Finite Group Databasebeta

← #29577082

Subgroups — #29577082

Order 15616.

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

normal subgroups only

This group has 470 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#21455760C211yesmaximal, characteristic
2#21454479C211yesmaximal, characteristic, complemented
3#21445911C211yesmaximal, characteristic, complemented
4#21444386C211yesmaximal, characteristic
5#21444180C211yesmaximal, characteristic
6#21443706C211yesmaximal, characteristic, complemented
7C61×#1305C211yesmaximal, characteristic, complemented
8C2×#14519751C2211yescharacteristic
9C2×#14519736C2211yescharacteristic
10#16840087—21no—
11#16840087—21no—
12#16840081—21no—
13#16840076—21no—
14#16839820—22no—
15#16839816—21no—
16#16839812—21no—
17#16839804C411yescharacteristic, complemented
18#16839804C411yescharacteristic, complemented
19#16839790—21no—
20#16839768C411yescharacteristic, complemented
21#16839768C411yescharacteristic, complemented
22C2×C22⋊#92418C2211yescharacteristic
23C2×C22⋊#92418C2211yescharacteristic
24#16839720—21no—
25#16839524—21no—
26#16839524—21no—
27#16839495—21no—
28#16839495—21no—
29#16839485—22no—
30C61×#407C2211yescharacteristic
31C61×#407C2211yescharacteristic
32C61×#380C2211yescharacteristic
33C22×D244C2×C411yescharacteristic
34C22×C22⋊D61—21no—
35C22×#92418—21no—
36C22×#92418—21no—
37#14519751—21no—
38#14519751—21no—
39#14519751—21no—
40#14519751—21no—
41#14519736—21no—
42#14519736—21no—
43#14519736C2×C411yescharacteristic
44#14519736C2×C411yescharacteristic
45#14519736D411yescharacteristic
46#14519736D411yescharacteristic
47C2×C4⋊Dic61C2×C411yescharacteristic
48C2×C4⋊Dic61D411yescharacteristic
49C2×C4⋊Dic61D411yescharacteristic
50C2×C22⋊Dic61—21no—
51C2×#11758198—21no—
52C2×C4×Dic61—21no—
53#14519711—21no—
54#14519711—21no—
55#14519709—21no—
56#14519709—21no—
57#14519706—22no—
58#14519706—22no—
59#14519699—21no—
60#14519699—21no—
61#14519699—21no—
62#14519699—21no—
63C22⋊#92418—21no—
64C22⋊#92418—21no—
65C22⋊#92418—21no—
66C22⋊#92418D411yescharacteristic
67C22⋊#92418D411yescharacteristic
68#14519687—22no—
69C8×Dic61—22no—
70C23×C4×C61C2311yescharacteristic
71C2×M16×C61—21no—
72C2×M16×C61C2×C411yescharacteristic
73C2×M16×C61C2×C411yescharacteristic
74C2×C4⋊C4×C61—21no—
75C2×C22⋊C4×C61—21no—
76C22⋊C8×C61—21no—
77C22⋊C8×C61—21no—
78C22⋊C8×C61—21no—
79C22⋊C8×C61D412yes—
80#95×C61—21no—
81C24×C61#4111yescharacteristic
82C23×D61—21no—
83C22×C4×C61—41no—
84C22×C4×C61—41no—
85C22×C4×C61—41no—
86C22×C4×C61C22×C411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61#4111yescharacteristic
92C2×C22⋊D61—41no—
93C2×C22⋊D61—41no—
94C2×C22⋊D61—81no—
95C2×C22⋊D61—81no—
96C22×Dic61—21no—
97C22×Dic61—21no—
98C22×Dic61—21no—
99C2×D244—21no—
100C2×D244—21no—
101C2×D244—41no—
102C2×D244—41no—
103M16×C61—41no—
104M16×C61—41no—
105M16×C61—41no—
106M16×C61—41no—
107C2×C8×C61—21no—
108C2×C8×C61—21no—
109C2×C8×C61—22no—
110C4⋊C4×C61—41no—
111C4⋊C4×C61—41no—
112C22⋊C4×C61—81no—
113C22⋊Dic61—21no—
114C22⋊Dic61—21no—
115C22⋊Dic61—21no—
116C22⋊Dic61—21no—
117#11758198—41no—
118#11758198—41no—
119C4⋊Dic61—21no—
120C4⋊Dic61—21no—
121C4⋊Dic61—21no—
122C4⋊Dic61—21no—
123C4⋊Dic61C22⋊C411yescharacteristic
