Finite Group Databasebeta

← #30599720

Subgroups — #30599720

Order 15616.

438 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 98 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#21462465C211yesmaximal, characteristic
2#21461692C211yesmaximal, characteristic
3#21461389C211yesmaximal, characteristic, complemented
4#21461378C211yesmaximal, characteristic, complemented
5#21460716C211yesmaximal, characteristic, complemented
6#21460654C211yesmaximal, characteristic, complemented
7#21444200C211yesmaximal, characteristic
8C2×#14519865—21no—
9#16840769—21no—
10#16840769—22no—
11#16840766—21no—
12#16840766C2211yescharacteristic
13#16840766C2211yescharacteristic
14D61⋊(C4⋊C8)—21no—
15D61⋊(C4⋊C8)C2211yescharacteristic
16C2×#14519842—21no—
17C4×C4⋊#6923C2211yescharacteristic
18#16840632—21no—
19#16840626—22no—
20#16840626—21no—
21#16840625—21no—
22C4×#11758226C2211yescharacteristic
23#16839814C2211yescharacteristic
24C4×C8×D61C2211yescharacteristic
25Q8⋊#92418C411yescharacteristic, complemented
26#16839491C411yescharacteristic, complemented
27#16839489C411yescharacteristic, complemented
28C61×Q8⋊C8C411yescharacteristic, complemented
29#14519865—42no—
30#14519865—42no—
31C2×#11758213—22no—
32C2×#11758213—21no—
33C2×#11758213—21no—
34C2×D61⋊C8—22no—
35C2×D61⋊C8—21no—
36C2×C4⋊#6923—21no—
37C2×C4×#6923—21no—
38#14519844—22no—
39#14519844—21no—
40C4⋊#92420—22no—
41C4⋊#92420—21no—
42C4⋊#92420—21no—
43C4⋊#92420—21no—
44C4⋊#92420D411yescharacteristic
45C4⋊#92420D411yescharacteristic
46C4⋊#92420D412yes—
47#14519842—42no—
48(C2×#6923)⋊C4—21no—
49C2×#11758226D411yescharacteristic
50C2×C8×D61—21no—
51Q8×Dic61Q811yescharacteristic
52C4×D244—21no—
53C4×D244C2×C411yescharacteristic
54#14519739—21no—
55#14519739Q811yescharacteristic
56#14519735—21no—
57#14519735D411yescharacteristic
58C42×D61—21no—
59C42×D61C2311yescharacteristic
60C4⋊#92418C2×C411yescharacteristic
61#14519689—21no—
62C4×#92418C2×C411yescharacteristic
63C8×Dic61—21no—
64C8×Dic61—21no—
65C4×Q8×C61C2×C411yescharacteristic
66C4⋊C8×C61C2×C411yescharacteristic
67C4×C8×C61C2×C411yescharacteristic
68C22×#6923—41no—
69C2×Q8×C61C4⋊C411yescharacteristic
70#11758226—21no—
71#11758226—21no—
72#11758226D812yes—
73#11758226Q1612yes—
74C2×D244—21no—
75C2×D244C4⋊C411yescharacteristic
76C2×C4×D61—21no—
77C2×C4×D61—21no—
78C2×C4×D61—21no—
79C2×C4×D61C2×D411yescharacteristic
80C2×C4×D61#4111yescharacteristic
81C2×C4×D61#4111yescharacteristic
82C2×#92420—21no—
83C2×#92420—22no—
84C2×#92420—21no—
85C2×#92420—41no—
86C4⋊#6923—22no—
87C4⋊#6923—41no—
88C4×#6923—41no—
89C4×#6923—41no—
90#11758213—44no—
91#11758213—41no—
92#11758213—41no—
93#11758213—41no—
94#11758213—41no—
95D61⋊C8—44no—
96D61⋊C8—41no—
97D61⋊C8—41no—
98C2×C8×C61—21no—
99C2×C8×C61—21no—
100C4⋊C4×C61—21no—
101C4⋊C4×C61C4⋊C411yescharacteristic
102C42×C61—21no—
103C42×C61C22×C411yescharacteristic
104#11758198—21no—
105#11758198—21no—
106#11758198—21no—
107C4⋊Dic61—21no—
