Finite Group Databasebeta

← #29663872

Subgroups — #29663872

Order 15616.

628 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#21456378C211yesmaximal, characteristic
2#21455949C211yesmaximal, characteristic, complemented
3#21455941C211yesmaximal, characteristic
4#21445277C212yesmaximal, complemented
5#21445263C212yesmaximal, complemented
6#21445201C212yesmaximal, complemented
7#21445167C212yesmaximal, complemented
8#21444582C211yesmaximal, characteristic, complemented
9#21444517C211yesmaximal, characteristic, complemented
10#21444512C211yesmaximal, characteristic
11C2×C8×Q8×C61C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13#16840312C2211yescharacteristic
14#16840160C2211yescharacteristic
15#16840135C2211yescharacteristic
16#16840113C2211yescharacteristic
17#16840109C2211yescharacteristic
18#16840092—21no—
19#16840075—21no—
20#16840057C2212yes—
21#16840047C2212yes—
22#16839991—21no—
23#16839991—21no—
24#16839972C2212yes—
25#16839962C2212yes—
26#16839918—21no—
27#16839901—21no—
28#16839867—22no—
29#16839860—22no—
30C4×#11758190—21no—
31C4×#11758190—21no—
32#16839819C2211yescharacteristic
33#16839819C2211yescharacteristic
34#16839515C2212yes—
35#16839513C2212yes—
36#16839510C2212yes—
37#16839509C2212yes—
38#16839507C2212yes—
39#16839505C2212yes—
40C2×C4×Q8×C61C2211yescharacteristic
41C8×Q8×C61—21no—
42C8×Q8×C61—21no—
43C8×Q8×C61C2212yes—
44C8×Q8×C61C2212yes—
45C2×C4⋊C8×C61—21no—
46C2×C4⋊C8×C61C2211yescharacteristic
47C2×C4×C8×C61—21no—
48C2×C4×C8×C61C2211yescharacteristic
49C2×#11758222—21no—
50C2×#11758202—22no—
51C2×#11758190—21no—
52C2×#11758190—21no—
53C2×#11758201—22no—
54#14519777C2311yescharacteristic
55#14519777C2311yescharacteristic
56#14519777C2311yescharacteristic
57#14519774C2311yescharacteristic
58#14519769—21no—
59#14519765—21no—
60#14519756—21no—
61#14519756—21no—
62#14519751—21no—
63#14519751—21no—
64C4×Dic122—21no—
65C4×Dic122—21no—
66C4×D244—21no—
67C4×D244—21no—
68#14519736—21no—
69#14519710—22no—
70#14519709—21no—
71#14519709—21no—
72#14519708—22no—
73C4⋊#92418C2312yes—
74C4⋊#92418C2312yes—
75#14519702—21no—
76#14519702—21no—
77#14519701—22no—
78#14519701—22no—
79#14519700—22no—
80#14519699—21no—
81#14519699—21no—
82#14519697—22no—
83C22⋊#92418—21no—
84C22⋊#92418—21no—
85#14519688—21no—
86#14519688—21no—
87C22×Q8×C61D411yescharacteristic
88C22×C8×C61—21no—
89C22×C8×C61D411yescharacteristic
90C22×C8×C61D411yescharacteristic
91C4×Q8×C61—21no—
92C4×Q8×C61—21no—
93C4×Q8×C61D412yes—
94C4×Q8×C61C2312yes—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61D411yescharacteristic
97C2×C42×C61—21no—
98C2×C42×C61C2311yescharacteristic
99C4⋊C8×C61—22no—
100C4⋊C8×C61—22no—
101C4⋊C8×C61—21no—
102C4⋊C8×C61C2312yes—
103C4×C8×C61—22no—
104C4×C8×C61—22no—
105C4×C8×C61D411yescharacteristic
106C4×C8×C61D411yescharacteristic
107C4×C8×C61C2311yescharacteristic
108C4×C8×C61C2311yescharacteristic
109C2×Q8×C61—24no—
110C2×Q8×C61—21no—
111C2×Q8×C61C2×D412yes—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61#4111yescharacteristic
117C2×C22⋊D61—41no—
118#11758222—82no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244—21no—
122C2×C4×D61—41no—
123C2×Dic122—21no—
124C2×Dic122—21no—
125C2×Dic122—21no—
126C2×C8×C61—22no—
127C2×C8×C61—22no—
128C2×C8×C61—41no—
129C2×C8×C61—42no—
130C2×C8×C61C2×D411yescharacteristic
131C2×C8×C61C2×D411yescharacteristic
132C2×C8×C61C2×D411yescharacteristic
133C2×C8×C61C2×D411yescharacteristic
