Finite Group Databasebeta

← #29582586

Subgroups — #29582586

Order 15616.

594 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840049C211yesmaximal, characteristic
2C2×#16839964C211yesmaximal, characteristic
3#21446613C211yesmaximal, characteristic
4C2×#16839511C211yesmaximal, characteristic, complemented
5C2×#16839508C211yesmaximal, characteristic, complemented
6#21443654C211yesmaximal, characteristic, complemented
7C2×C61×#498C211yesmaximal, characteristic, complemented
8C22×#11758201—21no—
9C2×C4×D244C2211yescharacteristic
10#16840049—22no—
11#16840049—22no—
12#16839964—22no—
13#16839964—22no—
14C2×#14519708—21no—
15C2×C4⋊#92418C2211yescharacteristic
16C2×C4⋊#92418C2211yescharacteristic
17C2×C4⋊#92418C2211yescharacteristic
18C2×#14519701—21no—
19C2×#14519697—21no—
20C2×#14519689—22no—
21#16839511—22no—
22#16839508—22no—
23C2×C4⋊Q8×C61C2211yescharacteristic
24C2×C4⋊Q8×C61C2211yescharacteristic
25C61×#498—22no—
26C61×#498—22no—
27C2×#106×C61—22no—
28C2×C4×C8×C61C2211yescharacteristic
29C22×D244D411yescharacteristic
30C22×C4×D61—21no—
31C2×#11758201—24no—
32C2×#11758201—42no—
33C2×#11758201—42no—
34C22×#92418—21no—
35C22×#92418—21no—
36C22×#92418—21no—
37C4×D244—21no—
38C4×D244—21no—
39C4×D244—21no—
40C4×D244—21no—
41C2×C4⋊Dic61D411yescharacteristic
42C2×#11758198—21no—
43#14519708—24no—
44C4⋊#92418—21no—
45C4⋊#92418—21no—
46C4⋊#92418—21no—
47C4⋊#92418—21no—
48C4⋊#92418D412yes—
49C4⋊#92418D412yes—
50#14519701—24no—
51#14519697—24no—
52#14519689—44no—
53C22×Q8×C61—21no—
54C22×Q8×C61—21no—
55C22×C8×C61D412yes—
56C4⋊Q8×C61—21no—
57C4⋊Q8×C61—21no—
58C4⋊Q8×C61—21no—
59C4⋊Q8×C61—21no—
60C4⋊Q8×C61—21no—
61C4⋊Q8×C61—21no—
62C4⋊Q8×C61D412yes—
63C4⋊Q8×C61D412yes—
64C2×C4⋊C4×C61—21no—
65C2×C4⋊C4×C61—21no—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C42×C61C2311yescharacteristic
69#106×C61—28no—
70C4×C8×C61—21no—
71C4×C8×C61—21no—
72C23×D61—21no—
73C2×Q8×C61—22no—
74C2×Q8×C61—22no—
75C2×Q8×C61—41no—
76C2×Q8×C61—41no—
77C2×Q8×C61—41no—
78C2×Q8×C61—41no—
79C2×Q8×C61—41no—
80C2×Q8×C61—41no—
81C2×Q8×C61—82no—
82C22×C4×C61—41no—
83C22×C4×C61—41no—
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61#4111yescharacteristic
87C22×Dic61—21no—
88C2×D244—21no—
89C2×D244—21no—
90C2×D244—21no—
91C2×D244—21no—
92C2×D244—21no—
93C2×D244SD1612yes—
94C2×C4×D61—41no—
95C2×C4×D61—41no—
96C2×C4×D61—42no—
97C2×C4×D61—42no—
98C2×C8×C61—22no—
99C2×C8×C61—22no—
100C2×C8×C61SD1614yescomplemented
101C4⋊C4×C61—22no—
102C4⋊C4×C61—22no—
103C4⋊C4×C61—22no—
104C4⋊C4×C61—22no—
105C4⋊C4×C61—41no—
106C4⋊C4×C61—41no—
107C4⋊C4×C61—41no—
108C4⋊C4×C61—41no—
109C42×C61C2×D411yescharacteristic
110C42×C61C2×D411yescharacteristic
111C42×C61C2×D411yescharacteristic
112C42×C61C2×D411yescharacteristic
113#11758201—48no—
114#11758198—41no—
115#11758198—41no—
116C4⋊Dic61—21no—
117C4⋊Dic61—21no—
118C2×#92418—22no—
119C2×#92418—22no—
120C2×#92418—41no—
121C2×#92418—41no—
122C2×#92418—41no—
123C2×#92418—41no—
124C2×#92418—41no—
125C2×#92418—42no—
126C23×C61#12111yescharacteristic
127C22×D61—22no—
128C22×D61—41no—
129C22×D61—42no—
