Finite Group Databasebeta

← #29580481

Subgroups — #29580481

Order 15616.

594 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456576C211yesmaximal, characteristic
2C2×#16840073C211yesmaximal, characteristic
3#21450304C211yesmaximal, characteristic, complemented
4#21445903C211yesmaximal, characteristic, complemented
5#21444390C211yesmaximal, characteristic, complemented
6#21443667C211yesmaximal, characteristic, complemented
7C61×#1964C211yesmaximal, characteristic
8C2×#14519756C2211yescharacteristic
9C22×C4⋊Dic61C2211yescharacteristic
10#16840092—21no—
11#16840078—21no—
12#16840073—22no—
13#16840073—22no—
14#16839989—22no—
15#16839913—21no—
16#16839900—21no—
17C2×#14519710—21no—
18C2×#14519698—21no—
19C2×C22⋊#92418C2211yescharacteristic
20C2×C22⋊#92418C2211yescharacteristic
21#16839702—22no—
22#16839668—21no—
23#16839628—21no—
24#16839623—22no—
25#16839524—22no—
26#16839498—22no—
27C2×#121×C61C2211yescharacteristic
28C2×#99×C61—22no—
29C61×#380C2211yescharacteristic
30C4×C22⋊C4×C61C2211yescharacteristic
31C22×Dic122D411yescharacteristic
32C23×Dic61—21no—
33C2×#11758200—22no—
34C22×#92418—21no—
35C22×#92418D412yes—
36#14519756—21no—
37#14519756—21no—
38#14519756—21no—
39#14519756—21no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61D412yes—
46C2×C4⋊Dic61D411yescharacteristic
47C2×#11758196—21no—
48C2×C22⋊Dic61—21no—
49#14519710—24no—
50#14519708—22no—
51#14519700—22no—
52#14519698—22no—
53#14519698—24no—
54C22⋊#92418—21no—
55C22⋊#92418—21no—
56C22⋊#92418—21no—
57C22⋊#92418D412yes—
58#14519684—24no—
59#14519684—24no—
60C22×D4×C61—21no—
61C22×D4×C61D411yescharacteristic
62C23×C4×C61C2311yescharacteristic
63#121×C61—21no—
64#121×C61—21no—
65#121×C61—21no—
66#121×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61D411yescharacteristic
69C2×C22⋊C4×C61—21no—
70C2×C22⋊C4×C61—21no—
71C2×C22⋊C4×C61D412yes—
72C2×C42×C61—21no—
73#99×C61—44no—
74#95×C61—21no—
75#95×C61—21no—
76C24×C61—21no—
77C24×C61C2×D411yescharacteristic
78C2×D4×C61—21no—
79C2×D4×C61—21no—
80C2×D4×C61—42no—
81C2×D4×C61—41no—
82C2×D4×C61—41no—
83C2×D4×C61—42no—
84C2×D4×C61—42no—
85C22×C4×C61—22no—
86C22×C4×C61—21no—
87C22×C4×C61—42no—
88C22×C4×C61—41no—
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61#4111yescharacteristic
95C22×Dic61—22no—
96C22×Dic61—21no—
97C22×Dic61—21no—
98C22×Dic61—41no—
99C22×Dic61—41no—
100C22×Dic61—42no—
101C22×Dic61—42no—
102C2×Dic122—21no—
103C2×Dic122—21no—
104C2×Dic122—42no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—42no—
108C22⋊C4×C61—22no—
109C22⋊C4×C61—22no—
110C22⋊C4×C61—41no—
111C22⋊C4×C61—41no—
112C22⋊C4×C61—42no—
113C42×C61—42no—
114C22⋊Dic61—41no—
115C22⋊Dic61—41no—
116#11758200—82no—
117C4⋊Dic61—21no—
118C4⋊Dic61—22no—
119C4⋊Dic61—22no—
120C4⋊Dic61—21no—
121C4⋊Dic61—41no—
122C4⋊Dic61—41no—
123#11758196—41no—
124#11758196—41no—
125C2×#92418—22no—
126C2×#92418—22no—
127C2×#92418—42no—
128C2×#92418—42no—
129C23×C61—41no—
130C23×C61—41no—
131C23×C61—41no—
