Finite Group Databasebeta

← #29656227

Subgroups — #29656227

Order 15616.

886 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456361C211yesmaximal, characteristic
2#21455949C211yesmaximal, characteristic
3#21455949C211yesmaximal, characteristic
4#21445096C214yesmaximal, complemented
5#21445061C214yesmaximal, complemented
6#21444536C211yesmaximal, characteristic
7#21444266C211yesmaximal, characteristic
8#21444266C211yesmaximal, characteristic
9C2×C61×#557C211yesmaximal, characteristic
10#16840319C2211yescharacteristic
11#16840168—22no—
12#16840165C2212yes—
13#16840158C2212yes—
14#16840092—21no—
15#16840092—21no—
16#16840047C2212yes—
17#16840047C2212yes—
18#16839972C2212yes—
19#16839972C2212yes—
20#16839918—21no—
21#16839918—21no—
22#16839867—22no—
23#16839867—22no—
24#16839819C2211yescharacteristic
25#16839819C2211yescharacteristic
26#16839802—22no—
27C4×#11758194C2211yescharacteristic
28#16839508C2214yes—
29Q8⋊#92418C2214yes—
30Q8⋊#92418C2214yes—
31C61×#557—22no—
32C61×#557—22no—
33C61×#557—22no—
34C61×#557C2214yes—
35C2×C4⋊Q8×C61—21no—
36C2×C4⋊Q8×C61C2211yescharacteristic
37C2×#125×C61—22no—
38C2×#125×C61—21no—
39C2×C4×Q8×C61—21no—
40C2×C4×Q8×C61—21no—
41C2×C4×Q8×C61C2211yescharacteristic
42C2×C4×Q8×C61C2211yescharacteristic
43C2×#11758202—22no—
44C2×#11758202—22no—
45C2×#11758194—21no—
46#14519777C2311yescharacteristic
47#14519777C2311yescharacteristic
48#14519769—21no—
49#14519756—21no—
50#14519756—21no—
51C4×Dic122—21no—
52C4×Dic122—21no—
53#14519736—21no—
54#14519710—24no—
55#14519708—22no—
56#14519708—22no—
57#14519708—24no—
58C4⋊#92418C2312yes—
59C4⋊#92418C2312yes—
60#14519701—22no—
61#14519701—22no—
62#14519700—22no—
63#14519700—22no—
64C22⋊#92418—21no—
65C22⋊#92418—21no—
66#14519688—21no—
67#14519688—21no—
68#14519688—21no—
69C4×#92418C2312yes—
70C22×Q8×C61—21no—
71C22×Q8×C61D411yescharacteristic
72C22×Q8×C61D411yescharacteristic
73C4⋊Q8×C61—24no—
74C4⋊Q8×C61—24no—
75C4⋊Q8×C61—21no—
76C4⋊Q8×C61—21no—
77C4⋊Q8×C61D412yes—
78C4⋊Q8×C61C2312yes—
79#125×C61—28no—
80#125×C61—24no—
81#125×C61—28no—
82#125×C61—24no—
83C4×Q8×C61—24no—
84C4×Q8×C61—24no—
85C4×Q8×C61—24no—
86C4×Q8×C61—24no—
87C4×Q8×C61—21no—
88C4×Q8×C61—21no—
89C4×Q8×C61—21no—
90C4×Q8×C61—21no—
91C4×Q8×C61D412yes—
92C4×Q8×C61D412yes—
93C4×Q8×C61C2312yes—
94C4×Q8×C61C2312yes—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61D412yes—
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C42×C61—22no—
105C2×C42×C61—21no—
106C2×C42×C61—21no—
107C2×C42×C61C2311yescharacteristic
108C2×Q8×C61—24no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—42no—
114C2×Q8×C61—44no—
115C2×Q8×C61C2×D412yes—
116C2×Q8×C61C2×D412yes—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—22no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C2×Dic122—21no—
126C2×Dic122—21no—
127C4⋊C4×C61—28no—
128C4⋊C4×C61—28no—
129C4⋊C4×C61—28no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61C2×D412yes—
140C4⋊C4×C61C2×D412yes—
