Finite Group Databasebeta

← #29580518

Subgroups — #29580518

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#21456577C211yesmaximal, characteristic
2C2×#16840097C211yesmaximal, characteristic, complemented
3#21450307C211yesmaximal, characteristic
4#21446803C211yesmaximal, characteristic, complemented
5#21444386C211yesmaximal, characteristic
6#21443666C211yesmaximal, characteristic, complemented
7C61×#1824C211yesmaximal, characteristic, complemented
8C2×#14519756C2211yescharacteristic
9C2×#14519751C2211yescharacteristic
10#16840097—22no—
11#16840097—22no—
12#16840093—21no—
13#16840081—21no—
14#16839991—22no—
15#16839915—21no—
16#16839903—21no—
17C2×#14519710—21no—
18C2×#14519700—21no—
19C2×C22⋊#92418C2211yescharacteristic
20#16839688—22no—
21#16839663—21no—
22#16839641—22no—
23#16839627—21no—
24C4×C22⋊Dic61C2211yescharacteristic
25#16839528—22no—
26#16839495—22no—
27C22×C4⋊C4×C61C2211yescharacteristic
28C2×C22⋊C8×C61C2211yescharacteristic
29C61×#380C2211yescharacteristic
30C61×#340—22no—
31C22×Dic122D411yescharacteristic
32C22×D244D411yescharacteristic
33C22×C22⋊D61—21no—
34C2×#11758190—22no—
35C22×#92418—21no—
36#14519756—21no—
37#14519756—21no—
38#14519756—21no—
39#14519756—21no—
40#14519751—21no—
41#14519751—21no—
42#14519751—21no—
43#14519751—21no—
44C2×C4⋊Dic61D411yescharacteristic
45C2×C4⋊Dic61D411yescharacteristic
46C2×#11758196—21no—
47C2×C22⋊Dic61—21no—
48C2×C22⋊Dic61D412yes—
49C2×#11758198—21no—
50C2×#11758198—24no—
51C2×C4×Dic61—21no—
52#14519710—24no—
53#14519709—22no—
54#14519702—22no—
55#14519700—24no—
56#14519699—22no—
57C22⋊#92418—21no—
58C22⋊#92418D412yes—
59#14519684—24no—
60#14519684—21no—
61C23×C4×C61—21no—
62C23×C4×C61C2311yescharacteristic
63C22×C8×C61D412yes—
64C2×C4⋊C4×C61—21no—
65C2×C4⋊C4×C61—21no—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61D412yes—
71C2×C22⋊C4×C61—21no—
72C22⋊C8×C61—21no—
73C22⋊C8×C61—21no—
74#95×C61—21no—
75C24×C61C2×D411yescharacteristic
76C23×D61—21no—
77C22×C4×C61—22no—
78C22×C4×C61—42no—
79C22×C4×C61—41no—
80C22×C4×C61—41no—
81C22×C4×C61—41no—
82C22×C4×C61—82no—
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61#4111yescharacteristic
89C2×C22⋊D61—41no—
90C2×C22⋊D61—41no—
91C2×C22⋊D61—42no—
92C2×C22⋊D61—42no—
93C22×Dic61—21no—
94C22×Dic61—22no—
95C22×Dic61—21no—
96C22×Dic61—21no—
97C2×D244—21no—
98C2×D244—21no—
99C2×D244—42no—
100C2×Dic122—21no—
101C2×Dic122—21no—
102C2×Dic122—42no—
103C2×C8×C61—22no—
104C2×C8×C61—42no—
105C4⋊C4×C61—22no—
106C4⋊C4×C61—22no—
107C4⋊C4×C61—42no—
108C4⋊C4×C61—41no—
109C4⋊C4×C61—41no—
110C22⋊C4×C61—42no—
111C22⋊Dic61—22no—
112C22⋊Dic61—22no—
113C22⋊Dic61—41no—
114C22⋊Dic61—41no—
115#11758198—41no—
116#11758198—41no—
117#11758198—84no—
118C4⋊Dic61—21no—
119C4⋊Dic61—21no—
120C4⋊Dic61—21no—
121C4⋊Dic61—21no—
122#11758196—41no—
123#11758196—41no—
124C4×Dic61—42no—
125C2×#92418—22no—
126C2×#92418—42no—
127#11758190—82no—
128C23×C61—41no—
129C23×C61—41no—
130C23×C61#12011yescharacteristic
131C23×C61#12111yescharacteristic
132C23×C61#12711yescharacteristic
133C23×C61#13512yes—
134C23×C61#13712yes—
135C22×D61—42no—
136C22×D61—42no—
137C22×D61—41no—
138C22×D61—42no—
139C2×C4×C61—22no—
140C2×C4×C61—22no—
141C2×C4×C61—22no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—44no—
146C2×C4×C61—42no—
147C2×C4×C61—82no—
148C2×C4×C61—82no—
149C2×C4×C61—81no—
150C2×C4×C61—81no—
151C2×C4×C61—81no—
152C2×C4×C61#12011yescharacteristic
153C2×C4×C61#12011yescharacteristic
154C2×C4×C61#12111yescharacteristic
155C2×C4×C61#12111yescharacteristic
156C22⋊D61—82no—
157C22⋊D61—81no—
158C22⋊D61—81no—
159C2×Dic61—42no—
160C2×Dic61—42no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—42no—
165C2×Dic61—42no—
166C2×Dic61—81no—
167C2×Dic61—81no—
168D244—42no—
169D244—81no—
170Dic122—42no—
171Dic122—42no—
172C8×C61—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#39211yescharacteristic
184C22×C61#45212yes—
185C22×C61#46412yes—
186C22×C61#49511yescharacteristic
187C22×C61#49711yescharacteristic
188C2×D61—84no—
189C2×D61—81no—
190C2×D61—81no—
191C2×D61—81no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—82no—
196C4×C61—161no—
197C4×C61#45812yes—
198Dic61—82no—
199Dic61—81no—
200Dic61—81no—
201Dic61—81no—
202#3187—611no—
203C2×#484—611no—
