Finite Group Databasebeta

← #29580517

Subgroups — #29580517

Order 15616.

594 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 101 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×#16840097C211yesmaximal, characteristic, complemented
3#21446804C211yesmaximal, characteristic, complemented
4#21445909C211yesmaximal, characteristic, complemented
5#21444387C211yesmaximal, characteristic
6#21443667C211yesmaximal, characteristic, complemented
7C61×#2504C211yesmaximal, characteristic
8C2×#14519756C2211yescharacteristic
9#16840097—22no—
10#16840097—22no—
11#16840092—21no—
12#16840078—21no—
13#16839989—22no—
14#16839913—21no—
15#16839900—21no—
16C2×#14519710—21no—
17C2×#14519700—21no—
18C2×C22⋊#92418C2211yescharacteristic
19C2×C22⋊#92418C2211yescharacteristic
20#16839622C2211yescharacteristic
21#16839528—22no—
22#16839524—22no—
23#16839496—22no—
24C2×#121×C61C2211yescharacteristic
25C22×C4⋊C4×C61C2211yescharacteristic
26C61×#397—22no—
27C61×#390—21no—
28C61×#386—22no—
29C61×#382—21no—
30C4×C22⋊C4×C61C2211yescharacteristic
31C22×Dic122D411yescharacteristic
32C2×#11758200—22no—
33C22×#92418—21no—
34C22×#92418D412yes—
35#14519756—21no—
36#14519756—21no—
37#14519756—21no—
38#14519756—21no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61D411yescharacteristic
41C2×#11758196—21no—
42C2×C22⋊Dic61—21no—
43C2×C22⋊Dic61—21no—
44#14519710—24no—
45#14519708—22no—
46#14519700—22no—
47#14519700—24no—
48#14519698—22no—
49C22⋊#92418—21no—
50C22⋊#92418—21no—
51C22⋊#92418—21no—
52C22⋊#92418D412yes—
53#14519684—21no—
54C22×D4×C61—21no—
55C22×D4×C61D411yescharacteristic
56C23×C4×C61—21no—
57C23×C4×C61C2311yescharacteristic
58#121×C61—21no—
59#121×C61—21no—
60#121×C61—21no—
61#121×C61—21no—
62C2×C4⋊C4×C61—21no—
63C2×C4⋊C4×C61—21no—
64C2×C4⋊C4×C61—21no—
65C2×C4⋊C4×C61—21no—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61D411yescharacteristic
68C2×C4⋊C4×C61D412yes—
69C2×C22⋊C4×C61—21no—
70C2×C22⋊C4×C61—24no—
71C2×C22⋊C4×C61D412yes—
72C2×C42×C61—21no—
73#95×C61—24no—
74#95×C61—21no—
75C24×C61—21no—
76C24×C61C2×D411yescharacteristic
77C2×D4×C61—21no—
78C2×D4×C61—21no—
79C2×D4×C61—42no—
80C2×D4×C61—41no—
81C2×D4×C61—41no—
82C2×D4×C61—42no—
83C2×D4×C61—42no—
84C22×C4×C61—21no—
85C22×C4×C61—22no—
86C22×C4×C61—22no—
87C22×C4×C61—42no—
88C22×C4×C61—41no—
89C22×C4×C61—41no—
90C22×C4×C61—82no—
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61#4111yescharacteristic
97C22×Dic61—21no—
98C22×Dic61—21no—
99C22×Dic61—41no—
100C2×Dic122—21no—
101C2×Dic122—21no—
102C2×Dic122—42no—
103C4⋊C4×C61—22no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—22no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—41no—
108C4⋊C4×C61—41no—
109C22⋊C4×C61—22no—
110C22⋊C4×C61—22no—
111C22⋊C4×C61—41no—
112C22⋊C4×C61—41no—
113C22⋊C4×C61—84no—
114C42×C61—42no—
115C22⋊Dic61—41no—
116C22⋊Dic61—41no—
117C22⋊Dic61—42no—
118#11758200—82no—
119C4⋊Dic61—21no—
120C4⋊Dic61—21no—
121C4⋊Dic61—42no—
122#11758196—41no—
123#11758196—41no—
124C2×#92418—22no—
125C2×#92418—22no—
126C2×#92418—42no—
127C2×#92418—42no—
128C23×C61—41no—
129C23×C61—41no—
130C23×C61—42no—
131C23×C61—42no—
132C23×C61—41no—
133C23×C61—42no—
134C23×C61#12011yescharacteristic
135C23×C61#12111yescharacteristic
136C23×C61#12711yescharacteristic
137C23×C61#13512yes—
138C23×C61#13712yes—
139D4×C61—42no—
140D4×C61—81no—
141D4×C61—82no—
142D4×C61—81no—
143D4×C61—81no—
144C2×C4×C61—22no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—42no—
151C2×C4×C61—42no—
152C2×C4×C61—44no—
153C2×C4×C61—41no—
154C2×C4×C61—42no—
155C2×C4×C61—42no—
156C2×C4×C61—42no—
157C2×C4×C61—82no—
158C2×C4×C61—82no—
159C2×C4×C61#12011yescharacteristic
160C2×C4×C61#12011yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—81no—
166C2×Dic61—81no—
167C2×Dic61—81no—
168C2×Dic61—81no—
169C2×Dic61—81no—
170Dic122—42no—
171Dic122—42no—
172#92418—42no—
173#92418—42no—
174#9958—611nomaximal, Sylow
175C22×C61—21no—
176C22×C61—21no—
177C22×C61—21no—
178C22×C61—21no—
179C22×C61—21no—
180C22×C61—21no—
181C22×C61—42no—
182C22×C61—42no—
183C22×C61—84no—
184C22×C61—81no—
185C22×C61—81no—
186C22×C61—81no—
187C22×C61#39211yescharacteristic
188C22×C61#45212yes—
189C22×C61#46412yes—
190C22×C61#49511yescharacteristic
191C22×C61#49711yescharacteristic
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—82no—
196C4×C61—82no—
197C4×C61—81no—
198C4×C61#45812yes—