124C4⋊Dic61C22⋊C411yescharacteristic
125C4⋊Dic61C22⋊C411yescharacteristic
126C4⋊Dic61C22⋊C411yescharacteristic
127C4×Dic61—21no—
128C4×Dic61—21no—
129C4×Dic61—21no—
130C4×Dic61—21no—
131C2×#92418—21no—
132C2×#92418—21no—
133C2×#92418—41no—
134C2×#92418—41no—
135C2×#92418—41no—
136C2×#92418—41no—
137C2×#92418—41no—
138C23×C61—21no—
139C23×C61—21no—
140C23×C61—41no—
141C23×C61—41no—
142C23×C61C4×D411yescharacteristic
143C23×C61#12111yescharacteristic
144C23×C61#12311yescharacteristic
145C22×D61—41no—
146C22×D61—42no—
147C22×D61—81no—
148C22×D61—81no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—81no—
159C2×C4×C61—81no—
160C2×C4×C61—81no—
161C2×C4×C61C2×C22⋊C411yescharacteristic
162C2×C4×C61C4×D411yescharacteristic
163C2×C4×C61#12011yescharacteristic
164C2×C4×C61#12111yescharacteristic
165C22⋊D61—82no—
166C22⋊D61—81no—
167C22⋊D61—81no—
168C2×Dic61—21no—
169C2×Dic61—21no—
170C2×Dic61—21no—
171C2×Dic61—21no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—81no—
176C2×Dic61—81no—
177D244—41no—
178D244—41no—
179D244—81no—
180C8×C61—42no—
181C8×C61—41no—
182C8×C61—41no—
183#92418—41no—
184#92418—41no—
185#92418—81no—
186#9348—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#38611yescharacteristic
198C22×C61#47111yescharacteristic
199C22×C61#47211yescharacteristic
200C2×D61—81no—
201C2×D61—81no—
202C2×D61—81no—
203C2×D61—162no—
204C4×C61—21no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—161no—
208C4×C61#41011yescharacteristic
209C4×C61#41011yescharacteristic
210Dic61—41no—
211Dic61—41no—
212Dic61—41no—
213Dic61—41no—
214Dic61—81no—
215#3094—611no—
216#2924—611no—
217#1825—611no—
218#1519—611no—
219#1471—611no—
220#1305D6111yescharacteristic, complemented
221#1298—611no—
222C2×C61—21no—
223C2×C61—21no—
224C2×C61—41no—
225C2×C61—41no—
226C2×C61#188111yescharacteristic
227C2×C61#189911yescharacteristic
228C2×C61#190711yescharacteristic
229D61—161no—
230C2×#121—611no—
231C2×#117—611no—
232#482—1221no—
233#482—1221no—
234#481—1221no—
235#480—1221no—
236#433—1222no—
237#432—1221no—
238#431—1221no—
239#428—611no—
240#428—611no—
241#424—1221no—
242#418—611no—
243#418—611no—
244#407C2×D6111yescharacteristic
245#407C2×D6111yescharacteristic
246C2×C22⋊C8—611no—
247C2×C22⋊C8—611no—
248#405—1221no—
249#380C2×D6111yescharacteristic
250#340—1221no—
251#340—1221no—
252#327—1221no—
253#327—1221no—
254#323—1222no—
255C61#934811yescharacteristic, complemented, Sylow
256C22×D4—611no—
257C22×D4—1221no—
258C23×C4C22×D6111yescharacteristic
259C2×M16—21no—
260C2×M16C4×D6111yescharacteristic
261C2×M16C4×D6111yescharacteristic
262C22×C8—1221no—
263C22×C8—1221no—
264#121—1221no—
265#121—1221no—
266#121—1221no—
267#121—1221no—
268#117—611no—
269#117—611no—
270#117—611no—
271#117—611no—
272#117—1221no—
273#117—1221no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—611no—
276C2×C4⋊C4—611no—
277C2×C4⋊C4—611no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—1221no—
280C2×C22⋊C4—1221no—
281C2×C42—1221no—
282#107—1221no—
283#107—1221no—
284#106—1221no—
285#106—1221no—
286C4⋊C8—1222no—
287C4⋊C8—1222no—
288#102—1221no—
289#102—1221no—
290#102—1221no—
291#102—1221no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—611no—