132C23×C61—42no—
133C23×C61—81no—
134C23×C61—81no—
135C23×C61#12011yescharacteristic
136C23×C61#12111yescharacteristic
137C23×C61#12711yescharacteristic
138C23×C61#13512yes—
139C23×C61AGL(1,ℤ/8)12yes—
140D4×C61—42no—
141D4×C61—42no—
142D4×C61—82no—
143D4×C61—81no—
144D4×C61—81no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61—22no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—42no—
152C2×C4×C61—42no—
153C2×C4×C61—41no—
154C2×C4×C61—42no—
155C2×C4×C61—42no—
156C2×C4×C61—81no—
157C2×C4×C61—81no—
158C2×C4×C61—81no—
159C2×C4×C61#12011yescharacteristic
160C2×C4×C61#12011yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×Dic61—42no—
164C2×Dic61—42no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—42no—
168C2×Dic61—44no—
169C2×Dic61—82no—
170C2×Dic61—81no—
171C2×Dic61—81no—
172Dic122—42no—
173Dic122—81no—
174#92418—42no—
175#92418—42no—
176#9953—611nomaximal, Sylow
177C22×C61—21no—
178C22×C61—21no—
179C22×C61—21no—
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—42no—
184C22×C61—42no—
185C22×C61—81no—
186C22×C61—81no—
187C22×C61—81no—
188C22×C61—162no—
189C22×C61#39211yescharacteristic
190C22×C61#44912yes—
191C22×C61#46312yes—
192C22×C61#49511yescharacteristic
193C22×C61#49611yescharacteristic
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—82no—
198C4×C61—81no—
199C4×C61—161no—
200C4×C61#45812yes—
201Dic61—81no—
202Dic61—82no—
203Dic61—81no—
204#3187—611no—
205C2×#480—611no—
206#2503—611no—
207#1964D6111yescharacteristic
208#1823—611no—
209#1520—611no—
210#1282—611no—
211C2×C61—22no—
212C2×C61—21no—
213C2×C61—22no—
214C2×C61—21no—
215C2×C61—161no—
216C2×C61#200911yescharacteristic
217C2×C61#201211yescharacteristic
218C2×C61#202211yescharacteristic
219C2×C22⋊Q8—611no—
220C2×#121C2×D6111yescharacteristic
221C22×C4⋊C4—611no—
222#483—1221no—
223#481—1221no—
224#480—1222no—
225#480—1222no—
226#465—1222no—
227#450—1221no—
228#448—1221no—
229C2×#107—1221no—
230C2×#102—1221no—
231C2×#99—22no—
232C2×C22⋊C8—611no—
233C2×C22⋊C8—611no—
234#400—1222no—
235#391—1221no—
236#382—1221no—
237#381—1222no—
238#380C2×D6111yescharacteristic
239C4×C22⋊C4C2×D6111yescharacteristic
240#340—1222no—
241#328—1222no—
242C61#995311yescharacteristic, complemented, Sylow
243C22×Q8—611no—
244C22×D4—21no—
245C22×D4D24411yescharacteristic
246C23×C4—1221no—
247C23×C4C22×D6111yescharacteristic
248C2×SD16—1222no—
249C22×C8—612no—
250C22×C8—1221no—
251C22⋊Q8—1221no—
252C22⋊Q8—1221no—
253C22⋊Q8—1221no—
254C22⋊Q8—1221no—
255#121—21no—
256#121—21no—
257#121—21no—
258#121—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—612no—
261C2×C4⋊C4—611no—
262C2×C4⋊C4—1221no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4—1221no—
266C2×C4⋊C4—1221no—
267C2×C4⋊C4—1221no—
268C2×C4⋊C4D24411yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—1221no—
272C2×C22⋊C4C22⋊D6112yes—
273C2×C42—21no—
274#107—1224no—
275#106—1222no—
276#103—1222no—
277#102—1224no—
278#102—1222no—
279#99—44no—
280C22⋊C8—612no—
