Finite Group Databasebeta

← #29658629

Subgroups — #29658629

Order 15616.

878 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456963C211yesmaximal, characteristic
2#21455858C211yesmaximal, characteristic
3#21455624C211yesmaximal, characteristic
4#21445096C214yesmaximal, complemented
5#21444977C214yesmaximal, complemented
6#21444491C211yesmaximal, characteristic
7#21444248C211yesmaximal, characteristic
8#21444214C211yesmaximal, characteristic
9C2×C61×#557C211yesmaximal, characteristic
10#16840319C2211yescharacteristic
11#16840158C2212yes—
12#16840114C2212yes—
13#16840097—21no—
14#16840092—22no—
15#16840069—22no—
16#16840057C2212yes—
17#16840057C2212yes—
18#16840047C2212yes—
19#16840044—22no—
20#16839972C2212yes—
21#16839918—21no—
22C4×#11758202—22no—
23#16839815C2211yescharacteristic
24#16839815C2211yescharacteristic
25#16839802—21no—
26#16839786C2211yescharacteristic
27#16839771—21no—
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×#11758194—21no—
45#14519777C2311yescharacteristic
46#14519777C2311yescharacteristic
47#14519769—21no—
48#14519756—21no—
49#14519756—21no—
50C4×Dic122—21no—
51C4×Dic122—21no—
52#14519736—21no—
53#14519710—22no—
54#14519710—22no—
55#14519708—24no—
56#14519708—22no—
57C4⋊#92418—21no—
58C4⋊#92418—21no—
59C4⋊#92418C2312yes—
60C4⋊#92418C2312yes—
61#14519701—22no—
62#14519701—22no—
63#14519701—22no—
64#14519700—22no—
65#14519700—24no—
66C22⋊#92418—21no—
67C22⋊#92418—21no—
68C4×#92418—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—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—42no—
116C2×Q8×C61—42no—
117C2×Q8×C61C2×D412yes—
118C2×Q8×C61C2×D412yes—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—22no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C2×Dic122—21no—
128C2×Dic122—21no—
129C4⋊C4×C61—28no—
130C4⋊C4×C61—28no—
131C4⋊C4×C61—28no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61C2×D412yes—
142C4⋊C4×C61C2×D412yes—
143C4⋊C4×C61C2×D414yes—
144C42×C61—28no—
145C42×C61—22no—
146C42×C61—22no—
147C42×C61—22no—
148C42×C61—22no—
149C42×C61C2×D411yescharacteristic
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61C2411yescharacteristic
153C22⋊Dic61—41no—
154#11758202—44no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159#11758196—41no—
160#11758196—41no—
161C4×Dic61—41no—
162#11758194—42no—
163C2×#92418—22no—
164C2×#92418—22no—
165C2×#92418—22no—
166C23×C61#12011yescharacteristic
167Q8×C61—24no—
168Q8×C61—22no—
169Q8×C61—22no—
170Q8×C61—22no—
171Q8×C61—42no—
172Q8×C61—42no—
173Q8×C61—42no—
174Q8×C61—44no—
175Q8×C61#13714yes—
176C2×C4×C61—24no—
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—41no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61—44no—
188C2×C4×C61#12011yescharacteristic
189C2×C4×C61#12011yescharacteristic
190C2×C4×C61#12011yescharacteristic
191C2×C4×C61#13512yes—
192C2×C4×C61#13712yes—
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C22×D411yescharacteristic
196C2×Dic61—41no—
197C2×Dic61—41no—
198Dic122—42no—
199Dic122—42no—
200#92418—42no—
201#92418—42no—
202#92418—42no—
203#18761—611nomaximal, Sylow
204C22×C61—21no—
205C22×C61—21no—
206C22×C61#45212yes—
207C22×C61C2×#12011yescharacteristic
208C4×C61—24no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—42no—
213C4×C61—44no—
214C4×C61—42no—
215C4×C61—42no—
216C4×C61#45212yes—
217C4×C61C2×#13511yescharacteristic
218C4×C61C2×#13711yescharacteristic
219C4×C61C2×#13712yes—
220Dic61—81no—
221Dic61—81no—
222C2×#557D6111yescharacteristic
223#3229—611no—
224#3104—611no—
225#3078—611no—
226#1653—614no—
227#1636—614no—
228#1546—611no—
229#1487—611no—
230#1479—611no—
231C2×C61—22no—
232C2×C61C2×#45211yescharacteristic
233C2×C61#300411yescharacteristic
234C2×C61#302011yescharacteristic
235#557—22no—
236#557—22no—
237#557—22no—
238#557C2×D6114yes—
239#533—611no—
240C2×C4⋊Q8—21no—
241C2×C4⋊Q8C2×D6111yescharacteristic
242C2×#125—22no—
243C2×#125—21no—
244C2×C4×Q8—21no—
245C2×C4×Q8—21no—
