Finite Group Databasebeta

← #29580342

Subgroups — #29580342

Order 15616.

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

normal subgroups only

This group has 568 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#21456621C211yesmaximal, characteristic
2#21455759C211yesmaximal, characteristic
3#21446654C211yesmaximal, characteristic
4#21445918C211yesmaximal, characteristic, complemented
5#21444339C211yesmaximal, characteristic, complemented
6#21443725C211yesmaximal, characteristic, complemented
7C61×#2625C211yesmaximal, characteristic, complemented
8C2×#14519751C2211yescharacteristic
9#16840095—21no—
10#16840086—22no—
11#16840080—21no—
12#16840075—21no—
13#16840075—21no—
14#16840010—22no—
15#16839912—21no—
16#16839901—21no—
17#16839805—21no—
18#16839766—21no—
19C2×#14519691—22no—
20#16839726C2211yescharacteristic
21#16839726C2211yescharacteristic
22C4×C22⋊Dic61C2211yescharacteristic
23#16839521—22no—
24#16839499—22no—
25C2×C22⋊Q8×C61C2211yescharacteristic
26C2×#117×C61C2211yescharacteristic
27C61×#400—21no—
28C61×#398—21no—
29C61×#397—22no—
30C61×#391—21no—
31C61×#389—21no—
32C61×#380C2211yescharacteristic
33C22×D244D411yescharacteristic
34C22×C22⋊D61—21no—
35C2×#11758201—22no—
36C2×#11758194—21no—
37C2×#11758194D412yes—
38#14519751—21no—
39#14519751—21no—
40#14519751—21no—
41#14519751—21no—
42C2×C4⋊Dic61D411yescharacteristic
43C2×C22⋊Dic61—21no—
44C2×#11758198—21no—
45C2×C4×Dic61—21no—
46#14519710—22no—
47#14519710—22no—
48#14519708—22no—
49#14519701—22no—
50#14519697—22no—
51#14519697—22no—
52#14519697—22no—
53#14519691—44no—
54C22⋊#92418—21no—
55C22⋊#92418—21no—
56C22⋊#92418—21no—
57C22⋊#92418D412yes—
58#14519684—21no—
59C22×Q8×C61D411yescharacteristic
60C23×C4×C61C2311yescharacteristic
61C22⋊Q8×C61—21no—
62C22⋊Q8×C61—21no—
63C22⋊Q8×C61—21no—
64C22⋊Q8×C61—21no—
65#117×C61—21no—
66#117×C61—21no—
67#117×C61—21no—
68#117×C61—21no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—22no—
72C2×C4⋊C4×C61D411yescharacteristic
73C2×C4⋊C4×C61D411yescharacteristic
74C2×C4⋊C4×C61D411yescharacteristic
75C2×C22⋊C4×C61—21no—
76C2×C22⋊C4×C61—21no—
77C2×C22⋊C4×C61D412yes—
78C2×C42×C61—21no—
79#95×C61—22no—
80#95×C61—22no—
81#95×C61—22no—
82#95×C61—21no—
83C24×C61C2×D411yescharacteristic
84C23×D61—21no—
85C2×Q8×C61—21no—
86C2×Q8×C61—21no—
87C2×Q8×C61—42no—
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61—22no—
91C22×C4×C61—21no—
92C22×C4×C61—21no—
93C22×C4×C61—42no—
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61#4111yescharacteristic
100C2×C22⋊D61—42no—
101C2×C22⋊D61—42no—
102C2×C22⋊D61—41no—
103C2×C22⋊D61—41no—
104C22×Dic61—21no—
105C22×Dic61—41no—
106C2×D244—21no—
107C2×D244—21no—
108C2×D244—42no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61—21no—
115C4⋊C4×C61—42no—
116C4⋊C4×C61—41no—
117C4⋊C4×C61—41no—
118C4⋊C4×C61—82no—
119C22⋊C4×C61—41no—
120C22⋊C4×C61—41no—
121C22⋊C4×C61—41no—
122C22⋊C4×C61—41no—
123C22⋊C4×C61—42no—
124C42×C61—41no—
125C42×C61—41no—
126C22⋊Dic61—22no—
127C22⋊Dic61—41no—
128#11758201—82no—