108C4⋊Dic61C4⋊C411yescharacteristic
109C4×Dic61—21no—
110C4×Dic61—21no—
111C4×Dic61C2×D411yescharacteristic
112C4×Dic61C2×D411yescharacteristic
113C4×Dic61C2×Q811yescharacteristic
114C2×#92418—21no—
115C2×#92418—21no—
116C8×D61—42no—
117C22×D61—21no—
118C22×D61#12311yescharacteristic
119C2×#6923—42no—
120C2×#6923—81no—
121C2×#6923—81no—
122Q8×C61—21no—
123Q8×C61#10712yes—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61C2×C4⋊C411yescharacteristic
127C2×C4×C61C4×D411yescharacteristic
128C2×C4×C61C4×D411yescharacteristic
129C2×Dic61—21no—
130C2×Dic61#12011yescharacteristic
131C2×Dic61C22⋊Q811yescharacteristic
132C2×Dic61C22⋊Q811yescharacteristic
133D244—22no—
134D244—22no—
135D244—21no—
136D244#10712yes—
137C4×D61—21no—
138C4×D61—21no—
139C4×D61—22no—
140C4×D61—22no—
141C4×D61—41no—
142C4×D61—41no—
143C4×D61—41no—
144C4×D61—81no—
145C4×D61C2×D811yescharacteristic
146C4×D61C2×Q1611yescharacteristic
147C4×D61AGL(1,ℤ/8)11yescharacteristic
148C4×D61#13711yescharacteristic
149#92420—41no—
150#92420—42no—
151#92420—41no—
152#92420—81no—
153C8×C61—41no—
154C8×C61—41no—
155#92418—22no—
156#92418—41no—
157#8902—611nomaximal, Sylow
158C22×C61#38511yescharacteristic
159C2×D61—21no—
160C2×D61—21no—
161C2×D61—22no—
162C2×D61—41no—
163C2×D61—41no—
164C2×D61#48011yescharacteristic
165C2×D61#48411yescharacteristic
166#6923—82no—
167C4×C61—22no—
168C4×C61—21no—
169C4×C61—41no—
170C4×C61C2×#10711yescharacteristic
171C4×C61#42811yescharacteristic
172C4×C61#44311yescharacteristic
173C4×C61#44411yescharacteristic
174Dic61—21no—
175Dic61—21no—
176Dic61—41no—
177Dic61#44711yescharacteristic
178Dic61#45111yescharacteristic
179#2906—611no—
180#1806—611no—
181#1571—611no—
182#1567—611no—
183#1476—611no—
184#1303—611no—
185#1279—611no—
186C2×C61#185211yescharacteristic
187C2×C61#186111yescharacteristic
188C2×C61#186611yescharacteristic
189D61—21no—
190D61—21no—
191D61—42no—
192C2×#131—1221no—
193C4×#41—611no—
194#431—611no—
195#424—1222no—
196#424—1221no—
197#423—611no—
198#423—611no—
199#423—1221no—
200C2×C4⋊C8—611no—
201C2×C4⋊C8—1221no—
202C2×#104—1221no—
203C2×C4×C8—611no—
204C4×C4⋊C4—611no—
205#339—1221no—
206#335—1222no—
207#335—1221no—
208#335—1221no—
209Q8⋊C8—611no—
210Q8⋊C8#692311yescharacteristic, complemented
211D4⋊C8—611no—
212D4⋊C8—611no—
213C61#890211yescharacteristic, complemented, Sylow
214C2×#41—611no—
215#131—2442no—
216#131—2442no—
217C2×M16—1221no—
218C2×M16—1222no—
219C2×M16—1221no—
220C22×C8—1221no—
221C22×C8—1221no—
222C22×C8—1222no—
223C4×Q8—611no—
224C4×Q8C2×#692311yescharacteristic
225C4×D4—611no—
226C4×D4—611no—
227C4×D4—1221no—
228C4×D4—1221no—
229#117—611no—
230#117—1221no—
231C2×C4⋊C4—1221no—
232C2×C42—611no—
233C2×C42—1221no—
234C2×C42—1221no—
235C4⋊C8—611no—
236C4⋊C8—611no—
237C4⋊C8—612no—
238C4⋊C8—611no—
239C4⋊C8—1221no—
240C4⋊C8—1221no—
241C4⋊C8—1221no—
242C4⋊C8—1221no—
243C4⋊C8—1222no—
244C4⋊C8—1222no—
245C4⋊C8C2×#692311yescharacteristic
