Finite Group Databasebeta

← #29656169

Subgroups — #29656169

Order 15616.

908 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456215C211yesmaximal, characteristic
2#21455941C211yesmaximal, characteristic
3#21455941C211yesmaximal, characteristic
4#21444977C214yesmaximal, complemented
5#21444915C214yesmaximal, complemented
6#21444536C211yesmaximal, characteristic, complemented
7#21444266C211yesmaximal, characteristic, complemented
8#21444266C211yesmaximal, characteristic, complemented
9C2×C61×#557C211yesmaximal, characteristic, complemented
10#16840312C2211yescharacteristic
11#16840124—22no—
12#16840114C2212yes—
13#16840110C2212yes—
14#16840075—21no—
15#16840075—21no—
16#16840057C2212yes—
17#16840057C2212yes—
18#16839962C2212yes—
19#16839962C2212yes—
20#16839901—21no—
21#16839901—21no—
22#16839860—22no—
23#16839860—22no—
24#16839819C2211yescharacteristic
25#16839819C2211yescharacteristic
26#16839761—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×#11758222—21no—
44C2×#11758201—22no—
45C2×#11758201—22no—
46C2×#11758194—21no—
47#14519777C2311yescharacteristic
48#14519774C2311yescharacteristic
49#14519765—21no—
50#14519751—21no—
51#14519751—21no—
52C4×D244—21no—
53C4×D244—21no—
54#14519710—22no—
55#14519710—22no—
56C4⋊#92418C2312yes—
57C4⋊#92418C2312yes—
58#14519701—22no—
59#14519701—22no—
60#14519700—24no—
61#14519697—22no—
62#14519697—22no—
63#14519697—24no—
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×C22⋊D61—41no—
126#11758222—44no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244—21no—
130C2×C4×D61—41no—
131C2×Dic122—21no—
132C4⋊C4×C61—28no—
133C4⋊C4×C61—28no—
134C4⋊C4×C61—28no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61C2×D412yes—
145C4⋊C4×C61C2×D412yes—
146C4⋊C4×C61C2×D414yes—
147C42×C61—28no—
148C42×C61—22no—
149C42×C61—22no—
150C42×C61—22no—
151C42×C61—22no—
152C42×C61C2×D411yescharacteristic
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C42×C61C2411yescharacteristic
156#11758201—44no—
157#11758201—44no—
158#11758198—41no—
159#11758198—41no—
160C4⋊Dic61—21no—
161C4⋊Dic61—21no—
162#11758194—42no—
163C2×#92418—22no—
164C2×#92418—22no—
165C2×#92418—22no—
166C23×C61#12011yescharacteristic
167C22×D61—41no—
168Q8×C61—24no—
169Q8×C61—22no—
170Q8×C61—22no—
171Q8×C61—22no—
172Q8×C61—42no—
173Q8×C61—44no—
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—22no—
183C2×C4×C61—22no—
184C2×C4×C61—24no—
185C2×C4×C61—42no—
186C2×C4×C61—44no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61#12011yescharacteristic
190C2×C4×C61#12011yescharacteristic
191C2×C4×C61#12011yescharacteristic
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C22⋊D61—82no—
196C2×Dic61—41no—
197D244—42no—
198D244—42no—
199D244—42no—
200C4×D61—82no—
201Dic122—42no—
202#92418—42no—
203#92418—42no—
204#92418—42no—
205#18535—611nomaximal, Sylow
206C22×C61—21no—
207C22×C61—21no—
208C22×C61#45412yes—
209C22×C61C2×#12011yescharacteristic
210C2×D61—81no—
211C2×D61—81no—
212C2×D61—81no—
213C4×C61—24no—
214C4×C61—21no—
215C4×C61—21no—
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—42no—
219C4×C61—44no—
220C4×C61—42no—
221C4×C61—42no—
222C4×C61C2×#13712yes—
223C4×C61#57512yes—
224Dic61—81no—
225C2×#557D6111yescharacteristic, complemented
226#3141—611no—
227#3112—611no—
228#3112—611no—
229#1636—614no—
230#1626—614no—
231#1556—611no—
232#1491—611no—
233#1491—611no—
234C2×C61—22no—
235C2×C61#300411yescharacteristic
236C2×C61#300511yescharacteristic
237C2×C61C2×#45411yescharacteristic
238D61—161no—
239#557—22no—
240#557—22no—
241#557—22no—
242#557C2×D6114yes—
243#532—611no—
244C2×C4⋊Q8—21no—
245C2×C4⋊Q8C2×D6111yescharacteristic
246C2×#125—22no—
247C2×#125—21no—
248C2×C4×Q8—21no—
249C2×C4×Q8—21no—
250C2×C4×Q8C2×D6111yescharacteristic
251C2×C4×Q8C2×D6111yescharacteristic
252#489—1222no—
253#487—612no—
254#486—612no—
255#480—1221no—
256#480—1221no—
257#477—612no—
258#477—612no—
259#460—612no—