129#11758198—41no—
130#11758198—41no—
131C4⋊Dic61—21no—
132C4⋊Dic61—21no—
133C4×Dic61—42no—
134#11758194—42no—
135#11758194—42no—
136C2×#92418—22no—
137C2×#92418—22no—
138C23×C61—21no—
139C23×C61—21no—
140C23×C61—41no—
141C23×C61—41no—
142C23×C61#12011yescharacteristic
143C23×C61#12111yescharacteristic
144C23×C61#12711yescharacteristic
145C22×D61—41no—
146C22×D61—42no—
147C22×D61—81no—
148C22×D61—81no—
149Q8×C61—42no—
150Q8×C61—81no—
151C2×C4×C61—22no—
152C2×C4×C61—22no—
153C2×C4×C61—22no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—42no—
158C2×C4×C61—42no—
159C2×C4×C61—42no—
160C2×C4×C61—42no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—42no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61#12011yescharacteristic
171C2×C4×C61#12011yescharacteristic
172C2×C4×C61#12111yescharacteristic
173C2×C4×C61#12111yescharacteristic
174C22⋊D61—82no—
175C22⋊D61—81no—
176C22⋊D61—81no—
177C2×Dic61—42no—
178C2×Dic61—42no—
179C2×Dic61—41no—
180C2×Dic61—81no—
181C2×Dic61—81no—
182D244—42no—
183D244—42no—
184#92418—42no—
185#92418—42no—
186#9933—611nomaximal, Sylow
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—81no—
196C22×C61—81no—
197C22×C61#39211yescharacteristic
198C22×C61#45311yescharacteristic
199C22×C61#45411yescharacteristic
200C2×D61—81no—
201C2×D61—81no—
202C2×D61—81no—
203C2×D61—162no—
204C4×C61—21no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—82no—
208C4×C61—81no—
209C4×C61—81no—
210C4×C61—81no—
211C4×C61—81no—
212C4×C61#45812yes—
213Dic61—82no—
214Dic61—81no—
215#3193—611no—
216#3094—611no—
217#2625D6111yescharacteristic, complemented
218#1964—611no—
219#1826—611no—
220#1508—611no—
221#1305—611no—
222C2×C61—21no—
223C2×C61—21no—
224C2×C61—41no—
225C2×C61—41no—
226C2×C61#200611yescharacteristic
227C2×C61#200911yescharacteristic
228C2×C61#202211yescharacteristic
229D61—161no—
230C2×C22⋊Q8C2×D6111yescharacteristic
231C2×#121—611no—
232C2×#117C2×D6111yescharacteristic
233#484—1221no—
234#482—1222no—
235#481—1221no—
236#480—1221no—
237#480—1221no—
238#468—1222no—
239#450—1221no—
240#448—1221no—
241#428—1221no—
242#418—1221no—
243C2×#99—1222no—
244#407—611no—
245#407—611no—
246#400—21no—
247#398—21no—
248#397—22no—
249#391—21no—
250#389—21no—
251#380C2×D6111yescharacteristic
252C4×C22⋊C4—611no—
253#339—1222no—
254#328—1222no—
255C61#993311yescharacteristic, complemented, Sylow
256C22×Q8C22⋊D6111yescharacteristic
257C22×D4—611no—
258C22×D4—1221no—
259C23×C4C22×D6111yescharacteristic
260C2×SD16—1222no—
261C2×M16—612no—
262C2×M16—1221no—
263C22⋊Q8—21no—
264C22⋊Q8—21no—
265C22⋊Q8—21no—
266C22⋊Q8—21no—
267#121—1221no—
268#121—1221no—
269#121—1221no—
270#121—1221no—
271#117—21no—
272#117—21no—
273#117—21no—
274#117—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—611no—
279C2×C4⋊C4C22⋊D6111yescharacteristic
280C2×C4⋊C4C22⋊D6111yescharacteristic
281C2×C4⋊C4C22⋊D6111yescharacteristic
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—1221no—
285C2×C22⋊C4—1221no—
286C2×C22⋊C4C22⋊D6112yes—
287C2×C42—21no—