199Dic61—81no—
200Dic61—81no—
201Dic61—161no—
202#3187—611no—
203C2×#484—611no—
204#2504D6111yescharacteristic
205#1996—611no—
206#1824—611no—
207#1519—611no—
208#1282—611no—
209C2×C61—21no—
210C2×C61—22no—
211C2×C61—22no—
212C2×C61—21no—
213C2×C61—161no—
214C2×C61#200811yescharacteristic
215C2×C61#201511yescharacteristic
216C2×C61#202211yescharacteristic
217C2×C22⋊Q8—611no—
218C2×#121C2×D6111yescharacteristic
219C22×C4⋊C4C2×D6111yescharacteristic
220#484—1222no—
221#484—1222no—
222#483—1221no—
223#481—1221no—
224#465—1222no—
225#450—1221no—
226#448—1221no—
227C2×#107—1221no—
228C2×#103—1221no—
229C2×C22⋊C8—611no—
230C2×C22⋊C8—611no—
231#397—22no—
232#390—21no—
233#386—22no—
234#382—21no—
235#380—611no—
236C4×C22⋊C4C2×D6111yescharacteristic
237#342—1222no—
238#340—1222no—
239#327—1222no—
240C61#995811yescharacteristic, complemented, Sylow
241C22×Q8—611no—
242C22×D4—21no—
243C22×D4C22⋊D6111yescharacteristic
244C23×C4—21no—
245C23×C4C22×D6111yescharacteristic
246C2×SD16—1222no—
247C22×C8—612no—
248C22×C8—1221no—
249C22⋊Q8—1221no—
250C22⋊Q8—1221no—
251C22⋊Q8—1221no—
252C22⋊Q8—1221no—
253#121—21no—
254#121—21no—
255#121—21no—
256#121—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—611no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4C22⋊D6112yes—
266C2×C4⋊C4C22⋊D6111yescharacteristic
267C2×C22⋊C4—21no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—1221no—
270C2×C22⋊C4—1221no—
271C2×C22⋊C4C22⋊D6112yes—
272C2×C42—21no—
273#107—1224no—
274#106—1222no—
275#103—1224no—
276#103—1222no—
277#102—1222no—
278C22⋊C8—612no—
279C22⋊C8—1221no—
280C22⋊C8—1221no—
281C22⋊C8—1221no—
282#95—24no—
283#95—21no—
284#95—1221no—
285C24—21no—
286C24C2×C22⋊D6111yescharacteristic
287C2×Q8—1221no—
288C2×Q8—1221no—
289C2×Q8—2442no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—42no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—82no—
304C22×C4—1221no—
305C22×C4—1221no—
306C22×C4—2441no—
307C22×C4D4×D6111yescharacteristic
308C22×C4D4×D6111yescharacteristic
309C22×C4D4×D6111yescharacteristic
310C22×C4#1175822411yescharacteristic
311C22×C4C2×C22⋊D6111yescharacteristic
312C22×C4C2×C22⋊D6111yescharacteristic
313SD16—4882no—
314C2×C8—1222no—
315C2×C8—1222no—
316C2×C8—2442no—
317C2×C8—2442no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—1221no—
325C4⋊C4—1221no—
326C4⋊C4—2442no—
327C4⋊C4—2441no—
328C4⋊C4—2441no—
329C22⋊C4—22no—
330C22⋊C4—22no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—84no—
334C22⋊C4—2442no—
335C22⋊C4—2441no—
336C22⋊C4—2441no—
337C42—42no—
338C23—41no—
339C23—41no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—41no—
344C23#1451974611yescharacteristic
345C23#1451974811yescharacteristic
346C23#1451977311yescharacteristic
347C23#1451979712yes—
348C23#1451980712yes—
349Q8—2442no—
350Q8—2442no—
351D4—42no—
352D4—81no—
353D4—82no—
354D4—81no—
355D4—81no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—4881no—
372C2×C4—4881no—
373C2×C4—4881no—
374C2×C4—4881no—
375C2×C4—4881no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4#1451974511yescharacteristic
379C2×C4#1451974511yescharacteristic
380C2×C4#1451974911yescharacteristic
381C2×C4#1451974911yescharacteristic
382C8—2442no—
383C8—2442no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—42no—
391C22—42no—
392C22—84no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22#1683967011yescharacteristic
397C22#1683992412yes—
398C22#1683998112yes—
399C22#1684014411yescharacteristic
400C22#1684015711yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—82no—
405C4—81no—
406C4—82no—
407C4—9761no—
408C4—4881no—
409C4—4881no—
410C4#1683995112yes—
411C2—22no—
412C2—21no—
413C2—22no—
414C2—21no—
415C2—161no—
416C2#2144687211yescharacteristic
417C2#2144691311yescharacteristic
418C2#2144694711yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#2504C2
socleC22×C61#392
prime radicalC2×#121×C61C22
hypercenter#2504D61
hypocenterC61#9958
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2504

Complements

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

NH ≅ G/Ntype
2. C2×#16840097C2semidirect
3. #21446804C2semidirect
4. #21445909C2semidirect
6. #21443667C2semidirect
240. C61#9958semidirect

Maximal subgroups

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

subgroupindex
#214565762
C2×#168400972
#214468042
#214459092
#214443872
#214436672
C61×#25042
#995861