134C4⋊C4×C61—22no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61C2×D412yes—
138C42×C61—22no—
139C42×C61—22no—
140C42×C61C2×D411yescharacteristic
141C42×C61C2×D411yescharacteristic
142C42×C61#4111yescharacteristic
143C42×C61C2411yescharacteristic
144C22⋊Dic61—41no—
145#11758202—44no—
146#11758201—44no—
147#11758198—41no—
148#11758198—41no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155#11758196—41no—
156#11758196—41no—
157C4×Dic61—41no—
158C2×#92418—22no—
159C2×#92418—22no—
160#11758190—42no—
161#11758190—42no—
162C23×C61#12111yescharacteristic
163C22×D61—41no—
164Q8×C61—24no—
165Q8×C61—22no—
166Q8×C61#13514yes—
167C2×C4×C61—24no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—22no—
174C2×C4×C61—42no—
175C2×C4×C61#12111yescharacteristic
176C2×C4×C61#12111yescharacteristic
177C2×C4×C61#12111yescharacteristic
178C2×C4×C61C22×D411yescharacteristic
179C2×C4×C61C22×D411yescharacteristic
180C2×C4×C61C2×#4111yescharacteristic
181C22⋊D61—82no—
182C2×Dic61—41no—
183C2×Dic61—41no—
184C2×Dic61—41no—
185D244—42no—
186D244—41no—
187D244—41no—
188D244—81no—
189C4×D61—82no—
190Dic122—42no—
191Dic122—41no—
192Dic122—41no—
193Dic122—81no—
194C8×C61—21no—
195C8×C61—21no—
196C8×C61—21no—
197C8×C61—42no—
198C8×C61#13712yes—
199#92418—42no—
200#92418—42no—
201#19263—611nomaximal, Sylow
202C22×C61—21no—
203C22×C61—21no—
204C22×C61#47112yes—
205C22×C61C2×#12111yescharacteristic
206C2×D61—81no—
207C2×D61—81no—
208C2×D61—81no—
209C4×C61—21no—
210C4×C61—24no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—42no—
215C4×C61C2×#13512yes—
216C4×C61C2×#13711yescharacteristic
217C4×C61#57511yescharacteristic
218Dic61—81no—
219Dic61—81no—
220Dic61—81no—
221#3167—611no—
222#3113—611no—
223#3112—611no—
224C2×C8×Q8D6111yescharacteristic, complemented
225#1684—612no—
226#1682—612no—
227#1672—612no—
228#1666—612no—
229#1568—611no—
230#1552—611no—
231#1551—611no—
232C2×C61—22no—
233C2×C61#303711yescharacteristic
234C2×C61#305011yescharacteristic
235C2×C61C2×#47111yescharacteristic
236D61—161no—
237#533—611no—
238#532—611no—
239C2×C4×Q8C2×D6111yescharacteristic
240#498—611no—
241#492—611no—
242#487—611no—
243#486—611no—
244#483—1221no—
245#480—1221no—
246#477—612no—
247#475—612no—
248#465—1221no—
249#465—1221no—
250#462—612no—
251#460—612no—
252#451—1221no—
253#448—1221no—
254C8×Q8—21no—
255C8×Q8—21no—
256C8×Q8C2×D6112yes—
257C8×Q8C2×D6112yes—
258#441—1222no—
259#440—1222no—
260C4×SD16—1221no—
261C4×SD16—1221no—
262#432—611no—
263#432—611no—
264C2×C4⋊C8—21no—
265C2×C4⋊C8C2×D6111yescharacteristic
266C2×C4×C8—21no—
267C2×C4×C8C2×D6111yescharacteristic
268#335—612no—
269#334—612no—
270#333—612no—
271#332—612no—
272#332—612no—
273#331—612no—
274C61#1926311yescharacteristic, complemented, Sylow
275C2×#41—1221no—
276C22×Q8D24411yescharacteristic
277C2×Q16—1222no—
278C2×SD16—1222no—
279C2×SD16—1221no—
280C2×SD16—1221no—
281C22×C8—21no—
282C22×C8C22⋊D6111yescharacteristic
283C22×C8C22⋊D6111yescharacteristic
284C4⋊Q8—611no—
285C4⋊Q8—611no—
286C4⋊Q8—611no—
287#127—611no—
288#125—1221no—
289#124—1221no—
290C22⋊Q8—1221no—
291C22⋊Q8—1221no—
292#121—1221no—
293#121—1221no—
294C4×Q8—21no—
295C4×Q8—21no—
296C4×Q8—1221no—
297C4×Q8—1221no—
298C4×Q8D24412yes—
299C4×Q8C22×D6112yes—
300C4×D4—1221no—
301C4×D4—1221no—
302#117—1221no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4D24411yescharacteristic
305C2×C42—21no—
306C2×C42C22×D6111yescharacteristic
307#107—1222no—
308#106—1221no—
309#106—1221no—
