Finite Group Databasebeta

← #29582579

Subgroups — #29582579

Order 15616.

594 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840039C211yesmaximal, characteristic
2C2×#16839964C211yesmaximal, characteristic
3#21446613C211yesmaximal, characteristic
4C2×#16839508C211yesmaximal, characteristic, complemented
5C2×#16839508C211yesmaximal, characteristic, complemented
6#21443654C211yesmaximal, characteristic, complemented
7C2×C61×#487C211yesmaximal, characteristic, complemented
8C22×#11758201—21no—
9C2×C4×D244C2211yescharacteristic
10#16840039—22no—
11#16840039—22no—
12#16839964—22no—
13#16839964—22no—
14C2×#14519710—21no—
15C2×C4⋊#92418C2211yescharacteristic
16C2×C4⋊#92418C2211yescharacteristic
17C2×C4⋊#92418C2211yescharacteristic
18C2×#14519701—21no—
19C2×#14519697—21no—
20C2×#14519689—22no—
21#16839508—22no—
22#16839508—22no—
23C2×C4⋊Q8×C61C2211yescharacteristic
24C2×C4⋊Q8×C61C2211yescharacteristic
25C61×#487—22no—
26C61×#487—22no—
27C2×#103×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#14519710—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#103×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—42no—
79C2×Q8×C61—41no—
80C2×Q8×C61—42no—
81C2×Q8×C61—41no—
82C2×Q8×C61—41no—
83C22×C4×C61—41no—
84C22×C4×C61—41no—
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61#4111yescharacteristic
88C22×Dic61—21no—
89C2×D244—21no—
90C2×D244—21no—
91C2×D244—21no—
92C2×D244—21no—
93C2×D244—21no—
94C2×D244SD1612yes—
95C2×C4×D61—41no—
96C2×C4×D61—41no—
97C2×C4×D61—42no—
98C2×C4×D61—42no—
99C2×C8×C61—22no—
100C2×C8×C61—22no—
101C2×C8×C61—22no—
102C4⋊C4×C61—22no—
103C4⋊C4×C61—22no—
104C4⋊C4×C61—22no—
105C4⋊C4×C61—22no—
106C4⋊C4×C61—41no—
107C4⋊C4×C61—41no—
108C4⋊C4×C61—41no—
109C4⋊C4×C61—41no—
110C42×C61C2×D411yescharacteristic
111C42×C61C2×D411yescharacteristic
112C42×C61C2×D411yescharacteristic
113C42×C61C2×D411yescharacteristic
114#11758201—48no—
115#11758198—41no—
116#11758198—41no—
117C4⋊Dic61—21no—
118C4⋊Dic61—21no—
119C2×#92418—22no—
120C2×#92418—22no—
121C2×#92418—41no—
122C2×#92418—41no—
123C2×#92418—41no—
124C2×#92418—41no—
125C2×#92418—41no—
126C2×#92418—42no—
127C23×C61#12111yescharacteristic
128C22×D61—22no—
129C22×D61—41no—
130C22×D61—42no—
131C22×D61—41no—
132C22×D61—42no—
133Q8×C61—44no—
134Q8×C61—42no—
135Q8×C61—42no—
136Q8×C61—81no—
137Q8×C61—81no—
138Q8×C61—81no—
139Q8×C61—81no—
140C2×C4×C61—22no—
141C2×C4×C61—42no—
142C2×C4×C61—42no—
143C2×C4×C61—82no—
144C2×C4×C61—81no—
145C2×C4×C61—81no—
146C2×C4×C61#12011yescharacteristic
147C2×C4×C61#12011yescharacteristic
148C2×C4×C61#12011yescharacteristic
149C2×C4×C61#12111yescharacteristic
150C2×C4×C61#12111yescharacteristic
151C2×C4×C61#12711yescharacteristic
152C2×C4×C61C2×SD1611yescharacteristic
153C2×C4×C61#13511yescharacteristic
154C2×C4×C61#13512yes—
155C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C61AGL(1,ℤ/8)12yes—
157C2×C4×C61#13711yescharacteristic
158C2×Dic61—41no—
159C2×Dic61—42no—
160D244—22no—
161D244—22no—
162D244—41no—
163D244—41no—
164C4×D61—82no—
165C4×D61—81no—
166C4×D61—81no—
167C8×C61—42no—
168#92418—42no—
169#92418—42no—
170#92418—81no—
171#10270—611nomaximal, Sylow
172C22×C61#39211yescharacteristic
173C22×C61#46312yes—
174C22×C61#46511yescharacteristic
175C22×C61#46811yescharacteristic
176C22×C61#47312yes—
177C2×D61—44no—
178C2×D61—42no—
179C2×D61—82no—
180C2×D61—81no—
181C2×D61—81no—
182C4×C61—21no—
183C4×C61—21no—
184C4×C61—82no—
185C4×C61—82no—
186C4×C61#44811yescharacteristic
187C4×C61#44911yescharacteristic
188C4×C61#44912yes—
189C4×C61#45011yescharacteristic
190C4×C61#45211yescharacteristic
191C4×C61#49511yescharacteristic
192C4×C61#49611yescharacteristic
193Dic61—81no—
194C2×#487D6111yescharacteristic, complemented
195C2×#474—611no—
196C2×#461—611no—
197#1954—611no—
198C2×#332—611no—
199C2×#332—611no—
200#1277—611no—
201C2×C61#167912yes—
202C2×C61#169912yes—
203C2×C61#200011yescharacteristic
204C2×C61#201211yescharacteristic
205C2×C61#202011yescharacteristic
206D61—82no—
207C22×SD16—1221no—
208C2×C4⋊Q8C2×D6111yescharacteristic
209C2×C4⋊Q8C2×D6111yescharacteristic
210C2×C4×D4—611no—
211#487—22no—
212#487—22no—
213#474—1222no—