130C22×D61—41no—
131C22×D61—42no—
132Q8×C61—42no—
133Q8×C61—42no—
134Q8×C61—82no—
135Q8×C61—82no—
136Q8×C61—81no—
137Q8×C61—81no—
138C2×C4×C61—22no—
139C2×C4×C61—42no—
140C2×C4×C61—42no—
141C2×C4×C61—81no—
142C2×C4×C61—81no—
143C2×C4×C61—82no—
144C2×C4×C61#12011yescharacteristic
145C2×C4×C61#12011yescharacteristic
146C2×C4×C61#12011yescharacteristic
147C2×C4×C61#12111yescharacteristic
148C2×C4×C61#12111yescharacteristic
149C2×C4×C61#12711yescharacteristic
150C2×C4×C61C2×SD1611yescharacteristic
151C2×C4×C61C2×SD1612yes—
152C2×C4×C61#13511yescharacteristic
153C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C61#13711yescharacteristic
155C2×C4×C61#13712yes—
156C2×Dic61—41no—
157C2×Dic61—42no—
158D244—22no—
159D244—22no—
160D244—41no—
161D244—41no—
162C4×D61—82no—
163C4×D61—81no—
164C4×D61—81no—
165C8×C61—24no—
166#92418—42no—
167#92418—42no—
168#92418—81no—
169#10271—611nomaximal, Sylow
170C22×C61#39211yescharacteristic
171C22×C61#46112yes—
172C22×C61#46511yescharacteristic
173C22×C61#46811yescharacteristic
174C22×C61#47112yes—
175C2×D61—44no—
176C2×D61—42no—
177C2×D61—82no—
178C2×D61—81no—
179C2×D61—81no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—82no—
183C4×C61—82no—
184C4×C61#44811yescharacteristic
185C4×C61#44812yes—
186C4×C61#44911yescharacteristic
187C4×C61#45011yescharacteristic
188C4×C61#45211yescharacteristic
189C4×C61#49211yescharacteristic
190C4×C61#49711yescharacteristic
191Dic61—81no—
192C2×#498D6111yescharacteristic, complemented
193C2×#476—611no—
194C2×#461—611no—
195#1954—611no—
196C2×#333—611no—
197C2×#332—611no—
198#1277—611no—
199C2×C61#167112yes—
200C2×C61#169112yes—
201C2×C61#200011yescharacteristic
202C2×C61#201311yescharacteristic
203C2×C61#201911yescharacteristic
204D61—82no—
205C22×SD16—1221no—
206C2×C4⋊Q8C2×D6111yescharacteristic
207C2×C4⋊Q8C2×D6111yescharacteristic
208C2×C4×D4—611no—
209#498—22no—
210#498—22no—
211#476—1222no—
212#476—1222no—
213#461—1222no—
214#461—1222no—
215C2×#106—22no—
216C2×#106—1221no—
217C2×C4⋊C8—611no—
218C2×C4⋊C8—611no—
219C2×C4⋊C8—611no—
220C2×#103—1221no—
221C2×#102—1221no—
222C2×C22⋊C8—1222no—
223C2×C4×C8C2×D6111yescharacteristic
224#333—1222no—
225#332—1222no—
226C61#1027111yescharacteristic, complemented, Sylow
227C22×Q8—21no—
228C22×Q8—21no—
229C22×D4—611no—
230C23×C4—1221no—
231C2×SD16—1224no—
232C2×SD16—2442no—
233C2×SD16—2442no—
234C22×C8—1221no—
235C22×C8—1221no—
236C22×C8—1221no—
237C22×C8C22⋊D6112yes—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8C22⋊D6112yes—
245C4⋊Q8C22⋊D6112yes—
246C4×D4—1221no—
247C4×D4—1221no—
248C4×D4—1221no—
249C4×D4—1221no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—611no—
255C2×C22⋊C4—1221no—
256C2×C42C22×D6111yescharacteristic
257#106—28no—
258#106—1224no—
259C4⋊C8—612no—
260C4⋊C8—612no—
261C4⋊C8—1221no—
262C4⋊C8—1221no—
263C4⋊C8—1221no—
264C4⋊C8—1221no—
265#103—1224no—
266#102—1224no—
267C22⋊C8—2444no—
268C4×C8—21no—
269C4×C8—21no—
270C24—1221no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—82no—
280C2×D4—612no—
281C2×D4—1221no—
282C2×D4—1221no—