204#2504—611no—
205#1996—611no—
206#1824D6111yescharacteristic, complemented
207#1519—611no—
208#1282—611no—
209C2×C61—22no—
210C2×C61—21no—
211C2×C61—22no—
212C2×C61—21no—
213C2×C61#200811yescharacteristic
214C2×C61#201511yescharacteristic
215C2×C61#202211yescharacteristic
216D61—161no—
217C2×C22⋊Q8—611no—
218C2×#121—611no—
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⋊C8C2×D6111yescharacteristic
231#397—1222no—
232#390—1221no—
233#386—1222no—
234#382—1221no—
235#380C2×D6111yescharacteristic
236C4×C22⋊C4—611no—
237#342—1222no—
238#340—22no—
239#327—1222no—
240C61#995811yescharacteristic, complemented, Sylow
241C22×Q8—611no—
242C22×D4—611no—
243C22×D4—1221no—
244C23×C4—21no—
245C23×C4C22×D6111yescharacteristic
246C2×SD16—1222no—
247C22×C8—1221no—
248C22×C8C22⋊D6112yes—
249C22⋊Q8—1221no—
250C22⋊Q8—1221no—
251C22⋊Q8—1221no—
252C22⋊Q8—1221no—
253#121—1221no—
254#121—1221no—
255#121—1221no—
256#121—1221no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—611no—
264C2×C4⋊C4—611no—
265C2×C4⋊C4—1221no—
266C2×C4⋊C4D24412yes—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—612no—
269C2×C22⋊C4—1224no—
270C2×C22⋊C4—1221no—
271C2×C22⋊C4—1221no—
272C2×C42—1221no—
273#107—1224no—
274#106—1222no—
275#103—1224no—
276#103—1222no—
277#102—1222no—
278C22⋊C8—21no—
279C22⋊C8—21no—
280C22⋊C8—612no—
281C22⋊C8—1221no—
282#95—21no—
283#95—1224no—
284#95—1221no—
285C24—1221no—
286C24D4×D6111yescharacteristic
287C2×Q8—1221no—
288C2×Q8—1221no—
289C2×Q8—2442no—
290C2×D4—1221no—
291C2×D4—1221no—
292C2×D4—2442no—
293C2×D4—2442no—
294C2×D4—2441no—
295C2×D4—2441no—
296C2×D4—2442no—
297C22×C4—22no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—82no—
303C22×C4—1221no—
304C22×C4—1221no—
305C22×C4—1221no—
306C22×C4—1222no—
307C22×C4C2×D24411yescharacteristic
308C22×C4#1175822211yescharacteristic
309C22×C4D4×D6111yescharacteristic
310C22×C4D4×D6111yescharacteristic
311C22×C4D4×D6111yescharacteristic
312C22×C4C2×C22⋊D6111yescharacteristic
313SD16—4882no—
314C2×C8—22no—
315C2×C8—42no—
316C2×C8—1222no—
317C2×C8—2442no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—42no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—1221no—
324C4⋊C4—1221no—
325C4⋊C4—1221no—
326C4⋊C4—1221no—
327C4⋊C4—2441no—
328C4⋊C4—2441no—
329C22⋊C4—42no—
330C22⋊C4—1222no—
331C22⋊C4—1222no—
332C22⋊C4—4884no—
333C22⋊C4—2441no—
334C22⋊C4—2441no—
335C22⋊C4—2441no—
336C22⋊C4—2441no—
337C42—2442no—
338C23—41no—
339C23—41no—
340C23—2442no—
341C23—2442no—
342C23—2441no—
343C23—2442no—
344C23#1451974711yescharacteristic
345C23#1451975011yescharacteristic
346C23#1451977311yescharacteristic
347C23#1451979812yes—
348C23#1451980912yes—
349Q8—2442no—
350Q8—2442no—
351D4—4881no—
352D4—4882no—
353D4—4881no—
354D4—4881no—
355D4—2442no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—4881no—
370C2×C4—4881no—
371C2×C4—2442no—
372C2×C4—2442no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4—2442no—
377C2×C4—2442no—
378C2×C4#1451974511yescharacteristic
379C2×C4#1451974711yescharacteristic
380C2×C4#1451975011yescharacteristic
381C2×C4#1451975111yescharacteristic
382C8—42no—
383C8—2442no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—42no—
391C22—42no—
392C22—4884no—
393C22—4881no—
394C22—4881no—
395C22—4881no—
396C22#1683967111yescharacteristic
397C22#1683992712yes—
398C22#1683998712yes—
399C22#1684014311yescharacteristic
400C22#1684015511yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—82no—
405C4—161no—
406C4—4882no—
407C4—4881no—
408C4—4881no—
409C4—4881no—
410C4#1683995112yes—
411C2—21no—
412C2—22no—
413C2—22no—
414C2—21no—
415C2—9761no—
416C2#2144687411yescharacteristic
417C2#2144691411yescharacteristic
418C2#2144695011yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1824C2
socleC22×C61#392
prime radicalC2×#14519751C22
hypercenter#1824D61
hypocenterC61#9958
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
2. C2×#16840097C2semidirect
4. #21446803C2semidirect
6. #21443666C2semidirect
7. C61×#1824C2semidirect
206. #1824D61semidirect
240. C61#9958semidirect

Maximal subgroups

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

subgroupindex
#214565772
C2×#168400972
#214503072
#214468032
#214443862
#214436662
C61×#18242
#995861