Finite Group Databasebeta

← #29577037

Subgroups — #29577037

Order 15616.

534 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 89 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#21456621C211yesmaximal, characteristic
2#21448973C211yesmaximal, characteristic
3#21444352C211yesmaximal, characteristic, complemented
4#21444334C211yesmaximal, characteristic
5#21443725C211yesmaximal, characteristic, complemented
6#21443672C211yesmaximal, characteristic, complemented
7C61×#1996C211yesmaximal, characteristic, complemented
8C2×#14519751C2211yescharacteristic
9#16840095—21no—
10#16840075C412yescomplemented
11#16840010—22no—
12#16839912—21no—
13#16839901C412yescomplemented
14#16839726C2211yescharacteristic
15#16839726C2211yescharacteristic
16#16839662—21no—
17#16839652—21no—
18#16839647—21no—
19#16839640—21no—
20#16839638—21no—
21#16839633—21no—
22#16839625C2211yescharacteristic
23#16839499—21no—
24#16839499—21no—
25#16839495—22no—
26C2×C22⋊Q8×C61C2211yescharacteristic
27C61×#380C2211yescharacteristic
28C4×C22⋊C4×C61C2211yescharacteristic
29C61×#342—21no—
30C61×#342—21no—
31C22×D244D411yescharacteristic
32C22×C22⋊D61—21no—
33C22×C4×D61—21no—
34C2×#11758201—22no—
35C2×#11758194—21no—
36C2×#11758194—21no—
37#14519751—21no—
38#14519751—21no—
39#14519751C2×C412yes—
40#14519751D412yes—
41C2×C4⋊Dic61D411yescharacteristic
42C2×#11758196—21no—
43C2×C22⋊Dic61—21no—
44C2×#11758198—21no—
45C2×#11758198—21no—
46C2×#11758198—21no—
47C2×#11758198—21no—
48C2×C4×Dic61—21no—
49#14519710—22no—
50#14519701—22no—
51#14519697—22no—
52C22⋊#92418—21no—
53C22⋊#92418—21no—
54C22⋊#92418—21no—
55C22⋊#92418C2×C412yescomplemented
56#14519684—21no—
57#14519684—21no—
58#14519684—21no—
59#14519684D411yescharacteristic
60#14519684D411yescharacteristic
61C22×Q8×C61C2×C411yescharacteristic, complemented
62C23×C4×C61C2311yescharacteristic
63C22⋊Q8×C61—21no—
64C22⋊Q8×C61—21no—
65C22⋊Q8×C61—21no—
66C22⋊Q8×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
70C2×C22⋊C4×C61—21no—
71C2×C22⋊C4×C61—21no—
72C2×C22⋊C4×C61D411yescharacteristic
73C2×C22⋊C4×C61D411yescharacteristic
74C2×C42×C61—21no—
75#95×C61—21no—
76#95×C61—21no—
77C24×C61C2×D411yescharacteristic
78C23×D61—21no—
79C2×Q8×C61—21no—
80C2×Q8×C61—21no—
81C2×Q8×C61—81no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—41no—
87C22×C4×C61—81no—
88C22×C4×C61C22×C411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61#4111yescharacteristic
93C22×C4×C61#4111yescharacteristic
94C2×C22⋊D61—22no—
95C2×C22⋊D61—22no—
96C2×C22⋊D61—81no—
97C2×C22⋊D61—81no—
98C22×Dic61—21no—
99C22×Dic61—21no—
100C22×Dic61—21no—
101C2×D244—21no—
102C2×D244—81no—
103C2×D244C22⋊C412yes—
104C2×C4×D61—81no—
105C2×C4×D61—81no—
106C2×C4×D61—81no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—41no—
110C4⋊C4×C61—41no—
111C4⋊C4×C61—81no—
112C22⋊C4×C61—22no—
113C22⋊C4×C61—41no—
114C22⋊C4×C61—41no—
115C22⋊C4×C61—41no—
116C22⋊C4×C61—41no—
117C22⋊C4×C61—41no—
118C42×C61—81no—
119C22⋊Dic61—81no—
120#11758201—82no—
121#11758198—22no—
122#11758198—22no—
123#11758198—81no—
124#11758198—81no—
125#11758198—81no—
126C4⋊Dic61—21no—
127C4⋊Dic61C22⋊C412yes—
128#11758196—81no—
129C4×Dic61—81no—