246C2×C4×Q8C2×D6111yescharacteristic
247C2×C4×Q8C2×D6111yescharacteristic
248#498—612no—
249#487—612no—
250#484—1221no—
251#483—1222no—
252#479—1222no—
253#477—612no—
254#477—612no—
255#475—612no—
256#475—1222no—
257#462—612no—
258#451—1221no—
259C4×Q16—1222no—
260#431—611no—
261#431—611no—
262#428—1221no—
263#423—611no—
264#419—1221no—
265#332—614no—
266Q8⋊C8—614no—
267Q8⋊C8—614no—
268C61#1876111yescharacteristic, complemented, Sylow
269C22×Q8—21no—
270C22×Q8C22⋊D6111yescharacteristic
271C22×Q8C22⋊D6111yescharacteristic
272C2×Q16—1222no—
273C2×M16—1221no—
274C4⋊Q8—24no—
275C4⋊Q8—24no—
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8—611no—
279C4⋊Q8—611no—
280C4⋊Q8C22⋊D6112yes—
281C4⋊Q8C22×D6112yes—
282#125—28no—
283#125—24no—
284#125—28no—
285#125—24no—
286#125—1221no—
287C22⋊Q8—1221no—
288C22⋊Q8—1221no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×Q8—21no—
296C4×Q8—21no—
297C4×Q8—1221no—
298C4×Q8—1221no—
299C4×Q8C22⋊D6112yes—
300C4×Q8C22⋊D6112yes—
301C4×Q8C22×D6112yes—
302C4×Q8C22×D6112yes—
303#117—1221no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4C22⋊D6111yescharacteristic
311C2×C4⋊C4C22⋊D6111yescharacteristic
312C2×C4⋊C4C22⋊D6112yes—
313C2×C42—22no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42C22×D6111yescharacteristic
317#107—1222no—
318#107—1222no—
319#106—1222no—
320#106—1224no—
321C4⋊C8—612no—
322C4⋊C8—612no—
323C4⋊C8—1221no—
324C4⋊C8—1221no—
325#103—1222no—
326#103—1222no—
327#103—1222no—
328#103—1222no—
329#103—1224no—
330C22⋊C8—1221no—
331C22⋊C8—1221no—
332C4×C8—612no—
333C4×C8—1221no—
334C2×Q8—24no—
335C2×Q8—22no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—1221no—
344C2×Q8—1221no—
345C2×Q8C2×C22⋊D6112yes—
346C2×Q8C2×C22⋊D6112yes—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4C2×C22⋊D6111yescharacteristic
353C22×C4C2×C22⋊D6111yescharacteristic
354C22×C4C2×C22⋊D6111yescharacteristic
355Q16—2444no—
356M16—2442no—
357C2×C8—1222no—
358C2×C8—1222no—
359C2×C8—1222no—
360C4⋊C4—28no—
361C4⋊C4—28no—
362C4⋊C4—28no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—22no—
372C4⋊C4—1221no—
373C4⋊C4—1221no—
374C4⋊C4—1221no—
375C4⋊C4—1221no—
376C4⋊C4—2441no—
377C4⋊C4—2441no—
378C4⋊C4C2×C22⋊D6114yes—
379C4⋊C4C2×C22⋊D6112yes—
380C4⋊C4C2×C22⋊D6112yes—
381C22⋊C4—2441no—
382C42—28no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—2441no—
388C42C2×C22⋊D6111yescharacteristic
389C42C2×C22⋊D6111yescharacteristic
390C42C2×C22⋊D6111yescharacteristic
391C42C23×D6111yescharacteristic
392C23#1451974611yescharacteristic
393Q8—24no—
394Q8—22no—
395Q8—22no—
396Q8—22no—
397Q8—42no—
398Q8—42no—
399Q8—42no—
400Q8—44no—
401Q8—2442no—
402Q8—2442no—
403Q8#1451980814yes—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—2441no—
417C2×C4—2441no—
418C2×C4#1451974611yescharacteristic
419C2×C4#1451974611yescharacteristic
420C2×C4#1451974611yescharacteristic
421C2×C4#1451979712yes—
422C2×C4#1451980712yes—
423C2×C4C22×C22⋊D6111yescharacteristic
424C2×C4C22×C22⋊D6111yescharacteristic
425C2×C4C22×C22⋊D6111yescharacteristic
426C8—2442no—
427C8—2442no—
428C8—2442no—
429C22—21no—
430C22—21no—
431C22#1683992412yes—
432C22C2×#1451974611yescharacteristic
433C4—21no—
434C4—21no—
435C4—24no—
436C4—21no—
437C4—42no—
438C4—42no—
439C4—42no—
440C4—44no—
441C4—4881no—
442C4—4881no—
443C4#1683992412yes—
444C4C2×#1451979711yescharacteristic
445C4C2×#1451980711yescharacteristic
446C4C2×#1451980812yes—
447C2—22no—
448C2C2×#1683992411yescharacteristic
449C2#2145505311yescharacteristic
450C2#2145518311yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#557C2
socleC22×C61C2×#120
prime radicalC23×C61#120
hypercenterC2×#557D61
hypocenterC61#18761
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. #21444977C2semidirect
268. C61#18761semidirect

Maximal subgroups

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

subgroupindex
#214569632
#214558582
#214556242
#214450962
#214449772
#214444912
#214442482
#214442142
C2×C61×#5572
#1876161