310#106—1222no—
311C4⋊C8—22no—
312C4⋊C8—22no—
313C4⋊C8—21no—
314C4⋊C8—612no—
315C4⋊C8—612no—
316C4⋊C8C22×D6112yes—
317#103—1222no—
318#103—1222no—
319#103—1221no—
320#103—1221no—
321#103—1222no—
322#102—1221no—
323#102—1221no—
324#102—1222no—
325C22⋊C8—1221no—
326C22⋊C8—1221no—
327#97—1221no—
328#97—1221no—
329C4×C8—22no—
330C4×C8—22no—
331C4×C8C22⋊D6111yescharacteristic
332C4×C8C22⋊D6111yescharacteristic
333C4×C8C22×D6111yescharacteristic
334C4×C8C22×D6111yescharacteristic
335#41—4882no—
336C2×Q8—24no—
337C2×Q8—21no—
338C2×Q8—1221no—
339C2×Q8—1221no—
340C2×Q8—1221no—
341C2×Q8C2×D24412yes—
342C2×D4—1221no—
343C2×D4—1221no—
344C2×D4—1221no—
345C2×D4—2441no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—2441no—
349C22×C4C2×D24411yescharacteristic
350C22×C4#1175822211yescharacteristic
351C22×C4C2×C22⋊D6111yescharacteristic
352Q16—2444no—
353SD16—2442no—
354SD16—2444no—
355SD16—2442no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—41no—
359C2×C8—42no—
360C2×C8—1222no—
361C2×C8—1222no—
362C2×C8C2×C22⋊D6111yescharacteristic
363C2×C8C2×C22⋊D6111yescharacteristic
364C2×C8C2×C22⋊D6111yescharacteristic
365C2×C8C2×C22⋊D6111yescharacteristic
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—21no—
369C4⋊C4—1221no—
370C4⋊C4—1221no—
371C4⋊C4—1221no—
372C4⋊C4—1221no—
373C4⋊C4—1221no—
374C4⋊C4—1221no—
375C4⋊C4—2441no—
376C4⋊C4—2441no—
377C4⋊C4C2×D24412yes—
378C22⋊C4—2441no—
379C22⋊C4—2441no—
380C22⋊C4—2441no—
381C42—22no—
382C42—22no—
383C42—2441no—
384C42C2×D24411yescharacteristic
385C42#1175822211yescharacteristic
386C42C2×C22⋊D6111yescharacteristic
387C42C23×D6111yescharacteristic
388C23—2441no—
389C23#1451975111yescharacteristic
390Q8—24no—
391Q8—22no—
392Q8—4881no—
393Q8—2442no—
394Q8—2441no—
395Q8—2441no—
396Q8#1451979914yes—
397D4—4881no—
398D4—4882no—
399D4—2442no—
400D4—2441no—
401D4—2441no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—42no—
410C2×C4—4882no—
411C2×C4—2441no—
412C2×C4—2441no—
413C2×C4—2441no—
414C2×C4#1451975111yescharacteristic
415C2×C4#1451975111yescharacteristic
416C2×C4#1451975111yescharacteristic
417C2×C4C22×C22⋊D6111yescharacteristic
418C2×C4C22×D24411yescharacteristic
419C2×C4C2×#1175822211yescharacteristic
420C8—21no—
421C8—21no—
422C8—21no—
423C8—42no—
424C8—2442no—
425C8—2442no—
426C8#1451980812yes—
427C22—21no—
428C22—21no—
429C22—4881no—
430C22—4881no—
431C22—4881no—
432C22#1684002512yes—
433C22C2×#1451975111yescharacteristic
434C4—24no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—42no—
440C4—4881no—
441C4—4881no—
442C4—4881no—
443C4C2×#1451979912yes—
444C4C2×#1451980811yescharacteristic
445C4#1684056011yescharacteristic
446C2—22no—
447C2—9761no—
448C2#2145529111yescharacteristic
449C2#2145540011yescharacteristic
450C2C2×#1684002511yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C8×Q8×C61C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenterC2×C8×Q8D61
hypocenterC61#19263
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
2. #21455949C2semidirect
4. #21445277C2semidirect
5. #21445263C2semidirect
6. #21445201C2semidirect
7. #21445167C2semidirect
8. #21444582C2semidirect
9. #21444517C2semidirect
11. C2×C8×Q8×C61C2semidirect
224. C2×C8×Q8D61semidirect
274. C61#19263semidirect

Maximal subgroups

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

subgroupindex
#214563782
#214559492
#214559412
#214452772
#214452632
#214452012
#214451672
#214445822
#214445172
#214445122
C2×C8×Q8×C612
#1926361