Finite Group Databasebeta

← #29653936

Subgroups — #29653936

Order 15616.

940 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456591C212yesmaximal
2#21451632C211yesmaximal, characteristic
3#21444861C214yesmaximal, complemented
4#21444793C214yesmaximal, complemented
5#21444391C212yesmaximal
6C2×#16839488C211yesmaximal, characteristic
7C2×C61×#544C211yesmaximal, characteristic
8C2×#14519756C2211yescharacteristic
9#16840097—22no—
10#16840095C2214yes—
11#16840001—24no—
12#16839918—22no—
13#16839917C2214yes—
14C2×C22⋊#92418C2212yes—
15#16839701C2212yes—
16#16839695—22no—
17#16839682C2212yes—
18#16839635—21no—
19#16839625—21no—
20#16839622C2211yescharacteristic
21#16839499C2218yes—
22#16839488C2214yes—
23C61×#544—24no—
24C61×#544—22no—
25C61×#544C2214yes—
26C2×C4⋊Q8×C61—22no—
27C2×#125×C61—22no—
28C2×C22⋊Q8×C61—21no—
29C2×C22⋊Q8×C61C2212yes—
30C2×#117×C61—21no—
31C22×C4⋊C4×C61C2211yescharacteristic
32C22×Dic122C2311yescharacteristic
33C2×#11758202—24no—
34C22×#92418—22no—
35#14519756—22no—
36#14519756—22no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61C2311yescharacteristic
39C2×#11758196—21no—
40C2×#11758196—22no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61—21no—
43#14519710—24no—
44#14519701—24no—
45#14519700—24no—
46C22⋊#92418—22no—
47C22⋊#92418C2314yes—
48#14519684—22no—
49#14519684—22no—
50#14519684—22no—
51#14519684C2312yes—
52C22×Q8×C61—21no—
53C22×Q8×C61D412yes—
54C23×C4×C61D411yescharacteristic
55C23×C4×C61D411yescharacteristic
56C23×C4×C61C2311yescharacteristic
57C4⋊Q8×C61—28no—
58C4⋊Q8×C61—28no—
59#125×C61—28no—
60#125×C61—28no—
61C22⋊Q8×C61—24no—
62C22⋊Q8×C61—22no—
63C22⋊Q8×C61—22no—
64C22⋊Q8×C61—24no—
65C22⋊Q8×C61D414yes—
66C22⋊Q8×C61C2314yes—
67#117×C61—24no—
68#117×C61—24no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61—22no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61—22no—
74C2×C4⋊C4×C61—22no—
75C2×C4⋊C4×C61—22no—
76C2×C4⋊C4×C61D412yes—
77C2×C4⋊C4×C61D412yes—
78C2×C4⋊C4×C61C2312yes—
79C2×C22⋊C4×C61—22no—
80C2×C42×C61—22no—
81C24×C61C2×D411yescharacteristic
82C2×Q8×C61—24no—
83C2×Q8×C61—22no—
84C2×Q8×C61—82no—
85C2×Q8×C61—82no—
86C2×Q8×C61C2×D414yes—
87C22×C4×C61—21no—
88C22×C4×C61—21no—
89C22×C4×C61—22no—
90C22×C4×C61—22no—
91C22×C4×C61—22no—
92C22×C4×C61—21no—
93C22×C4×C61—21no—
94C22×C4×C61—81no—
95C22×C4×C61—81no—
96C22×C4×C61Q1614yescomplemented
97C22×C4×C61C2×D412yes—
98C22×C4×C61C2×D412yes—
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D412yes—
101C22×C4×C61C2×D412yes—
102C22×C4×C61C2411yescharacteristic
103C22×Dic61—21no—
104C22×Dic61—21no—
105C22×Dic61—22no—
106C2×Dic122—22no—
107C2×Dic122—81no—
108C4⋊C4×C61—24no—
109C4⋊C4×C61—24no—
110C4⋊C4×C61—24no—
111C4⋊C4×C61—28no—
112C4⋊C4×C61—24no—
113C4⋊C4×C61—24no—
114C4⋊C4×C61—22no—
115C4⋊C4×C61—24no—