130#11758194—81no—
131#11758194—81no—
132C2×#92418—22no—
133C2×#92418—41no—
134C23×C61—21no—
135C23×C61—21no—
136C23×C61—41no—
137C23×C61—41no—
138C23×C61C4×D411yescharacteristic
139C23×C61#12011yescharacteristic
140C23×C61#12111yescharacteristic
141C22×D61—22no—
142C22×D61—41no—
143C22×D61—41no—
144C22×D61—41no—
145C22×D61—41no—
146C22×D61—41no—
147C22×D61—41no—
148Q8×C61—42no—
149Q8×C61—81no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
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×C61—42no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—81no—
166C2×C4×C61—81no—
167C2×C4×C61—81no—
168C2×C4×C61—81no—
169C2×C4×C61C2×C22⋊C411yescharacteristic
170C2×C4×C61C4×D411yescharacteristic
171C2×C4×C61#12111yescharacteristic
172C2×C4×C61#12311yescharacteristic
173C22⋊D61—82no—
174C22⋊D61—82no—
175C2×Dic61—22no—
176C2×Dic61—41no—
177C2×Dic61—41no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61—41no—
182C2×Dic61—41no—
183C2×Dic61—41no—
184D244—42no—
185D244—81no—
186C4×D61—84no—
187#92418—42no—
188#92418—81no—
189#9341—611nomaximal, Sylow
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—21no—
198C22×C61—81no—
199C22×C61—81no—
200C22×C61#38611yescharacteristic
201C22×C61#45511yescharacteristic
202C22×C61#45611yescharacteristic
203C2×D61—42no—
204C2×D61—42no—
205C2×D61—42no—
206C2×D61—81no—
207C2×D61—81no—
208C2×D61—81no—
209C2×D61—81no—
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—81no—
215C4×C61—81no—
216C4×C61—82no—
217C4×C61—81no—
218C4×C61—81no—
219Dic61—42no—
220Dic61—81no—
221Dic61—81no—
222#3193—611no—
223#2377—611no—
224#1996D6111yescharacteristic, complemented
225#1511—611no—
226#1507—611no—
227#1305—611no—
228#1284—611no—
229C2×C61—21no—
230C2×C61—21no—
231C2×C61—41no—
232C2×C61—41no—
233C2×C61#188211yescharacteristic
234C2×C61#190411yescharacteristic
235C2×C61#191311yescharacteristic
236D61—82no—
237C2×C22⋊Q8C2×D6111yescharacteristic
238C2×#121—611no—
239#484—1221no—
240#480—612no—
241#468—1222no—
242#450—1221no—
243#448—612no—
244#407—611no—
245#407—611no—
246#390—1221no—
247#388—1221no—
248#387—1221no—
249#386—1221no—
250#386—1221no—
251#385—1221no—
252#381—611no—
253#380C2×D6111yescharacteristic
254C4×C22⋊C4C2×D6111yescharacteristic
255#342—21no—
256#342—21no—
257#328—1221no—
258#328—1221no—
259#327—1222no—
260C61#934111yescharacteristic, complemented, Sylow
261C22×Q8C4×D6111yescharacteristic, complemented
262C22×D4—611no—
263C22×D4—1221no—
264C23×C4—1221no—
265C23×C4C22×D6111yescharacteristic
266C2×SD16—1222no—
267C2×M16—1221no—
268C2×M16—1221no—
269C22⋊Q8—21no—
270C22⋊Q8—21no—
271C22⋊Q8—21no—
272C22⋊Q8—21no—
273#121—612no—
274#121—612no—
275#121—1221no—
276#121—1221no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—611no—
280C2×C4⋊C4—1221no—
281C2×C4⋊C4C4×D6111yescharacteristic, complemented
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—1221no—
285C2×C22⋊C4—1221no—
286C2×C22⋊C4—1221no—
287C2×C22⋊C4—1221no—
288C2×C22⋊C4—1221no—
289C2×C22⋊C4C22⋊D6111yescharacteristic
290C2×C22⋊C4C22⋊D6111yescharacteristic
291C2×C42—21no—
292C2×C42—1221no—
293#107—1222no—
294#103—1222no—
295#102—1222no—
296C22⋊C8—612no—
297C22⋊C8—1221no—
