Finite Group Databasebeta

← #29667914

Subgroups — #29667914

Order 15616.

720 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456942C211yesmaximal, characteristic
2#21456379C211yesmaximal, characteristic
3C2×#16840158C211yesmaximal, characteristic, complemented
4#21456215C211yesmaximal, characteristic
5#21456201C211yesmaximal, characteristic, complemented
6#21445482C212yesmaximal, complemented
7#21445468C212yesmaximal, complemented
8#21445422C212yesmaximal, complemented
9#21445413C212yesmaximal, complemented
10#21444100C211yesmaximal, characteristic, complemented
11C61×#3209C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13C2×#14519777C2211yescharacteristic
14#16840167C2212yes—
15#16840158—22no—
16#16840158C2211yescharacteristic
17#16840158C2212yes—
18#16840158C2212yes—
19#16840136C2211yescharacteristic
20#16840132—22no—
21#16840124—22no—
22#16840114C2212yes—
23#16840114C2212yes—
24#16840114C2212yes—
25#16840112C2211yescharacteristic
26#16840110C2212yes—
27#16840110C2211yescharacteristic
28#16840109C2212yes—
29#16840076—22no—
30#16840043—22no—
31#16839993—22no—
32C4×#11758201—22no—
33C2×#14519708—22no—
34#16839770—22no—
35#16839761—22no—
36C4×#11758194C2211yescharacteristic
37C2×C4×#92418C2211yescharacteristic
38#16839484C2212yes—
39C8×#92418C2212yes—
40C61×#533C2211yescharacteristic
41C2×C4⋊Q8×C61C2211yescharacteristic
42C61×#494C2212yes—
43C61×#487C2212yes—
44C61×#462—22no—
45C61×#451—22no—
46C61×#431C2211yescharacteristic
47C2×#11758222—21no—
48C22×Dic122—21no—
49C2×#11758201—22no—
50C2×#11758194D412yes—
51C22×#92418D412yes—
52#14519777—21no—
53#14519777—21no—
54#14519777—21no—
55#14519777C2312yes—
56#14519777C2311yescharacteristic
57#14519774C2311yescharacteristic
58#14519765—21no—
59#14519751—21no—
60#14519751—21no—
61C4×D244—21no—
62C4×D244—21no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—21no—
65#14519711—24no—
66#14519708—22no—
67#14519708—24no—
68#14519708—24no—
69#14519701—22no—
70#14519700—24no—
71#14519700—24no—
72#14519700—24no—
73#14519699—24no—
74#14519697—22no—
75#14519697—24no—
76#14519688D412yes—
77C4×#92418D412yes—
78C4×#92418C2311yescharacteristic
79C4×#92418C2311yescharacteristic
80C4×#92418C2312yes—
81C22×Q8×C61—21no—
82C2×Q16×C61—24no—
83C4⋊Q8×C61—21no—
84C4⋊Q8×C61—21no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61C2312yes—
87C4⋊Q8×C61C2311yescharacteristic
88C4⋊Q8×C61C2311yescharacteristic
89#125×C61—21no—
90C22⋊Q8×C61—21no—
91C22⋊Q8×C61—21no—
92C4×Q8×C61—21no—
93C4×Q8×C61—21no—
94#117×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C42×C61C2311yescharacteristic
98C4⋊C8×C61D412yes—
99#103×C61—24no—
100C22⋊C8×C61D412yes—
101C4×C8×C61C2312yes—
102C2×Q8×C61—22no—
103C2×Q8×C61—21no—
104C2×Q8×C61—21no—
105C2×Q8×C61—41no—
106C2×Q8×C61—82no—
107C22×C4×C61—41no—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C2×C22⋊D61—41no—
112C22×Dic61—41no—
113#11758222—44no—
114C2×D244—21no—
115C2×D244—21no—
116C2×D244—21no—
117C2×C4×D61—41no—
118C2×Dic122—22no—
119C2×Dic122—21no—
120C2×Dic122—44no—
121C2×Dic122—41no—
122Q16×C61—84no—
123C2×C8×C61C2×D414yes—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—41no—
