4881no—
388Q8—2441no—
389Q8—2441no—
390Q8—2441no—
391D4—4881no—
392D4—2441no—
393D4—2441no—
394D4—2441no—
395D4—2441no—
396D4—2441no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—1221no—
404C2×C4—1221no—
405C2×C4—1221no—
406C2×C4—4881no—
407C2×C4—4882no—
408C2×C4—4881no—
409C2×C4—4881no—
410C2×C4—4882no—
411C2×C4—2441no—
412C2×C4—2441no—
413C2×C4—2442no—
414C2×C4—2441no—
415C2×C4—2441no—
416C2×C4—2441no—
417C2×C4—2441no—
418C2×C4C22⋊C4×D6111yescharacteristic
419C2×C4C4×D24411yescharacteristic
420C2×C4#1451974711yescharacteristic
421C2×C4#1451975011yescharacteristic
422C8—41no—
423C8—41no—
424C8—41no—
425C8—41no—
426C8—81no—
427C8—2441no—
428C8—2441no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—2441no—
441C22—2441no—
442C22—2441no—
443C22#1683964011yescharacteristic
444C22#1683987811yescharacteristic
445C22#1683988311yescharacteristic
446C22#1684002611yescharacteristic
447C22#1684003811yescharacteristic
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—4882no—
453C4—4881no—
454C4—4881no—
455C4—2441no—
456C4—2441no—
457C4—2441no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—21no—
462C2—41no—
463C2—4881no—
464C2#2144614711yescharacteristic
465C2#2144633311yescharacteristic
466C2#2144636011yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1298C2
socleC22×C61#386
prime radical#16840273C22
cosocleC23×C4×C61C23
hypercenter#1298D61
hypocenterC61#9263
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1298

Complements

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

NH ≅ G/Ntype
4. C2×#16839500C2semidirect
5. #21444184C2semidirect
6. #21443661C2semidirect
7. C61×#1298C2semidirect
11. #16840093C4semidirect
12. #16840076C4semidirect
16. #16839920C4semidirect
17. #16839903C4semidirect
230. #1298D61semidirect
268. C61#9263semidirect

Maximal subgroups

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

subgroupindex
#214566022
#214548732
#214443242
C2×#168395002
#214441842
#214436612
C61×#12982
#926361