Finite Group Databasebeta

← #29592365

Subgroups — #29592365

Order 15616.

398 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 59 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#21447043C211yesmaximal, characteristic, complemented
2#21444855C211yesmaximal, characteristic, complemented
3#21444753C211yesmaximal, characteristic
4#21443846C211yesmaximal, characteristic, complemented
5#21443822C211yesmaximal, characteristic, complemented
6#21443765C211yesmaximal, characteristic, complemented
7C61×#2795C211yesmaximal, characteristic, complemented
8#16840343C2211yescharacteristic
9#16840080—21no—
10#16839896—21no—
11C2×#14519705—21no—
12#16839761—21no—
13#16839531C2211yescharacteristic
14#16839495—21no—
15#16839495—21no—
16#16839488C2211yescharacteristic
17#16839488C2211yescharacteristic
18C61×#400C2211yescharacteristic
19C61×#396—21no—
20C61×#394—21no—
21C61×#394C2211yescharacteristic
22C61×#388—21no—
23C61×#388—21no—
24C61×#388—21no—
25C61×#387—21no—
26C61×#384C2211yescharacteristic
27C2×#11758222—21no—
28C2×#11758199—41no—
29C2×#11758194—21no—
30#14519757—41no—
31#14519752—41no—
32#14519751—21no—
33#14519751—21no—
34#14519708—41no—
35#14519705—42no—
36#14519700—41no—
37#14519698—41no—
38#14519697—41no—
39#14519697—41no—
40C22⋊#92418—21no—
41C22⋊#92418—21no—
42C22×D4×C61D411yescharacteristic
43C2×C4⋊C4×C61—21no—
44C2×C4⋊C4×C61—21no—
45C2×C4⋊C4×C61—21no—
46C2×C4⋊C4×C61D411yescharacteristic
47C2×C4⋊C4×C61D411yescharacteristic
48C2×C4⋊C4×C61C2311yescharacteristic
49C2×C22⋊C4×C61—21no—
50C2×C22⋊C4×C61—21no—
51C2×C22⋊C4×C61—21no—
52C2×C22⋊C4×C61—21no—
53C2×C22⋊C4×C61—21no—
54C2×C22⋊C4×C61—21no—
55C2×C42×C61—21no—
56C2×C42×C61D411yescharacteristic
57#95×C61—21no—
58#95×C61—21no—
59#95×C61—21no—
60#95×C61—21no—
61#95×C61D411yescharacteristic
62#95×C61D411yescharacteristic
63C24×C61—21no—
64C2×D4×C61—41no—
65C2×D4×C61—81no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61C2×D411yescharacteristic
72C22×C4×C61C2×D411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C2×C22⋊D61—41no—
75#11758222—41no—
76#11758222—41no—
77#11758222—41no—
78#11758222—41no—
79C2×D244—21no—
80C2×D244—21no—
81C2×D244—21no—
82C2×C4×D61—41no—
83C2×Dic122—21no—
84C4⋊C4×C61—41no—
85C4⋊C4×C61—42no—
86C4⋊C4×C61—41no—
87C4⋊C4×C61—41no—
88C4⋊C4×C61—81no—
89C4⋊C4×C61—81no—
90C22⋊C4×C61—42no—
91C22⋊C4×C61—42no—
92C22⋊C4×C61—41no—
93C22⋊C4×C61—41no—
94C22⋊C4×C61—81no—
95C22⋊C4×C61—81no—
96C22⋊C4×C61—81no—
97C42×C61—21no—
98C42×C61—21no—
99C42×C61—81no—
100#11758199—82no—
101#11758198—41no—
102#11758198—41no—
103C4⋊Dic61—21no—
104C4⋊Dic61—21no—
105#11758194—41no—
106#11758194—41no—
107C2×#92418—41no—
108C2×#92418—41no—
109C2×#92418—41no—
110C23×C61—41no—
111C23×C61—41no—
112C23×C61—41no—
113C23×C61—41no—
114C23×C61—41no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61#12011yescharacteristic
118C22×D61—41no—
119D4×C61—42no—
120D4×C61—81no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—22no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61#13511yescharacteristic
143C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
144C22⋊D61—81no—
145C22⋊D61—81no—
146C2×Dic61—41no—
147D244—41no—
148D244—41no—
149D244—81no—
150D244—81no—
151C4×D61—81no—
152C4×D61—81no—
153Dic122—41no—
154Dic122—41no—
155#92418—81no—
156#92418—81no—
157#92418—81no—
158#12144—611nomaximal, Sylow
159C22×C61—21no—
160C22×C61—21no—
161C22×C61—81no—
162C22×C61—81no—
163C22×C61—81no—
164C22×C61—81no—
165C22×C61—81no—
166C22×C61—81no—
167C22×C61—81no—
168C22×C61#44911yescharacteristic
169C22×C61#45411yescharacteristic
170C22×C61B(4,𝔽2)11yescharacteristic
171C2×D61—81no—
172C2×D61—81no—
173C2×D61—81no—
174C4×C61—21no—
175C4×C61—21no—
176C4×C61—42no—
177C4×C61—41no—
178C4×C61—41no—
179C4×C61—81no—
180C4×C61—81no—
