Finite Group Databasebeta

← #29574121

Subgroups — #29574121

Order 15616.

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

normal subgroups only

This group has 464 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#21454869C211yesmaximal, characteristic
2#21446121C211yesmaximal, characteristic
3#21446002C211yesmaximal, characteristic, complemented
4#21444311C211yesmaximal, characteristic
5#21443704C211yesmaximal, characteristic, complemented
6#21443665C211yesmaximal, characteristic, complemented
7C61×#1855C211yesmaximal, characteristic, complemented
8#16840273C2211yescharacteristic
9#16839838—21no—
10#16839831—21no—
11#16839830—21no—
12#16839825—21no—
13#16839820—21no—
14C2×#14519706—21no—
15C2×#14519703—21no—
16#16839727—21no—
17C2×C22⋊#92418C2211yescharacteristic
18C4×C22⋊Dic61C2211yescharacteristic
19C4×C22⋊Dic61C2211yescharacteristic
20#16839523—21no—
21#16839516—21no—
22#16839500C411yescharacteristic, complemented
23#16839500C411yescharacteristic, complemented
24#16839497C411yescharacteristic, complemented
25#16839497C411yescharacteristic, complemented
26C22×M16×C61C2211yescharacteristic
27C2×#108×C61—21no—
28C61×#407C2211yescharacteristic
29C2×C22⋊C8×C61C2211yescharacteristic
30C61×#341—21no—
31C2×#11758222D411yescharacteristic
32C2×#11758189—21no—
33C22×#92418D411yescharacteristic
34C22×#92418D411yescharacteristic
35C2×C8×D61—21no—
36#14519760—21no—
37#14519756C2×C411yescharacteristic
38#14519756Q811yescharacteristic
39#14519751C2×C411yescharacteristic
40#14519751Q811yescharacteristic
41#14519746—21no—
42C4×C22⋊D61—21no—
43C4×C22⋊D61—21no—
44#14519736D411yescharacteristic
45C2×C22⋊Dic61—21no—
46C2×C22⋊Dic61—21no—
47C2×C4×Dic61—21no—
48C2×C4×Dic61—21no—
49#14519706—21no—
50#14519706—21no—
51#14519706—42no—
52#14519703—42no—
53C22⋊#92418—21no—
54C22⋊#92418—21no—
55#14519689—21no—
56#14519689—21no—
57#14519687—21no—
58C8×Dic61—21no—
59#14519684—21no—
60#14519684C2×C411yescharacteristic
61#14519684C2×C411yescharacteristic
62C23×C4×C61C2311yescharacteristic
63C2×M16×C61—21no—
64C2×M16×C61—21no—
65C2×M16×C61—42no—
66C2×M16×C61D411yescharacteristic
67C2×M16×C61D411yescharacteristic
68C22×C8×C61—21no—
69C22×C8×C61—21no—
70#108×C61—42no—
71C22⋊C8×C61—21no—
72C22⋊C8×C61—21no—
73C22⋊C8×C61—21no—
74C22⋊C8×C61C2×C411yescharacteristic
75C22⋊C8×C61C2×C411yescharacteristic
76C24×C61#4111yescharacteristic
77C22×C4×C61—41no—
78C22×C4×C61—41no—
79C22×C4×C61C22×C411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×Q811yescharacteristic
84C22×C4×C61#4111yescharacteristic
85C2×C22⋊D61—21no—
86C2×C22⋊D61—21no—
87C2×C22⋊D61—21no—
88C22×Dic61—21no—
89C22×Dic61—21no—
90C22×Dic61—41no—
91#11758222—41no—
92#11758222—41no—
93C2×D244C4⋊C411yescharacteristic
94C2×C4×D61—21no—
95C2×Dic122C4⋊C411yescharacteristic
96M16×C61—22no—
97M16×C61—22no—
98M16×C61—41no—
99M16×C61—41no—
100M16×C61—41no—
101M16×C61—41no—
102C2×C8×C61—22no—
103C2×C8×C61—21no—
104C2×C8×C61—21no—
105C2×C8×C61—21no—
106C2×C8×C61—21no—
107C2×C8×C61—41no—
108C2×C8×C61—41no—
109C2×C8×C61—41no—
110C2×C8×C61—41no—
111C22⋊Dic61—21no—
112C22⋊Dic61—21no—
113C22⋊Dic61—21no—
114C22⋊Dic61—22no—
115C22⋊Dic61—42no—
116C22⋊Dic61—41no—
117#11758198—21no—
118#11758198—21no—