246#104—2442no—
247C22⋊C8—1221no—
248C4×C8—611no—
249C4×C8—1221no—
250C4×C8—1221no—
251C4×C8C2×#692311yescharacteristic
252#95—1221no—
253#41—612no—
254#41—612no—
255#41—1221no—
256#41—1221no—
257C2×Q8C4⋊#692311yescharacteristic
258C2×D4—611no—
259C2×D4—1221no—
260C22×C4—611no—
261C22×C4—611no—
262C22×C4—611no—
263C22×C4—1221no—
264C22×C4—1221no—
265C22×C4—1221no—
266C22×C4—2441no—
267M16—2441no—
268M16—2441no—
269M16—2441no—
270M16—2441no—
271M16—2444no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—1221no—
275C2×C8—1221no—
276C2×C8—1221no—
277C2×C8—1222no—
278C2×C8—1221no—
279C2×C8—2442no—
280C2×C8—2441no—
281C2×C8—2441no—
282C2×C8—2441no—
283C2×C8—2444no—
284C4⋊C4—21no—
285C4⋊C4—611no—
286C4⋊C4—1221no—
287C4⋊C4—1222no—
288C4⋊C4—2441no—
289C4⋊C4C4⋊#692311yescharacteristic
290C22⋊C4—1221no—
291C22⋊C4—1221no—
292C22⋊C4—1221no—
293C42—21no—
294C42—611no—
295C42—611no—
296C42—611no—
297C42—1221no—
298C42—1221no—
299C42—2441no—
300C42—2441no—
301C42C22×#692311yescharacteristic
302C23—611no—
303C23—1221no—
304Q8—21no—
305Q8#1451984612yes—
306D4—612no—
307D4—1222no—
308D4—1222no—
309D4—1221no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—611no—
313C2×C4—611no—
314C2×C4—611no—
315C2×C4—611no—
316C2×C4—611no—
317C2×C4—611no—
318C2×C4—611no—
319C2×C4—1222no—
320C2×C4—1222no—
321C2×C4—1221no—
322C2×C4—1221no—
323C2×C4—1221no—
324C2×C4—4881no—
325C2×C4—4881no—
326C2×C4—4881no—
327C2×C4—2441no—
328C2×C4—2442no—
329C2×C4—2441no—
330C2×C4—2441no—
331C2×C4C2×C4⋊#692311yescharacteristic
332C2×C4D4×#692311yescharacteristic
333C2×C4D4×#692311yescharacteristic
334C8—41no—
335C8—41no—
336C8—1222no—
337C8—4881no—
338C8—2441no—
339C8—2441no—
340C8—2442no—
341C8—2441no—
342C22—611no—
343C22—611no—
344C22—1221no—
345C22—1221no—
346C22—1222no—
347C22—2441no—
348C22—2441no—
349C22#1684070911yescharacteristic
350C4—22no—
351C4—21no—
352C4—41no—
353C4—611no—
354C4—611no—
355C4—1221no—
356C4—1221no—
357C4—4882no—
358C4—2441no—
359C4C2×#1451984611yescharacteristic
360C4#1684077311yescharacteristic
361C4#1684080511yescharacteristic
362C4#1684080811yescharacteristic
363C2—1221no—
364C2—1221no—
365C2—2442no—
366C2#2146176811yescharacteristic
367C2#2146178111yescharacteristic
368C2#2146178911yescharacteristic
subgroupG/H
centerC22#16840709
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×Q8⋊C8C4
socleC22×C61#385
prime radicalC2×#11758226D4
cosocleC42×D61C23
hypercenterQ8⋊C8#6923
hypocenterC61#8902
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
3. #21461389C2semidirect
4. #21461378C2semidirect
5. #21460716C2semidirect
6. #21460654C2semidirect
25. Q8⋊#92418C4semidirect
26. #16839491C4semidirect
27. #16839489C4semidirect
28. C61×Q8⋊C8C4semidirect
210. Q8⋊C8#6923semidirect
213. C61#8902semidirect

Maximal subgroups

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

subgroupindex
#214624652
#214616922
#214613892
#214613782
#214607162
#214606542
#214442002
#890261