283C2×D4—1221no—
284C2×D4—1221no—
285C2×D4—1221no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—1221no—
289C22×C4—2442no—
290C22×C4—2441no—
291C22×C4—2441no—
292C22×C4—2442no—
293C22×C4D4×D6111yescharacteristic
294C22×C4#1175822411yescharacteristic
295C22×C4C2×C22⋊D6111yescharacteristic
296SD16—2448no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—1222no—
300C2×C8—1222no—
301C2×C8—2441no—
302C2×C8—2442no—
303C2×C8—2441no—
304C2×C8—2441no—
305C2×C8—2441no—
306C2×C8—2441no—
307C2×C8#1175820114yescomplemented
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—1221no—
317C4⋊C4—1221no—
318C22⋊C4—2441no—
319C22⋊C4—2441no—
320C42D4×D6111yescharacteristic
321C42D4×D6111yescharacteristic
322C42C2×C22⋊D6111yescharacteristic
323C42C2×C22⋊D6111yescharacteristic
324C23—1222no—
325C23—2442no—
326C23—2441no—
327C23—2442no—
328C23—2441no—
329C23#1451974811yescharacteristic
330Q8—42no—
331Q8—42no—
332Q8—82no—
333Q8—82no—
334Q8—81no—
335Q8—81no—
336D4—1222no—
337D4—1222no—
338D4—2441no—
339D4—2441no—
340C2×C4—22no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—82no—
346C2×C4—4881no—
347C2×C4—4881no—
348C2×C4—4882no—
349C2×C4—2442no—
350C2×C4—2441no—
351C2×C4#1451974511yescharacteristic
352C2×C4#1451974511yescharacteristic
353C2×C4#1451974611yescharacteristic
354C2×C4#1451974911yescharacteristic
355C2×C4#1451974911yescharacteristic
356C2×C4#1451977311yescharacteristic
357C2×C4SD16×D6111yescharacteristic
358C2×C4C2×#1175820112yes—
359C2×C4#1451979811yescharacteristic
360C2×C4#1451980411yescharacteristic
361C2×C4#1451980611yescharacteristic
362C2×C4#1451980812yes—
363C8—24no—
364C8—4881no—
365C8—2442no—
366C8—2442no—
367C22—4881no—
368C22—4882no—
369C22—4881no—
370C22—2444no—
371C22—2442no—
372C22#1683967011yescharacteristic
373C22#1683996412yes—
374C22#1683998811yescharacteristic
375C22#1684000511yescharacteristic
376C22#1684002212yes—
377C4—21no—
378C4—21no—
379C4—82no—
380C4—82no—
381C4—4881no—
382C4#1683989911yescharacteristic
383C4#1683990112yes—
384C4#1683990611yescharacteristic
385C4#1683991111yescharacteristic
386C4#1683992111yescharacteristic
387C4#1684013611yescharacteristic
388C4#1684015411yescharacteristic
389C2—4882no—
390C2#2144519112yes—
391C2#2144531512yes—
392C2#2144681911yescharacteristic
393C2#2144690211yescharacteristic
394C2#2144693111yescharacteristic
subgroupG/H
centerC23#14519748
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC2×C61×#498C2
socleC23×C61#121
prime radicalC22×D244D4
hypercenterC2×#498D61
hypocenterC61#10271
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#498

Complements

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

NH ≅ G/Ntype
4. C2×#16839511C2semidirect
5. C2×#16839508C2semidirect
6. #21443654C2semidirect
7. C2×C61×#498C2semidirect
100. C2×C8×C61SD16semidirect
192. C2×#498D61semidirect
226. C61#10271semidirect
307. C2×C8#11758201semidirect

Maximal subgroups

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

subgroupindex
C2×#168400492
C2×#168399642
#214466132
C2×#168395112
C2×#168395082
#214436542
C2×C61×#4982
#1027161