119C4⋊Dic61C4⋊C411yescharacteristic
120C4⋊Dic61C4⋊C411yescharacteristic
121#11758196—21no—
122#11758196—21no—
123C4×Dic61—21no—
124C4×Dic61—41no—
125C4×Dic61—41no—
126C4×Dic61—41no—
127C4×Dic61—41no—
128C2×#92418—21no—
129C2×#92418—21no—
130C2×#92418—41no—
131C2×#92418—41no—
132#11758189—81no—
133C8×D61—81no—
134C23×C61—21no—
135C23×C61—21no—
136C23×C61—41no—
137C23×C61—41no—
138C23×C61C4×D411yescharacteristic
139C23×C61C22⋊Q811yescharacteristic
140C23×C61#12311yescharacteristic
141C22×D61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61C2×C4⋊C411yescharacteristic
152C2×C4×C61C4×D411yescharacteristic
153C2×C4×C61#12011yescharacteristic
154C2×C4×C61C22⋊Q811yescharacteristic
155C22⋊D61—41no—
156C22⋊D61—41no—
157C22⋊D61—81no—
158C22⋊D61—81no—
159C2×Dic61—22no—
160C2×Dic61—21no—
161C2×Dic61—21no—
162C2×Dic61—21no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—42no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—81no—
170C2×Dic61—81no—
171D244—41no—
172C4×D61—41no—
173C4×D61—41no—
174Dic122—41no—
175C8×C61—41no—
176C8×C61—41no—
177C8×C61—41no—
178C8×C61—42no—
179C8×C61—41no—
180#92418—41no—
181#92418—41no—
182#8875—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—81no—
192C22×C61—81no—
193C22×C61#38511yescharacteristic
194C22×C61#44311yescharacteristic
195C22×C61#44411yescharacteristic
196C2×D61—41no—
197C2×D61—41no—
198C2×D61—41no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—21no—
202C4×C61B(4,𝔽2)11yescharacteristic
203C4×C61#45811yescharacteristic
204Dic61—42no—
205Dic61—41no—
206Dic61—41no—
207Dic61—41no—
208Dic61—81no—
209Dic61—82no—
210#2976—611no—
211#1871—611no—
212#1855D6111yescharacteristic, complemented
213#1851—611no—
214#1501—611no—
215#1297—611no—
216#1282—611no—
217C2×C61—21no—
218C2×C61—21no—
219C2×C61—41no—
220C2×C61—41no—
221C2×C61#186111yescharacteristic
222C2×C61#186611yescharacteristic
223C2×C61#186711yescharacteristic
224D61—81no—
225C22×M16C2×D6111yescharacteristic
226#525—611no—
227#436—1221no—
228#435—1221no—
229#435—1221no—
230C8×D4—1221no—
231#433—1221no—
232C2×#108—21no—
233C2×C4⋊C8—1221no—
234C2×#104—1221no—
235#408—1221no—
236#407C2×D6111yescharacteristic
237C2×C22⋊C8—611no—
238C2×C22⋊C8C2×D6111yescharacteristic
239C4×C22⋊C4—611no—
240C4×C22⋊C4—611no—
241#341—21no—
242#339—1221no—
243#336—1221no—
244#328—611no—
245#328—611no—
246#327—611no—
247#327—611no—
248C61#887511yescharacteristic, complemented, Sylow
249C2×#41—611no—
250C23×C4C22×D6111yescharacteristic
251C2×M16—21no—
252C2×M16—21no—
253C2×M16—42no—
254C2×M16—1221no—
255C2×M16C22⋊D6111yescharacteristic
256C2×M16C22⋊D6111yescharacteristic
257C22×C8—21no—
258C22×C8—21no—
259C22×C8—611no—
260C22×C8—611no—
261C22×C8—1221no—
262#123—1221no—
263C22⋊Q8—611no—
264C22⋊Q8—611no—
265#121—611no—
266#121—611no—
267#120—1221no—
268C4×D4—1221no—
269C4×D4—1221no—
270#117—611no—
271C2×C22⋊C4—1221no—
272C2×C22⋊C4—1221no—
273C2×C42—1221no—
274C2×C42—1221no—
275#108—42no—
276C4⋊C8—1221no—
277C4⋊C8—1221no—
278C4⋊C8—2442no—
279#104—2442no—
280C22⋊C8—21no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283C22⋊C8—1221no—
284C22⋊C8—1221no—
285C22⋊C8—1221no—
286C22⋊C8—1221no—
287C22⋊C8C4×D6111yescharacteristic
288C22⋊C8C4×D6111yescharacteristic
289#97—1221no—
