Finite Group Databasebeta

← #29571788

Subgroups — #29571788

Order 15616.

782 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840473C211yesmaximal, characteristic
2#21445812C211yesmaximal, characteristic, complemented
3#21444504C211yesmaximal, characteristic, complemented
4#21444245C211yesmaximal, characteristic
5#21444243C211yesmaximal, characteristic
6#21444142C211yesmaximal, characteristic, complemented
7C61×#1549C211yesmaximal, characteristic, complemented
8#16840473—22no—
9#16840473—22no—
10#16840473—22no—
11#16840473—22no—
12C2×#14519768C412yes—
13C2×#14519757—21no—
14C2×#14519753—22no—
15C2×C4×Dic122C2211yescharacteristic
16C2×C4×D244C2211yescharacteristic
17C2×#14519739C412yes—
18C2×C4⋊C4×D61—21no—
19#16839815C2211yescharacteristic
20#16839786C2211yescharacteristic
21C2×#14519695—21no—
22#16839530—21no—
23#16839508—22no—
24#16839491—22no—
25#16839489—22no—
26#16839487—22no—
27C2×C4⋊Q8×C61C2211yescharacteristic
28C61×#431C2211yescharacteristic
29C61×#423C2211yescharacteristic
30C61×#332—22no—
31C22×Dic122D411yescharacteristic
32C22×D244C2×C411yescharacteristic
33C22×C4×D61—21no—
34C22×C4×D61—22no—
35C2×#11758194—21no—
36#14519768—24no—
37#14519768—24no—
38#14519757—24no—
39#14519757—24no—
40#14519753—28no—
41#14519753—28no—
42C4×Dic122—21no—
43C4×Dic122—21no—
44C4×Dic122—21no—
45C4×Dic122—21no—
46C4×D244—21no—
47C4×D244—21no—
48C4×D244D412yes—
49C4×D244D412yes—
50#14519739—24no—
51#14519739—24no—
52C2×C4⋊Dic61—22no—
53C2×C4⋊Dic61C2×C411yescharacteristic
54C2×C4⋊Dic61D411yescharacteristic
55C2×#11758196—22no—
56C2×#11758196D412yes—
57C4⋊C4×D61—24no—
58C4⋊C4×D61—24no—
59C2×#11758198—21no—
60C2×#11758198—22no—
61C2×C4×Dic61C2×C412yes—
62C4⋊#92418—21no—
63C4⋊#92418—21no—
64C4⋊#92418—21no—
65#14519695—42no—
66C22⋊#92418—21no—
67C4×#92418—21no—
68C22×Q8×C61—21no—
69C2×M16×C61—21no—
70C4⋊Q8×C61—21no—
71C4⋊Q8×C61—21no—
72C4⋊Q8×C61D412yes—
73C4⋊Q8×C61D412yes—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61C2×C412yes—
76C2×C42×C61C2311yescharacteristic
77C4⋊C8×C61—21no—
78C4⋊C8×C61—21no—
79C4⋊C8×C61—21no—
80C22⋊C8×C61—21no—
81C4×C8×C61—21no—
82C23×D61—21no—
83C2×Q8×C61—22no—
84C2×Q8×C61—22no—
85C2×Q8×C61—82no—
86C22×C4×C61—22no—
87C22×C4×C61C22×C411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×Dic61—21no—
91C22×Dic61C22⋊C412yes—
92C22×Dic61C22⋊C412yes—
93C2×D244—21no—
94C2×D244—24no—
95C2×D244C22⋊C412yes—
96C2×C4×D61—22no—
97C2×C4×D61—22no—
98C2×C4×D61—28no—
99C2×C4×D61—41no—
100C2×C4×D61—41no—
101C2×C4×D61—41no—
102C2×C4×D61—41no—
103C2×C4×D61—84no—
104C2×Dic122—21no—
105C2×Dic122—21no—
106C2×Dic122—21no—
107C2×Dic122—21no—
108C2×Dic122—21no—
109C2×Dic122—21no—
110M16×C61—41no—
111M16×C61—41no—
112C2×C8×C61—41no—
113C2×C8×C61—41no—
114C4⋊C4×C61—22no—
115C4⋊C4×C61—22no—
116C4⋊C4×C61—22no—
117C4⋊C4×C61C22⋊C414yes—
118C42×C61C2×D411yescharacteristic
119C42×C61C2×D411yescharacteristic
120C42×C61C2×D411yescharacteristic
121C42×C61C2×D411yescharacteristic
122#11758198—28no—
123#11758198—22no—
124#11758198—22no—
125C4⋊Dic61—28no—
126C4⋊Dic61—21no—
127C4⋊Dic61—21no—
128C4⋊Dic61—21no—
129C4⋊Dic61C22⋊C412yes—
130#11758196—28no—
131#11758196—22no—
132#11758196—22no—
133C4×Dic61—22no—
134C4×Dic61—22no—
135#11758194—41no—
136#11758194—41no—
137C2×#92418—41no—