181C4×C61—81no—
182C4×C61—81no—
183Dic61—81no—
184#2795D6111yescharacteristic, complemented
185#2039—611no—
186#1616—611no—
187#1598—611no—
188#1353—611no—
189#1345—611no—
190#1325—611no—
191C2×C61—21no—
192C2×C61—21no—
193C2×C61—161no—
194C2×C61#159711yescharacteristic
195C2×C61#220011yescharacteristic
196C2×C61#220411yescharacteristic
197D61—161no—
198#537—611no—
199#481—1221no—
200#447—1221no—
201C2×#104—1221no—
202#417—1221no—
203#400C2×D6111yescharacteristic
204#396—21no—
205#394—21no—
206#394C2×D6111yescharacteristic
207#388—21no—
208#388—21no—
209#388—21no—
210#387—21no—
211#384C2×D6111yescharacteristic
212#343—611no—
213#327—1221no—
214#327—1221no—
215#324—611no—
216#324—611no—
217C61#1214411yescharacteristic, complemented, Sylow
218C2×#41—1221no—
219C22×D4C22⋊D6111yescharacteristic
220C2×D8—2441no—
221C2×M16—1221no—
222C22⋊Q8—2441no—
223#121—1221no—
224#121—1221no—
225#121—2441no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4C22⋊D6111yescharacteristic
230C2×C4⋊C4C22⋊D6111yescharacteristic
231C2×C4⋊C4C22×D6111yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C42—21no—
239C2×C42C22⋊D6111yescharacteristic
240#106—2441no—
241#104—2442no—
242#103—2441no—
243#102—2441no—
244#102—2441no—
245#102—2441no—
246C22⋊C8—1221no—
247C22⋊C8—1221no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95C22⋊D6111yescharacteristic
253#95C22⋊D6111yescharacteristic
254C24—21no—
255#41—2441no—
256#41—2441no—
257#41—2441no—
258#41—2441no—
259C2×Q8—1221no—
260C2×D4—41no—
261C2×D4—81no—
262C2×D4—1221no—
263C2×D4—1221no—
264C2×D4—1221no—
265C2×D4—2441no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—2441no—
272C22×C4C2×C22⋊D6111yescharacteristic
273C22×C4C2×C22⋊D6111yescharacteristic
274C22×C4C2×C22⋊D6111yescharacteristic
275D8—4882no—
276M16—2441no—
277M16—2441no—
278C2×C8—2441no—
279C2×C8—2441no—
280C2×C8—2441no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—81no—
286C4⋊C4—81no—
287C4⋊C4—1221no—
288C4⋊C4—1221no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—81no—
294C22⋊C4—81no—
295C22⋊C4—81no—
296C22⋊C4—2441no—
297C22⋊C4—2441no—
298C42—21no—
299C42—21no—
300C42—81no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—2441no—
309C23#1451974611yescharacteristic
310Q8—2441no—
311Q8—2441no—
312D4—42no—
313D4—81no—
314D4—4881no—
315D4—4881no—
316D4—4881no—
317D4—4881no—
318D4—2441no—
319D4—2441no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—4881no—
342C2×C4—4881no—
343C2×C4—2441no—
344C2×C4#1451979711yescharacteristic
345C2×C4#1451980211yescharacteristic
346C8—4881no—
347C8—4881no—
348C8—4881no—
349C22—21no—
350C22—21no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—4881no—
359C22—4881no—
360C22—4881no—
361C22#1683990711yescharacteristic
362C22#1683993511yescharacteristic
363C22#1683994911yescharacteristic
364C4—21no—
365C4—21no—
366C4—42no—
367C4—41no—
368C4—41no—
369C4—81no—
370C4—81no—
371C4—81no—
372C4—81no—
373C4—4881no—
374C2—21no—
375C2—21no—
376C2—161no—
377C2—9761no—
378C2#2144474811yescharacteristic
379C2#2144782611yescharacteristic
380C2#2144784511yescharacteristic
subgroupG/H
centerC22#16839949
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2795C2
socleC22×C61B(4,𝔽2)
prime radical#21444753C2
hypercenter#2795D61
hypocenterC61#12144
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2795

Complements

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

NH ≅ G/Ntype
1. #21447043C2semidirect
2. #21444855C2semidirect
4. #21443846C2semidirect
5. #21443822C2semidirect
6. #21443765C2semidirect
7. C61×#2795C2semidirect
184. #2795D61semidirect
217. C61#12144semidirect

Maximal subgroups

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

subgroupindex
#214470432
#214448552
#214447532
#214438462
#214438222
#214437652
C61×#27952
#1214461