116C4⋊C4×C61—24no—
117C4⋊C4×C61C2×D414yes—
118C22⋊C4×C61—24no—
119C22⋊C4×C61—24no—
120C42×C61—28no—
121C22⋊Dic61—41no—
122C22⋊Dic61—41no—
123C22⋊Dic61—42no—
124#11758202—84no—
125C4⋊Dic61—22no—
126C4⋊Dic61—81no—
127#11758196—42no—
128#11758196—82no—
129C2×#92418—24no—
130C2×#92418—82no—
131C23×C61—41no—
132C23×C61—41no—
133C23×C61#12011yescharacteristic
134C23×C61#12011yescharacteristic
135C23×C61C2×Q1612yes—
136C23×C61#13712yes—
137C23×C61C22×D411yescharacteristic
138Q8×C61—44no—
139Q8×C61—82no—
140Q8×C61—84no—
141C2×C4×C61—22no—
142C2×C4×C61—22no—
143C2×C4×C61—24no—
144C2×C4×C61—24no—
145C2×C4×C61—24no—
146C2×C4×C61—24no—
147C2×C4×C61—22no—
148C2×C4×C61—42no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—42no—
152C2×C4×C61—44no—
153C2×C4×C61—42no—
154C2×C4×C61—44no—
155C2×C4×C61—42no—
156C2×C4×C61—81no—
157C2×C4×C61#12012yes—
158C2×C4×C61C2×Q1614yes—
159C2×C4×C61C22×D412yes—
160C2×Dic61—42no—
161C2×Dic61—42no—
162C2×Dic61—42no—
163C2×Dic61—42no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—42no—
167Dic122—44no—
168#92418—44no—
169#18295—611nomaximal, Sylow
170C22×C61—21no—
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—21no—
174C22×C61—21no—
175C22×C61—21no—
176C22×C61—42no—
177C22×C61—42no—
178C22×C61#45112yes—
179C22×C61#45112yes—
180C22×C61C2×#12011yescharacteristic
181C22×C61C22×Q1611yescharacteristic
182C22×C61C2×#13711yescharacteristic
183C4×C61—24no—
184C4×C61—22no—
185C4×C61—22no—
186C4×C61—44no—
187C4×C61—42no—
188C4×C61—44no—
189Dic61—82no—
190Dic61—81no—
191Dic61—81no—
192C2×#544D6111yescharacteristic
193#3189—612no—
194#2627—611no—
195#1617—614no—
196#1605—614no—
197#1520—612no—
198C2×#324—611no—
199C2×C61—21no—
200C2×C61—21no—
201C2×C61—22no—
202C2×C61—22no—
203C2×C61C2×#45111yescharacteristic
204C2×C61#302011yescharacteristic
205C2×C61#302611yescharacteristic
206#544—24no—
207#544—22no—
208#544C2×D6114yes—
209C2×C4⋊Q8—22no—
210C2×#125—22no—
211C2×C22⋊Q8—21no—
212C2×C22⋊Q8—611no—
213C2×C22⋊Q8C2×D6112yes—
214C2×#117—21no—
215C22×C4⋊C4C2×D6111yescharacteristic
216#484—614no—
217#484—1222no—
218#467—1224no—
219#451—614no—
220#451—1222no—
221C2×C22⋊C8—612no—
222#400—612no—
223#398—1222no—
224#395—612no—
225#385—1221no—
226#381—1221no—
227#380—611no—
228#328—618no—
229#324—614no—
230C61#1829511yescharacteristic, complemented, Sylow
231C22×Q8—21no—
232C22×Q8—611no—
233C22×Q8C22⋊D6112yes—
234C23×C4C22⋊D6111yescharacteristic
235C23×C4C22⋊D6111yescharacteristic
236C23×C4C22×D6111yescharacteristic
237C2×Q16—1224no—
238C22×C8—1222no—
239C4⋊Q8—28no—
240C4⋊Q8—28no—
241#125—28no—
242#125—28no—
243C22⋊Q8—24no—
244C22⋊Q8—24no—
245C22⋊Q8—22no—
246C22⋊Q8—22no—
247C22⋊Q8—1222no—
248C22⋊Q8—1222no—
249C22⋊Q8C22⋊D6114yes—
250C22⋊Q8C22×D6114yes—
251#117—24no—