281C22⋊C8—1221no—
282C22⋊C8—1221no—
283C22⋊C8—1221no—
284#95—21no—
285#95—21no—
286#95—1224no—
287#95—1224no—
288C24—21no—
289C24C2×C22⋊D6111yescharacteristic
290C2×Q8—1221no—
291C2×Q8—1221no—
292C2×Q8—2442no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—42no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—42no—
299C2×D4—42no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—1221no—
305C22×C4—1221no—
306C22×C4—1222no—
307C22×C4—2441no—
308C22×C4—2441no—
309C22×C4—2442no—
310C22×C4—2442no—
311C22×C4C2×D24411yescharacteristic
312C22×C4#1175822211yescharacteristic
313C22×C4D4×D6111yescharacteristic
314C22×C4D4×D6111yescharacteristic
315C22×C4D4×D6111yescharacteristic
316C22×C4D4×D6111yescharacteristic
317SD16—4882no—
318C2×C8—1222no—
319C2×C8—1222no—
320C2×C8—2442no—
321C2×C8—2442no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—42no—
325C4⋊C4—1221no—
326C4⋊C4—1222no—
327C4⋊C4—1222no—
328C4⋊C4—1221no—
329C4⋊C4—2441no—
330C4⋊C4—2441no—
331C4⋊C4—2441no—
332C4⋊C4—2441no—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—42no—
338C22⋊C4—2441no—
339C22⋊C4—2441no—
340C42—42no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—42no—
345C23—81no—
346C23—81no—
347C23#1451974511yescharacteristic
348C23#1451975111yescharacteristic
349C23#1451977311yescharacteristic
350C23#1451979712yes—
351C23#1451980312yes—
352Q8—4881no—
353Q8—2442no—
354D4—42no—
355D4—42no—
356D4—81no—
357D4—81no—
358D4—82no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—4882no—
374C2×C4—4881no—
375C2×C4—4881no—
376C2×C4—2442no—
377C2×C4—2442no—
378C2×C4—2444no—
379C2×C4—2441no—
380C2×C4—2441no—
381C2×C4—2442no—
382C2×C4#1451974711yescharacteristic
383C2×C4#1451974711yescharacteristic
384C2×C4#1451975011yescharacteristic
385C2×C4#1451975011yescharacteristic
386C8—2442no—
387C8—2442no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—42no—
395C22—42no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—162no—
400C22#1683967111yescharacteristic
401C22#1683990512yes—
402C22#1683997712yes—
403C22#1684014411yescharacteristic
404C22#1684014911yescharacteristic
405C4—21no—
406C4—21no—
407C4—21no—
408C4—82no—
409C4—81no—
410C4—161no—
411C4—4881no—
412C4—4881no—
413C4—4882no—
414C4#1683995312yes—
415C2—22no—
416C2—21no—
417C2—22no—
418C2—21no—
419C2—161no—
420C2#2144687811yescharacteristic
421C2#2144689511yescharacteristic
422C2#2144694911yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1964C2
socleC22×C61#392
prime radicalC2×#121×C61C22
hypercenter#1964D61
hypocenterC61#9953
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1964

Complements

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

NH ≅ G/Ntype
3. #21450304C2semidirect
4. #21445903C2semidirect
5. #21444390C2semidirect
6. #21443667C2semidirect
242. C61#9953semidirect

Maximal subgroups

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

subgroupindex
#214565762
C2×#168400732
#214503042
#214459032
#214443902
#214436672
C61×#19642
#995361