298C22⋊C8—1221no—
299C22⋊C8—1221no—
300#95—21no—
301#95—21no—
302#95—611no—
303#95—611no—
304#95—1221no—
305#95—1221no—
306#95—1221no—
307C24—1221no—
308C24C2×C22⋊D6111yescharacteristic
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—81no—
312C2×D4—612no—
313C2×D4—1222no—
314C2×D4—1222no—
315C2×D4—1221no—
316C2×D4—4881no—
317C2×D4—4881no—
318C2×D4—4881no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—41no—
324C22×C4—81no—
325C22×C4—1221no—
326C22×C4—1221no—
327C22×C4—1221no—
328C22×C4—4881no—
329C22×C4—4881no—
330C22×C4—4881no—
331C22×C4C2×C4×D6111yescharacteristic
332C22×C4#1175822211yescharacteristic
333C22×C4D4×D6111yescharacteristic
334C22×C4D4×D6111yescharacteristic
335C22×C4D4×D6111yescharacteristic
336C22×C4#1175822411yescharacteristic
337SD16—4882no—
338M16—4881no—
339M16—4881no—
340C2×C8—1222no—
341C2×C8—2441no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—81no—
347C4⋊C4—612no—
348C4⋊C4—1221no—
349C4⋊C4—4881no—
350C22⋊C4—22no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—1222no—
357C22⋊C4—1222no—
358C22⋊C4—4881no—
359C22⋊C4—4881no—
360C22⋊C4—4881no—
361C22⋊C4—4881no—
362C42—81no—
363C42—4881no—
364C23—21no—
365C23—21no—
366C23—41no—
367C23—41no—
368C23—1222no—
369C23—2441no—
370C23—2441no—
371C23—2441no—
372C23—2441no—
373C23—2441no—
374C23—2441no—
375C23C4×C22⋊D6111yescharacteristic
376C23#1451974511yescharacteristic
377C23#1451974811yescharacteristic
378Q8—42no—
379Q8—81no—
380D4—4881no—
381D4—4882no—
382D4—4882no—
383D4—2442no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—1222no—
404C2×C4—4884no—
405C2×C4—2441no—
406C2×C4—2441no—
407C2×C4—2441no—
408C2×C4—2441no—
409C2×C4—2441no—
410C2×C4—2441no—
411C2×C4—2441no—
412C2×C4—2441no—
413C2×C4C22⋊C4×D6111yescharacteristic
414C2×C4#1451973711yescharacteristic
415C2×C4#1451975011yescharacteristic
416C2×C4#1451975911yescharacteristic
417C8—4881no—
418C8—2442no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—81no—
428C22—81no—
429C22—4881no—
430C22—4881no—
431C22—4881no—
432C22—4881no—
433C22—2442no—
434C22—2442no—
435C22—2442no—
436C22#1683963811yescharacteristic
437C22#1683993911yescharacteristic
438C22#1683994411yescharacteristic
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—81no—
444C4—81no—
445C4—82no—
446C4—81no—
447C4—81no—
448C4—4881no—
449C4—4881no—
450C4—2442no—
451C2—21no—
452C2—21no—
453C2—41no—
454C2—41no—
455C2—4882no—
456C2#2144619111yescharacteristic
457C2#2144630911yescharacteristic
458C2#2144635811yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1996C2
socleC22×C61#386
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#1996D61
hypocenterC61#9341
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
3. #21444352C2semidirect
5. #21443725C2semidirect
6. #21443672C2semidirect
7. C61×#1996C2semidirect
10. #16840075C4semidirect
13. #16839901C4semidirect
55. C22⋊#92418C2×C4semidirect
61. C22×Q8×C61C2×C4semidirect
69. C2×C4⋊C4×C61C2×C4semidirect
224. #1996D61semidirect
260. C61#9341semidirect
261. C22×Q8C4×D61semidirect
281. C2×C4⋊C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214566212
#214489732
#214443522
#214443342
#214437252
#214436722
C61×#19962
#934161