141C4⋊C4×C61C2×D414yes—
142C42×C61—28no—
143C42×C61—22no—
144C42×C61—22no—
145C42×C61—22no—
146C42×C61—22no—
147C42×C61C2×D411yescharacteristic
148C42×C61C2×D411yescharacteristic
149C42×C61C2×D411yescharacteristic
150C42×C61C2411yescharacteristic
151C22⋊Dic61—41no—
152#11758202—44no—
153#11758202—44no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158#11758196—41no—
159#11758196—41no—
160C4×Dic61—41no—
161#11758194—24no—
162C2×#92418—22no—
163C2×#92418—22no—
164C2×#92418—22no—
165C23×C61#12011yescharacteristic
166Q8×C61—24no—
167Q8×C61—22no—
168Q8×C61—22no—
169Q8×C61—22no—
170Q8×C61—42no—
171Q8×C61—44no—
172Q8×C61—44no—
173Q8×C61#13714yes—
174C2×C4×C61—24no—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—24no—
183C2×C4×C61—42no—
184C2×C4×C61—44no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61#12011yescharacteristic
188C2×C4×C61#12011yescharacteristic
189C2×C4×C61#12011yescharacteristic
190C2×C4×C61C22×D411yescharacteristic
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C22×D411yescharacteristic
193C2×Dic61—41no—
194C2×Dic61—41no—
195Dic122—42no—
196Dic122—42no—
197#92418—24no—
198#92418—42no—
199#92418—42no—
200#18541—611nomaximal, Sylow
201C22×C61—21no—
202C22×C61—21no—
203C22×C61#45612yes—
204C22×C61C2×#12011yescharacteristic
205C4×C61—24no—
206C4×C61—21no—
207C4×C61—21no—
208C4×C61—21no—
209C4×C61—21no—
210C4×C61—42no—
211C4×C61—44no—
212C4×C61—42no—
213C4×C61—42no—
214C4×C61C2×#13712yes—
215C4×C61#57512yes—
216Dic61—81no—
217Dic61—81no—
218C2×#557D6111yescharacteristic
219#3165—611no—
220#3113—611no—
221#3113—611no—
222#1653—614no—
223#1648—614no—
224#1556—611no—
225#1491—611no—
226#1491—611no—
227C2×C61—22no—
228C2×C61#300411yescharacteristic
229C2×C61#300511yescharacteristic
230C2×C61C2×#45611yescharacteristic
231#557—22no—
232#557—22no—
233#557—22no—
234#557C2×D6114yes—
235#533—611no—
236C2×C4⋊Q8—21no—
237C2×C4⋊Q8C2×D6111yescharacteristic
238C2×#125—22no—
239C2×#125—21no—
240C2×C4×Q8—21no—
241C2×C4×Q8—21no—
242C2×C4×Q8C2×D6111yescharacteristic
243C2×C4×Q8C2×D6111yescharacteristic
244#501—1222no—
245#500—612no—
246#498—612no—
247#483—1221no—
248#483—1221no—
249#475—612no—
250#475—612no—
251#462—612no—
252#462—612no—
253#451—1221no—
254#451—1221no—
255#441—1222no—
256#441—1222no—
257#432—611no—
258#432—611no—
259#428—1222no—
260C4×M16—611no—
261#332—614no—
262Q8⋊C8—614no—
263Q8⋊C8—614no—
264C61#1854111yescharacteristic, complemented, Sylow
265C22×Q8—21no—
266C22×Q8C22⋊D6111yescharacteristic
267C22×Q8C22⋊D6111yescharacteristic
268C2×Q16—1222no—
269C2×Q16—1222no—
270C2×M16—1221no—
271C4⋊Q8—24no—
272C4⋊Q8—24no—
273C4⋊Q8—21no—
274C4⋊Q8—21no—
275C4⋊Q8—611no—
276C4⋊Q8—611no—
277C4⋊Q8C22⋊D6112yes—
278C4⋊Q8C22×D6112yes—
279#125—28no—
280#125—24no—
281#125—28no—
282#125—24no—
283#125—1221no—
284C22⋊Q8—1221no—
285C22⋊Q8—1221no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—21no—
291C4×Q8—21no—
292C4×Q8—21no—
293C4×Q8—21no—