296C22⋊C8—611no—
297C22⋊C8—1221no—
298C22⋊C8—1221no—
299C22⋊C8—1221no—
300C22⋊C8D24412yes—
301#97—1222no—
302C4×C8—1222no—
303#95—21no—
304C24—1221no—
305C24#1175822211yescharacteristic
306C2×D4—1221no—
307C2×D4—1221no—
308C2×D4—4881no—
309C2×D4—4881no—
310C2×D4—2441no—
311C2×D4—2441no—
312C2×D4—2441no—
313C2×D4—2441no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—1221no—
318C22×C4—1221no—
319C22×C4—1221no—
320C22×C4C2×C4×D6111yescharacteristic
321C22×C4C2×D24411yescharacteristic
322C22×C4D4×D6111yescharacteristic
323C22×C4D4×D6111yescharacteristic
324C22×C4D4×D6111yescharacteristic
325C22×C4#1175822611yescharacteristic
326M16—41no—
327M16—41no—
328M16—41no—
329M16—41no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—22no—
333C2×C8—1221no—
334C2×C8—1221no—
335C2×C8—2441no—
336C2×C8—2441no—
337C2×C8—2441no—
338C2×C8—2441no—
339C2×C8—2441no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—611no—
343C4⋊C4—611no—
344C4⋊C4—611no—
345C4⋊C4—611no—
346C4⋊C4—1221no—
347C4⋊C4—1221no—
348C4⋊C4—1221no—
349C4⋊C4—1221no—
350C22⋊C4—81no—
351C22⋊C4—1221no—
352C22⋊C4—1221no—
353C22⋊C4—1221no—
354C22⋊C4—1221no—
355C22⋊C4—2441no—
356C22⋊C4—2441no—
357C42—1221no—
358C42—1221no—
359C42—1221no—
360C42—1221no—
361C23—21no—
362C23—21no—
363C23—41no—
364C23—41no—
365C23—4881no—
366C23—4881no—
367C23—2442no—
368C23—2441no—
369C23C4×D24411yescharacteristic
370C23#1451975011yescharacteristic
371C23#1451976211yescharacteristic
372D4—4881no—
373D4—4882no—
374D4—4881no—
375D4—4881no—
376D4—2441no—
377D4—2441no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—1221no—
391C2×C4—1221no—
392C2×C4—1221no—
393C2×C4—1221no—
394C2×C4—4881no—
395C2×C4—4881no—
396C2×C4—2441no—
397C2×C4—2441no—
398C2×C4—2441no—
399C2×C4C22⋊C4×D6111yescharacteristic
400C2×C4#1451973911yescharacteristic
401C2×C4#1451974711yescharacteristic
402C2×C4#1451975211yescharacteristic
403C8—42no—
404C8—41no—
405C8—41no—
406C8—4881no—
407C8—2441no—
408C8—2441no—
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#1683964011yescharacteristic
424C22#1684002811yescharacteristic
425C22#1684003111yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—161no—
430C4—4881no—
431C4—2441no—
432C4—2441no—
433C4—2441no—
434C4—2441no—
435C4#1683973411yescharacteristic
436C4#1683973411yescharacteristic
437C2—21no—
438C2—21no—
439C2—41no—
440C2—41no—
441C2—9761no—
442C2#2144618011yescharacteristic
443C2#2144627611yescharacteristic
444C2#2144632811yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1305C2
socleC22×C61#386
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#1305D61
hypocenterC61#9348
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1305

Complements

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

NH ≅ G/Ntype
2. #21454479C2semidirect
3. #21445911C2semidirect
6. #21443706C2semidirect
7. C61×#1305C2semidirect
17. #16839804C4semidirect
18. #16839804C4semidirect
20. #16839768C4semidirect
21. #16839768C4semidirect
220. #1305D61semidirect
255. C61#9348semidirect

Maximal subgroups

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

subgroupindex
#214557602
#214544792
#214459112
#214443862
#214441802
#214437062
C61×#13052
#934861