252#117—24no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—611no—
261C2×C4⋊C4—1221no—
262C2×C4⋊C4—1222no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4C22⋊D6112yes—
265C2×C4⋊C4C22⋊D6112yes—
266C2×C4⋊C4C22×D6112yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—1221no—
269C2×C22⋊C4—1221no—
270C2×C42—22no—
271#107—1224no—
272#103—1224no—
273#103—1224no—
274C22⋊C8—614no—
275C22⋊C8—1222no—
276#95—612no—
277#95—1222no—
278#95—1222no—
279#95—1222no—
280C24C2×C22⋊D6111yescharacteristic
281C2×Q8—24no—
282C2×Q8—22no—
283C2×Q8—82no—
284C2×Q8—82no—
285C2×Q8—1222no—
286C2×Q8—4881no—
287C2×Q8C2×C22⋊D6114yes—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4—1222no—
298C22×C4—1221no—
299C22×C4—1221no—
300C22×C4#1175820214yescomplemented
301C22×C4C2×C22⋊D6112yes—
302C22×C4C2×C22⋊D6112yes—
303C22×C4C2×C22⋊D6112yes—
304C22×C4C2×C22⋊D6112yes—
305C22×C4C2×C22⋊D6111yescharacteristic
306C22×C4C23×D6111yescharacteristic
307Q16—4884no—
308C2×C8—1224no—
309C2×C8—4882no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—28no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—22no—
319C4⋊C4—1222no—
320C4⋊C4—4882no—
321C4⋊C4—4881no—
322C4⋊C4—2442no—
323C4⋊C4C2×C22⋊D6114yes—
324C22⋊C4—24no—
325C22⋊C4—24no—
326C22⋊C4—2442no—
327C22⋊C4—2441no—
328C22⋊C4—2441no—
329C42—28no—
330C23—41no—
331C23—41no—
332C23#1451974611yescharacteristic
333C23#1451974611yescharacteristic
334C23C2×#1175820212yes—
335C23#1451980812yes—
336C23C22×C22⋊D6111yescharacteristic
337Q8—44no—
338Q8—82no—
339Q8—84no—
340Q8—2444no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—81no—
357C2×C4—2442no—
358C2×C4—2442no—
359C2×C4—2442no—
360C2×C4—2442no—
361C2×C4—2441no—
362C2×C4—2441no—
363C2×C4—2442no—
364C2×C4#1451974612yes—
365C2×C4C2×#1175820214yes—
366C2×C4C22×C22⋊D6112yes—
367C8—2444no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—42no—
375C22—42no—
376C22#1683991812yes—
377C22#1683991812yes—
378C22C2×#1451974611yescharacteristic
379C22C22×#1175820211yescharacteristic
380C22C2×#1451980811yescharacteristic
381C4—22no—
382C4—22no—
383C4—24no—
384C4—44no—
385C4—42no—
386C4—44no—
387C4—4882no—
388C4—4881no—
389C4—4881no—
390C2—21no—
391C2—21no—
392C2—22no—
393C2—22no—
394C2C2×#1683991811yescharacteristic
395C2#2145518311yescharacteristic
396C2#2145521611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#544C2
socleC22×C61C2×#120
prime radicalC24×C61C2×D4
hypercenterC2×#544D61
hypocenterC61#18295
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #21444861C2semidirect
4. #21444793C2semidirect
96. C22×C4×C61Q16semidirect
230. C61#18295semidirect
300. C22×C4#11758202semidirect

Maximal subgroups

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

subgroupindex
#214565912
#214516322
#214448612
#214447932
#214443912
C2×#168394882
C2×C61×#5442
#1829561