131C4⋊C4×C61—41no—
132C4⋊C4×C61—41no—
133C4⋊C4×C61—41no—
134C22⋊C4×C61—41no—
135C42×C61—41no—
136C42×C61C2×D411yescharacteristic
137C42×C61C2×D411yescharacteristic
138C42×C61C2×D411yescharacteristic
139C42×C61C2411yescharacteristic
140#11758201—44no—
141#11758198—41no—
142#11758198—41no—
143C4⋊Dic61—22no—
144C4⋊Dic61—21no—
145C4⋊Dic61—22no—
146C4⋊Dic61—21no—
147C4⋊Dic61—41no—
148C4⋊Dic61—41no—
149#11758194—24no—
150C2×#92418—24no—
151C2×#92418C2×D412yes—
152C2×#92418C2×D414yes—
153C2×#92418C2×D412yes—
154C23×C61#12711yescharacteristic
155C22×D61—41no—
156Q8×C61—42no—
157Q8×C61—42no—
158Q8×C61—44no—
159Q8×C61—82no—
160C2×C4×C61—22no—
161C2×C4×C61—22no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—42no—
165C2×C4×C61—82no—
166C2×C4×C61#12711yescharacteristic
167C2×C4×C61#12711yescharacteristic
168C2×C4×C61#12711yescharacteristic
169C2×C4×C61#13514yes—
170C2×C4×C61C22×D411yescharacteristic
171C2×C4×C61C22×D411yescharacteristic
172C2×C4×C61C22×D411yescharacteristic
173C22⋊D61—82no—
174C2×Dic61—42no—
175C2×Dic61—41no—
176C2×Dic61—82no—
177D244—42no—
178D244—42no—
179D244—42no—
180C4×D61—82no—
181Dic122—44no—
182Dic122—42no—
183Dic122—82no—
184C8×C61—24no—
185#92418—24no—
186#92418—22no—
187#92418#13714yes—
188#19626—611nomaximal, Sylow
189C22×C61—21no—
190C22×C61—21no—
191C22×C61#49512yes—
192C22×C61C2×#12711yescharacteristic
193C2×D61—81no—
194C2×D61—81no—
195C2×D61—81no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—81no—
199C4×C61—82no—
200C4×C61—81no—
201C4×C61#49512yes—
202C4×C61C2×#13512yes—
203C4×C61C2×#13712yes—
204C4×C61#57712yes—
205Dic61—82no—
206Dic61—81no—
207#3226—611no—
208#3209D6111yescharacteristic, complemented
209#3167—611no—
210C2×#498—611no—
211#3141—611no—
212#3139—611no—
213#1720—612no—
214#1718—612no—
215#1708—612no—
216#1707—612no—
217C8×M16—611no—
218C2×C61—22no—
219C2×C61C2×#49511yescharacteristic
220C2×C61#315211yescharacteristic
221C2×C61#315411yescharacteristic
222D61—161no—
223#533C2×D6111yescharacteristic
224#532—611no—
225C2×C4⋊Q8—611no—
226C2×C4⋊Q8C2×D6111yescharacteristic
227#500—612no—
228#498—612no—
229#498—612no—
230#498—611no—
231#498—1222no—
232#494C2×D6112yes—
233#492—611no—
234#491—1222no—
235#489—1222no—
236#487—612no—
237#487—612no—
238#487—612no—
239#487—611no—
240#487C2×D6112yes—
241#486—612no—
242#486—611no—
243#486—612no—
244#480—1222no—
245#474—1222no—
246#465—1222no—
247#462—22no—
248#451—22no—
249C4×SD16—1222no—
250#431C2×D6111yescharacteristic
251C2×#106—1222no—
252#419—1222no—
253#417—1222no—
254C4×M16—611no—
255C2×C4×C8—611no—
256#322—612no—
257C82—612no—
258C61#1962611yescharacteristic, complemented, Sylow
259C2×#41—1221no—
260C22×Q8—21no—
261C22×Q8—1221no—
262C2×Q16—24no—
263C2×SD16—1222no—
264C2×M16—612no—
265C22×C8—612no—
266C4⋊Q8—21no—
267C4⋊Q8—21no—
268C4⋊Q8—21no—
269C4⋊Q8—612no—
270C4⋊Q8—611no—
271C4⋊Q8—1221no—
272C4⋊Q8—1221no—
273C4⋊Q8—1221no—
274C4⋊Q8C22×D6112yes—
275C4⋊Q8C22×D6111yescharacteristic
276C4⋊Q8C22×D6111yescharacteristic
277#127—611no—
278#125—21no—
279#124—1221no—
280C22⋊Q8—21no—
281C22⋊Q8—21no—
282#121—1221no—
283#121—1221no—
284C4×Q8—21no—
285C4×Q8—21no—
286C4×D4—1221no—
287C4×D4—1221no—