138C2×#92418—41no—
139C23×C61C2×C22⋊C411yescharacteristic
140C22×D61—24no—
141C22×D61—22no—
142C22×D61—42no—
143C22×D61—41no—
144C22×D61—41no—
145Q8×C61—82no—
146Q8×C61—82no—
147C2×C4×C61—22no—
148C2×C4×C61—24no—
149C2×C4×C61—41no—
150C2×C4×C61—44no—
151C2×C4×C61C2×C22⋊C411yescharacteristic
152C2×C4×C61C2×C22⋊C411yescharacteristic
153C2×C4×C61#12011yescharacteristic
154C2×C4×C61#12011yescharacteristic
155C2×C4×C61#12011yescharacteristic
156C2×C4×C61#12011yescharacteristic
157C2×C4×C61#13511yescharacteristic
158C2×C4×C61#13511yescharacteristic
159C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C61#13711yescharacteristic
161C2×Dic61—22no—
162C2×Dic61—22no—
163C2×Dic61—22no—
164C2×Dic61—22no—
165C2×Dic61—22no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—42no—
169D244—24no—
170D244—22no—
171D244#10114yescomplemented
172C4×D61—42no—
173C4×D61—42no—
174C4×D61—42no—
175C4×D61—42no—
176C4×D61—88no—
177Dic122—41no—
178Dic122—41no—
179Dic122—41no—
180Dic122—41no—
181C8×C61—81no—
182C8×C61—81no—
183#92418—81no—
184#92418—81no—
185#8398—611nomaximal, Sylow
186C22×C61—21no—
187C22×C61—21no—
188C22×C61#37911yescharacteristic
189C22×C61#41611yescharacteristic
190C22×C61#41711yescharacteristic
191C2×D61—24no—
192C2×D61—44no—
193C2×D61—44no—
194C2×D61—42no—
195C2×D61—42no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—44no—
200C4×C61C2×#10112yes—
201C4×C61#44911yescharacteristic
202C4×C61#44911yescharacteristic
203C4×C61#45211yescharacteristic
204C4×C61#45211yescharacteristic
205Dic61—42no—
206Dic61—42no—
207Dic61—42no—
208C2×#554—611no—
209#1801—611no—
210#1549—611no—
211#1549D6111yescharacteristic, complemented
212#1486—611no—
213#1486—611no—
214#1461—611no—
215C2×C61—21no—
216C2×C61—21no—
217C2×C61#178411yescharacteristic
218C2×C61#178811yescharacteristic
219C2×C61#179511yescharacteristic
220D61—44no—
221#554—1222no—
222#554—1222no—
223#554—1222no—
224#554—1222no—
225C2×C4⋊Q8C2×D6111yescharacteristic
226C2×#125—612no—
227C2×C22⋊Q8—1221no—
228C2×C22⋊Q8—1222no—
229C2×C4×Q8—611no—
230C2×C4×D4—612no—
231C2×C4×D4—611no—
232C22×C4⋊C4—1221no—
233#431—611no—
234#431C2×D6111yescharacteristic
235#423—611no—
236#423C2×D6111yescharacteristic
237C2×#101—1221no—
238#343—1221no—
239#332—22no—
240#332—1222no—
241D4⋊C8—1222no—
242D4⋊C8—1222no—
243#324—1222no—
244C61#839811yescharacteristic, complemented, Sylow
245C22×Q8—21no—
246C22×Q8—611no—
247C22×D4—611no—
248C23×C4—1222no—
249C23×C4—1221no—
250C2×M16—21no—
251C2×M16—1221no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8D24412yes—
255C4⋊Q8C22⋊D6112yes—
256#125—1224no—
257#125—1224no—
258C22⋊Q8—1224no—
259C22⋊Q8—1224no—
260C22⋊Q8—1228no—
261C22⋊Q8—1228no—
262C4×Q8—1221no—
263C4×Q8—1221no—
264C4×Q8—1221no—
265C4×Q8—1221no—
266C4×D4—612no—
267C4×D4—612no—
268C4×D4—1224no—
269C4×D4—1224no—
270C4×D4—1221no—
271C4×D4—1221no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—611no—
274C2×C4⋊C4—611no—
275C2×C4⋊C4—612no—
276C2×C4⋊C4—1224no—
277C2×C4⋊C4—1224no—
278C2×C4⋊C4—1222no—
279C2×C4⋊C4—1222no—
280C2×C4⋊C4C4×D6112yes—
281C2×C22⋊C4—1222no—
282C2×C22⋊C4—1221no—
283C2×C42—612no—
284C2×C42C22×D6111yescharacteristic
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—1221no—
289C4⋊C8—1221no—
290C4⋊C8—1221no—
291#101—2442no—
292C22⋊C8—21no—