214#474—1222no—
215#461—1222no—
216#461—1222no—
217C2×#107—1221no—
218C2×C4⋊C8—611no—
219C2×C4⋊C8—611no—
220C2×C4⋊C8—611no—
221C2×#103—22no—
222C2×#103—1221no—
223C2×#102—1221no—
224C2×C22⋊C8—1222no—
225C2×C4×C8C2×D6111yescharacteristic
226#332—1222no—
227#332—1222no—
228C61#1027011yescharacteristic, complemented, Sylow
229C22×Q8—21no—
230C22×Q8—21no—
231C22×D4—611no—
232C23×C4—1221no—
233C2×SD16—1224no—
234C2×SD16—2442no—
235C2×SD16—2442no—
236C22×C8—1221no—
237C22×C8—1221no—
238C22×C8—1221no—
239C22×C8C22⋊D6112yes—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8C22⋊D6112yes—
247C4⋊Q8C22⋊D6112yes—
248C4×D4—1221no—
249C4×D4—1221no—
250C4×D4—1221no—
251C4×D4—1221no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—611no—
257C2×C22⋊C4—1221no—
258C2×C42C22×D6111yescharacteristic
259#107—1224no—
260C4⋊C8—612no—
261C4⋊C8—612no—
262C4⋊C8—1221no—
263C4⋊C8—1221no—
264C4⋊C8—1221no—
265C4⋊C8—1221no—
266#103—28no—
267#103—1224no—
268#102—1224no—
269C22⋊C8—2444no—
270C4×C8—21no—
271C4×C8—21no—
272C24—1221no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—42no—
279C2×Q8—41no—
280C2×Q8—42no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×D4—612no—
284C2×D4—1221no—
285C2×D4—1221no—
286C2×D4—1221no—
287C2×D4—1221no—
288C2×D4—1221no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—1221no—
292C22×C4—2441no—
293C22×C4—2441no—
294C22×C4—2442no—
295C22×C4—2442no—
296C22×C4D4×D6111yescharacteristic
297C22×C4#1175822411yescharacteristic
298C22×C4C2×C22⋊D6111yescharacteristic
299SD16—2448no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—1222no—
304C2×C8—1222no—
305C2×C8—2441no—
306C2×C8—2442no—
307C2×C8—2441no—
308C2×C8—2441no—
309C2×C8—2441no—
310C2×C8—2441no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—1221no—
320C4⋊C4—1221no—
321C22⋊C4—2441no—
322C22⋊C4—2441no—
323C42D4×D6111yescharacteristic
324C42D4×D6111yescharacteristic
325C42C2×C22⋊D6111yescharacteristic
326C42C2×C22⋊D6111yescharacteristic
327C23—1222no—
328C23—2442no—
329C23—2441no—
330C23—2442no—
331C23—2441no—
332C23#1451974811yescharacteristic
333Q8—44no—
334Q8—42no—
335Q8—42no—
336Q8—81no—
337Q8—81no—
338Q8—81no—
339Q8—81no—
340D4—1222no—
341D4—1222no—
342D4—2441no—
343D4—2441no—
344C2×C4—22no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—82no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—4881no—
351C2×C4—4881no—
352C2×C4—4882no—
353C2×C4—2442no—
354C2×C4—2441no—
355C2×C4#1451974511yescharacteristic
356C2×C4#1451974511yescharacteristic
357C2×C4#1451974611yescharacteristic
358C2×C4#1451974911yescharacteristic
359C2×C4#1451974911yescharacteristic
360C2×C4#1451977311yescharacteristic
361C2×C4SD16×D6111yescharacteristic
362C2×C4#1451979712yes—
363C2×C4#1451979811yescharacteristic
364C2×C4#1451980212yes—
365C2×C4#1451980411yescharacteristic
366C2×C4#1451980611yescharacteristic
367C8—42no—
368C8—4881no—
369C8—2442no—
370C8—2442no—
371C22—4881no—
372C22—4881no—
373C22—4882no—
374C22—2444no—
375C22—2442no—
376C22#1683967011yescharacteristic
377C22#1683997412yes—
378C22#1683998811yescharacteristic
379C22#1684000511yescharacteristic
380C22#1684003412yes—
381C4—21no—
382C4—21no—
383C4—82no—
384C4—82no—
385C4—4881no—
386C4#1683989911yescharacteristic
387C4#1683990611yescharacteristic
388C4#1683990712yes—
389C4#1683991111yescharacteristic
390C4#1683992111yescharacteristic
391C4#1684014411yescharacteristic
392C4#1684015211yescharacteristic
393C2—4882no—
394C2#2144523912yes—
395C2#2144536312yes—
396C2#2144681911yescharacteristic
397C2#2144689611yescharacteristic
398C2#2144693811yescharacteristic
subgroupG/H
centerC23#14519748
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC2×C61×#487C2
socleC23×C61#121
prime radicalC22×D244D4
hypercenterC2×#487D61
hypocenterC61#10270
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
4. C2×#16839508C2semidirect
5. C2×#16839508C2semidirect
6. #21443654C2semidirect
7. C2×C61×#487C2semidirect
194. C2×#487D61semidirect
228. C61#10270semidirect

Maximal subgroups

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

subgroupindex
C2×#168400392
C2×#168399642
#214466132
C2×#168395082
#214436542
C2×C61×#4872
#1027061