294C4×Q8—1221no—
295C4×Q8—1221no—
296C4×Q8C22⋊D6112yes—
297C4×Q8C22⋊D6112yes—
298C4×Q8C22×D6112yes—
299C4×Q8C22×D6112yes—
300#117—1221no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4C22⋊D6111yescharacteristic
308C2×C4⋊C4C22⋊D6111yescharacteristic
309C2×C4⋊C4C22⋊D6112yes—
310C2×C42—22no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42C22×D6111yescharacteristic
314#107—1224no—
315#106—1224no—
316#106—1222no—
317#106—1222no—
318C4⋊C8—612no—
319C4⋊C8—612no—
320#103—1222no—
321#103—1222no—
322#103—1222no—
323#103—1222no—
324C22⋊C8—1221no—
325C22⋊C8—1221no—
326#97—1221no—
327#97—1221no—
328#97—1221no—
329C4×C8—612no—
330C2×Q8—24no—
331C2×Q8—22no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—22no—
335C2×Q8—42no—
336C2×Q8—44no—
337C2×Q8—1221no—
338C2×Q8—1221no—
339C2×Q8C2×C22⋊D6112yes—
340C2×Q8C2×C22⋊D6112yes—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4C2×C22⋊D6111yescharacteristic
347C22×C4C2×C22⋊D6111yescharacteristic
348C22×C4C2×C22⋊D6111yescharacteristic
349Q16—2444no—
350Q16—2444no—
351M16—1224no—
352C2×C8—1222no—
353C2×C8—1222no—
354C2×C8—1222no—
355C4⋊C4—28no—
356C4⋊C4—28no—
357C4⋊C4—28no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—1221no—
368C4⋊C4—1221no—
369C4⋊C4—1221no—
370C4⋊C4—1221no—
371C4⋊C4—2441no—
372C4⋊C4—2441no—
373C4⋊C4C2×C22⋊D6112yes—
374C4⋊C4C2×C22⋊D6112yes—
375C4⋊C4C2×C22⋊D6114yes—
376C22⋊C4—2441no—
377C42—28no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—2441no—
383C42C2×C22⋊D6111yescharacteristic
384C42C2×C22⋊D6111yescharacteristic
385C42C2×C22⋊D6111yescharacteristic
386C42C23×D6111yescharacteristic
387C23#1451974611yescharacteristic
388Q8—24no—
389Q8—22no—
390Q8—22no—
391Q8—22no—
392Q8—42no—
393Q8—44no—
394Q8—44no—
395Q8—2442no—
396Q8—2442no—
397Q8#1451980814yes—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—2441no—
412C2×C4—2441no—
413C2×C4#1451974611yescharacteristic
414C2×C4#1451974611yescharacteristic
415C2×C4#1451974611yescharacteristic
416C2×C4C22×C22⋊D6111yescharacteristic
417C2×C4C22×C22⋊D6111yescharacteristic
418C2×C4C22×C22⋊D6111yescharacteristic
419C8—1224no—
420C8—2442no—
421C8—2442no—
422C22—21no—
423C22—21no—
424C22#1683994512yes—
425C22C2×#1451974611yescharacteristic
426C4—21no—
427C4—21no—
428C4—24no—
429C4—21no—
430C4—21no—
431C4—42no—
432C4—42no—
433C4—42no—
434C4—44no—
435C4—4881no—
436C4—4881no—
437C4C2×#1451980812yes—
438C4#1684056012yes—
439C2—22no—
440C2#2145505311yescharacteristic
441C2#2145506111yescharacteristic
442C2C2×#1683994511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#557C2
socleC22×C61C2×#120
prime radicalC23×C61#120
hypercenterC2×#557D61
hypocenterC61#18541
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
4. #21445096C2semidirect
5. #21445061C2semidirect
264. C61#18541semidirect

Maximal subgroups

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

subgroupindex
#214563612
#214559492
#214450962
#214450612
#214445362
#214442662
C2×C61×#5572
#1854161