290C4×C8—1221no—
291#95—611no—
292#95—611no—
293#95—1221no—
294C24#1175822611yescharacteristic
295#41—2441no—
296#41—2441no—
297C2×Q8—611no—
298C2×D4—611no—
299C2×D4—1221no—
300C2×D4—1221no—
301C2×D4—1221no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—1221no—
305C22×C4—1221no—
306C22×C4—1221no—
307C22×C4—2441no—
308C22×C4C2×C4×D6111yescharacteristic
309C22×C4#1175822211yescharacteristic
310C22×C4D4×D6111yescharacteristic
311C22×C4D4×D6111yescharacteristic
312C22×C4Q8×D6111yescharacteristic
313C22×C4C2×C22⋊D6111yescharacteristic
314M16—22no—
315M16—22no—
316M16—41no—
317M16—41no—
318M16—41no—
319M16—41no—
320M16—4881no—
321C2×C8—22no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—1221no—
331C2×C8—1221no—
332C2×C8—4881no—
333C2×C8—2441no—
334C2×C8—2441no—
335C4⋊C4—611no—
336C4⋊C4—611no—
337C4⋊C4—1221no—
338C4⋊C4—1221no—
339C22⋊C4—1221no—
340C22⋊C4—1221no—
341C22⋊C4—1221no—
342C22⋊C4—1221no—
343C22⋊C4—1221no—
344C22⋊C4—1222no—
345C22⋊C4—2442no—
346C22⋊C4—2441no—
347C42—1221no—
348C42—2441no—
349C42—2441no—
350C42—2441no—
351C42—2441no—
352C23—21no—
353C23—21no—
354C23—41no—
355C23—41no—
356C23—1221no—
357C23#1451973911yescharacteristic
358C23#1451975311yescharacteristic
359C23#1451976211yescharacteristic
360Q8—2441no—
361D4—4881no—
362D4—4881no—
363D4—2441no—
364D4—2441no—
365D4—2441no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—1222no—
376C2×C4—1221no—
377C2×C4—1221no—
378C2×C4—1221no—
379C2×C4—4881no—
380C2×C4—4881no—
381C2×C4—2441no—
382C2×C4—2442no—
383C2×C4—2441no—
384C2×C4—2441no—
385C2×C4—2441no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4—2441no—
389C2×C4C4⋊C4×D6111yescharacteristic
390C2×C4C4×C22⋊D6111yescharacteristic
391C2×C4#1451974511yescharacteristic
392C2×C4#1451975511yescharacteristic
393C8—41no—
394C8—41no—
395C8—42no—
396C8—41no—
397C8—41no—
398C8—2441no—
399C8—2441no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—81no—
409C22—81no—
410C22—2441no—
411C22—2441no—
412C22—2441no—
413C22#1683963311yescharacteristic
414C22#1683987811yescharacteristic
415C22#1683988311yescharacteristic
416C4—21no—
417C4—21no—
418C4—21no—
419C4—4881no—
420C4—4882no—
421C4—2442no—
422C4—2441no—
423C4—2441no—
424C4—2441no—
425C4#1683994811yescharacteristic
426C4#1683995111yescharacteristic
427C2—21no—
428C2—21no—
429C2—41no—
430C2—41no—
431C2—4881no—
432C2#2144605611yescharacteristic
433C2#2144609111yescharacteristic
434C2#2144609811yescharacteristic
subgroupG/H
centerC2×C4#14519745
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1855C2
socleC22×C61#385
prime radical#16840273C22
cosocleC23×C4×C61C23
hypercenter#1855D61
hypocenterC61#8875
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1855

Complements

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

NH ≅ G/Ntype
3. #21446002C2semidirect
5. #21443704C2semidirect
6. #21443665C2semidirect
7. C61×#1855C2semidirect
22. #16839500C4semidirect
23. #16839500C4semidirect
24. #16839497C4semidirect
25. #16839497C4semidirect
212. #1855D61semidirect
248. C61#8875semidirect

Maximal subgroups

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

subgroupindex
#214548692
#214461212
#214460022
#214443112
#214437042
#214436652
C61×#18552
#887561