293C22⋊C8—1221no—
294C4×C8—21no—
295C4×C8—1221no—
296C24—1221no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—82no—
300C2×Q8—1221no—
301C2×Q8—1221no—
302C2×Q8—1221no—
303C2×Q8—1221no—
304C2×Q8—1221no—
305C2×Q8—1221no—
306C2×D4—612no—
307C2×D4—1224no—
308C2×D4—1221no—
309C22×C4—22no—
310C22×C4—612no—
311C22×C4—612no—
312C22×C4—1228no—
313C22×C4—1221no—
314C22×C4—1222no—
315C22×C4—1222no—
316C22×C4—4884no—
317C22×C4—2441no—
318C22×C4—2441no—
319C22×C4—2441no—
320C22×C4—2441no—
321C22×C4C2×C4×D6111yescharacteristic
322C22×C4D4×D6111yescharacteristic
323C22×C4D4×D6111yescharacteristic
324M16—41no—
325M16—41no—
326M16—2441no—
327M16—2441no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—2441no—
331C2×C8—2441no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—612no—
336C4⋊C4—1228no—
337C4⋊C4—1228no—
338C4⋊C4—1221no—
339C4⋊C4—1221no—
340C4⋊C4—1222no—
341C4⋊C4—1221no—
342C4⋊C4—1222no—
343C4⋊C4#1175819814yes—
344C22⋊C4—1228no—
345C22⋊C4—1222no—
346C22⋊C4—1222no—
347C42—1222no—
348C42—1222no—
349C42C2×D24411yescharacteristic
350C42D4×D6111yescharacteristic
351C42D4×D6111yescharacteristic
352C42C2×C22⋊D6111yescharacteristic
353C23—1224no—
354C23—1222no—
355C23—2442no—
356C23—2441no—
357C23—2441no—
358C23C22⋊C4×D6111yescharacteristic
359Q8—82no—
360Q8—82no—
361Q8—2441no—
362Q8—2441no—
363Q8—2441no—
364Q8—2441no—
365D4—614no—
366D4—1224no—
367D4—1222no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—41no—
371C2×C4—44no—
372C2×C4—1222no—
373C2×C4—1222no—
374C2×C4—1222no—
375C2×C4—1222no—
376C2×C4—1222no—
377C2×C4—4888no—
378C2×C4—2441no—
379C2×C4—2441no—
380C2×C4—2442no—
381C2×C4—2442no—
382C2×C4—2442no—
383C2×C4—2442no—
384C2×C4—2442no—
385C2×C4C22⋊C4×D6111yescharacteristic
386C2×C4C2×#1175819811yescharacteristic
387C2×C4#1451974511yescharacteristic
388C2×C4#1451974511yescharacteristic
389C2×C4#1451974711yescharacteristic
390C2×C4#1451974711yescharacteristic
391C2×C4#1451979811yescharacteristic
392C2×C4#1451980011yescharacteristic
393C2×C4#1451980411yescharacteristic
394C2×C4#1451980911yescharacteristic
395C8—81no—
396C8—81no—
397C8—4881no—
398C8—4881no—
399C22—21no—
400C22—21no—
401C22—1224no—
402C22—2444no—
403C22—2444no—
404C22—2442no—
405C22—2442no—
406C22#1683961711yescharacteristic
407C22#1683975711yescharacteristic
408C22#1683976211yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—44no—
413C4—2442no—
414C4—2442no—
415C4—2442no—
416C4#101×D6112yes—
417C4#1683990611yescharacteristic
418C4#1683991011yescharacteristic
419C4#1683992311yescharacteristic
420C4#1683992711yescharacteristic
421C2—21no—
422C2—21no—
423C2—2444no—
424C2#2144574111yescharacteristic
425C2#2144575811yescharacteristic
426C2#2144578611yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1549C2
socleC22×C61#379
prime radicalC22×D244C2×C4
cosocleC2×C42×C61C23
hypercenter#1549D61
hypocenterC61#8398
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
2. #21445812C2semidirect
3. #21444504C2semidirect
6. #21444142C2semidirect
7. C61×#1549C2semidirect
171. D244#101semidirect
211. #1549D61semidirect
244. C61#8398semidirect

Maximal subgroups

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

subgroupindex
C2×#168404732
#214458122
#214445042
#214442452
#214442432
#214441422
C61×#15492
#839861