288C2×C42—1221no—
289#107—1222no—
290#107—1222no—
291#106—1222no—
292#103—1222no—
293#102—1222no—
294#102—1222no—
295#102—1222no—
296#99—2444no—
297C22⋊C8—612no—
298C22⋊C8—1221no—
299C22⋊C8—1221no—
300C22⋊C8—1221no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—21no—
305#95—1221no—
306C24—1221no—
307C24C2×C22⋊D6111yescharacteristic
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—42no—
311C2×D4—1221no—
312C2×D4—1221no—
313C2×D4—2442no—
314C2×D4—2442no—
315C2×D4—2442no—
316C2×D4—2441no—
317C2×D4—2441no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—42no—
324C22×C4—1221no—
325C22×C4—2441no—
326C22×C4D4×D6111yescharacteristic
327C22×C4D4×D6111yescharacteristic
328C22×C4D4×D6111yescharacteristic
329C22×C4#1175822411yescharacteristic
330C22×C4C2×C22⋊D6111yescharacteristic
331C22×C4C2×C22⋊D6111yescharacteristic
332SD16—4882no—
333M16—2442no—
334M16—2442no—
335C2×C8—1222no—
336C2×C8—1222no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—42no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—82no—
347C4⋊C4—1221no—
348C4⋊C4—1221no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—42no—
354C22⋊C4—1222no—
355C22⋊C4—2441no—
356C22⋊C4—2441no—
357C22⋊C4—2441no—
358C42—41no—
359C42—41no—
360C42—2442no—
361C23—21no—
362C23—21no—
363C23—41no—
364C23—41no—
365C23—4881no—
366C23—4881no—
367C23—2442no—
368C23—2441no—
369C23#1451974611yescharacteristic
370C23#1451974811yescharacteristic
371C23#1451977311yescharacteristic
372Q8—42no—
373Q8—81no—
374D4—4882no—
375D4—4881no—
376D4—4881no—
377D4—2442no—
378D4—2442no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—4881no—
399C2×C4—4881no—
400C2×C4—2442no—
401C2×C4—2442no—
402C2×C4—2441no—
403C2×C4#1451974511yescharacteristic
404C2×C4#1451974511yescharacteristic
405C2×C4#1451974911yescharacteristic
406C2×C4#1451974911yescharacteristic
407C8—2442no—
408C8—2442no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—81no—
418C22—81no—
419C22—9762no—
420C22—4881no—
421C22—4881no—
422C22—4881no—
423C22#1683967011yescharacteristic
424C22#1683993011yescharacteristic
425C22#1683993511yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—82no—
430C4—81no—
431C4—81no—
432C4—81no—
433C4—81no—
434C4—4882no—
435C4—4881no—
436C4#1683995112yes—
437C2—21no—
438C2—21no—
439C2—41no—
440C2—41no—
441C2—9761no—
442C2#2144686111yescharacteristic
443C2#2144687911yescharacteristic
444C2#2144694711yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#2625C2
socleC22×C61#392
prime radicalC2×#14519751C22
hypercenter#2625D61
hypocenterC61#9933
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2625

Complements

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

NH ≅ G/Ntype
4. #21445918C2semidirect
5. #21444339C2semidirect
6. #21443725C2semidirect
7. C61×#2625C2semidirect
217. #2625D61semidirect
255. C61#9933semidirect

Maximal subgroups

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

subgroupindex
#214566212
#214557592
#214466542
#214459182
#214443392
#214437252
C61×#26252
#993361