260#460—612no—
261#448—1221no—
262#448—1221no—
263#440—1222no—
264#440—1222no—
265#432—611no—
266#432—611no—
267#417—1222no—
268C4×M16—611no—
269#332—614no—
270Q8⋊C8—614no—
271Q8⋊C8—614no—
272C61#1853511yescharacteristic, complemented, Sylow
273C2×#41—1221no—
274C22×Q8—21no—
275C22×Q8C22⋊D6111yescharacteristic
276C22×Q8C22⋊D6111yescharacteristic
277C2×SD16—1222no—
278C2×SD16—1222no—
279C2×M16—1221no—
280C4⋊Q8—24no—
281C4⋊Q8—24no—
282C4⋊Q8—21no—
283C4⋊Q8—21no—
284C4⋊Q8—611no—
285C4⋊Q8C22⋊D6112yes—
286C4⋊Q8C22×D6112yes—
287#127—611no—
288#125—28no—
289#125—24no—
290#125—28no—
291#125—24no—
292#124—1221no—
293#121—1221no—
294#121—1221no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×Q8—21no—
302C4×Q8—21no—
303C4×Q8C22⋊D6112yes—
304C4×Q8C22⋊D6112yes—
305C4×Q8C22×D6112yes—
306C4×Q8C22×D6112yes—
307C4×D4—1221no—
308C4×D4—1221no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4C22⋊D6111yescharacteristic
316C2×C4⋊C4C22⋊D6111yescharacteristic
317C2×C4⋊C4C22⋊D6112yes—
318C2×C42—22no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42C22×D6111yescharacteristic
322#107—1222no—
323#107—1222no—
324C4⋊C8—612no—
325C4⋊C8—612no—
326#103—1222no—
327#103—1222no—
328#103—1224no—
329#102—1224no—
330#102—1222no—
331#102—1222no—
332C22⋊C8—1221no—
333C22⋊C8—1221no—
334#97—1221no—
335#97—1221no—
336#97—1221no—
337C4×C8—612no—
338#41—2444no—
339C2×Q8—24no—
340C2×Q8—22no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—22no—
344C2×Q8—42no—
345C2×Q8—44no—
346C2×Q8—1221no—
347C2×Q8C2×C22⋊D6112yes—
348C2×Q8C2×C22⋊D6112yes—
349C2×D4—1221no—
350C2×D4—1221no—
351C2×D4—1221no—
352C2×D4—2441no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—2441no—
359C22×C4C2×C22⋊D6111yescharacteristic
360C22×C4C2×C22⋊D6111yescharacteristic
361C22×C4C2×C22⋊D6111yescharacteristic
362SD16—2444no—
363SD16—2444no—
364M16—2442no—
365C2×C8—1222no—
366C2×C8—1222no—
367C2×C8—1222no—
368C4⋊C4—28no—
369C4⋊C4—28no—
370C4⋊C4—28no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—22no—
380C4⋊C4—1221no—
381C4⋊C4—1221no—
382C4⋊C4C2×C22⋊D6114yes—
383C4⋊C4C2×C22⋊D6112yes—
384C4⋊C4C2×C22⋊D6112yes—
385C22⋊C4—2441no—
386C22⋊C4—2441no—
387C42—28no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42C2×C22⋊D6111yescharacteristic
393C42C2×C22⋊D6111yescharacteristic
394C42C2×C22⋊D6111yescharacteristic
395C42C23×D6111yescharacteristic
396C23—2441no—
397C23#1451974611yescharacteristic
398Q8—24no—
399Q8—22no—
400Q8—22no—
401Q8—22no—
402Q8—42no—
403Q8—44no—
404Q8—44no—
405Q8—2442no—
406Q8#1451980814yes—
407D4—4882no—
408D4—2442no—
409D4—2442no—
410D4—2442no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—4882no—
425C2×C4—2441no—
426C2×C4#1451974611yescharacteristic
427C2×C4#1451974611yescharacteristic
428C2×C4#1451974611yescharacteristic
429C2×C4C22×C22⋊D6111yescharacteristic
430C2×C4C22×C22⋊D6111yescharacteristic
431C2×C4C22×C22⋊D6111yescharacteristic
432C8—2442no—
433C8—2442no—
434C8—2442no—
435C22—21no—
436C22—21no—
437C22—4881no—
438C22—4881no—
439C22—4881no—
440C22#1683993512yes—
441C22C2×#1451974611yescharacteristic
442C4—21no—
443C4—21no—
444C4—24no—
445C4—21no—
446C4—21no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—44no—
451C4—4881no—
452C4C2×#1451980812yes—
453C4#1684056012yes—
454C2—22no—
455C2—9761no—
456C2#2145505311yescharacteristic
457C2#2145506111yescharacteristic
458C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#557C2
socleC22×C61C2×#120
prime radical#16840312C22
hypercenterC2×#557D61
hypocenterC61#18535
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. #21444977C2semidirect
5. #21444915C2semidirect
6. #21444536C2semidirect
7. #21444266C2semidirect
8. #21444266C2semidirect
9. C2×C61×#557C2semidirect
225. C2×#557D61semidirect
272. C61#18535semidirect

Maximal subgroups

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

subgroupindex
#214562152
#214559412
#214449772
#214449152
#214445362
#214442662
C2×C61×#5572
#1853561