288#117—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—1221no—
292C2×C4⋊C4—1221no—
293C2×C42C22×D6111yescharacteristic
294#107—1224no—
295#106—1224no—
296#106—1224no—
297#106—1222no—
298C4⋊C8C22⋊D6112yes—
299#103—24no—
300#103—1224no—
301#103—1224no—
302#103—1224no—
303#103—1222no—
304#102—1224no—
305#102—1222no—
306#102—1224no—
307C22⋊C8C22⋊D6112yes—
308#97—612no—
309C4×C8—612no—
310C4×C8—611no—
311C4×C8—612no—
312C4×C8—611no—
313C4×C8C22×D6112yes—
314#41—2444no—
315C2×Q8—21no—
316C2×Q8—22no—
317C2×Q8—21no—
318C2×Q8—41no—
319C2×Q8—82no—
320C2×Q8—1221no—
321C2×Q8—1222no—
322C2×Q8—2444no—
323C2×Q8—2441no—
324C2×D4—1221no—
325C2×D4—1221no—
326C2×D4—1221no—
327C2×D4—2441no—
328C22×C4—41no—
329C22×C4—2441no—
330C22×C4—2441no—
331C22×C4D4×D6111yescharacteristic
332C22×C4D4×D6111yescharacteristic
333C22×C4C2×C22⋊D6111yescharacteristic
334Q16—84no—
335SD16—2444no—
336M16—1224no—
337C2×C8—612no—
338C2×C8—612no—
339C2×C8—614no—
340C2×C8—1224no—
341C2×C8C2×C22⋊D6114yes—
342C4⋊C4—22no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—22no—
347C4⋊C4—21no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—1221no—
353C4⋊C4—1222no—
354C4⋊C4—1222no—
355C4⋊C4—1221no—
356C4⋊C4—2441no—
357C4⋊C4—2441no—
358C22⋊C4—41no—
359C22⋊C4—2441no—
360C22⋊C4—2441no—
361C42—41no—
362C42D4×D6111yescharacteristic
363C42D4×D6111yescharacteristic
364C42C2×C22⋊D6111yescharacteristic
365C42C23×D6111yescharacteristic
366C23—2441no—
367C23#1451977311yescharacteristic
368Q8—42no—
369Q8—42no—
370Q8—44no—
371Q8—82no—
372Q8—4882no—
373Q8—2444no—
374Q8—2442no—
375D4—4882no—
376D4—2442no—
377D4—2442no—
378D4—2442no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—82no—
385C2×C4—4882no—
386C2×C4—4882no—
387C2×C4—2441no—
388C2×C4—2442no—
389C2×C4#1451977311yescharacteristic
390C2×C4#1451977311yescharacteristic
391C2×C4#1451977311yescharacteristic
392C2×C4#1451980014yes—
393C2×C4C2×D4×D6111yescharacteristic
394C2×C4C2×D4×D6111yescharacteristic
395C2×C4C22×C22⋊D6111yescharacteristic
396C8—24no—
397C8—614no—
398C8—1224no—
399C8—1222no—
400C22—21no—
401C22—21no—
402C22—4881no—
403C22—4881no—
404C22—4881no—
405C22#1684014712yes—
406C22C2×#1451977311yescharacteristic
407C4—21no—
408C4—21no—
409C4—81no—
410C4—82no—
411C4—81no—
412C4—4881no—
413C4—4882no—
414C4#1684014712yes—
415C4C2×#1451980012yes—
416C4#137×D6112yes—
417C4#1684057312yes—
418C2—22no—
419C2—9761no—
420C2C2×#1684014711yescharacteristic
421C2#2145627411yescharacteristic
422C2#2145629111yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3209C2
socleC22×C61C2×#127
prime radical#16840312C22
hypercenter#3209D61
hypocenterC61#19626
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
3. C2×#16840158C2semidirect
5. #21456201C2semidirect
6. #21445482C2semidirect
7. #21445468C2semidirect
8. #21445422C2semidirect
9. #21445413C2semidirect
10. #21444100C2semidirect
11. C61×#3209C2semidirect
208. #3209D61semidirect
258. C61#19626semidirect

Maximal subgroups

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

subgroupindex
#214569422
#214563792
C2×#168401582
#214562152
#214562012
#214454822
#214454682
#214454222
#214454132
#214441002
C61×#32092
#1962661