Finite Group Databasebeta

← #29572596

Subgroups — #29572596

Order 15616.

794 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 119 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#21459622C211yesmaximal, characteristic
2#21446287C212yesmaximal, complemented
3#21444182C212yesmaximal, complemented
4#21444157C211yesmaximal, characteristic
5C61×#1870C211yesmaximal, characteristic
6#16840471—24no—
7#16840469—22no—
8D4×Dic122—22no—
9#16840393—21no—
10#16840388—21no—
11#16840388C412yes—
12#16840319—21no—
13C2×#14519756—21no—
14C2×#14519756C2212yes—
15#16840237C412yes—
16C2×#14519695—22no—
17#16839726C2212yes—
18#16839530—22no—
19#16839488—21no—
20#16839487—24no—
21#16839486—21no—
22C22×M16×C61C2211yescharacteristic
23C2×C4×D4×C61C2211yescharacteristic
24C2×C22⋊C8×C61—21no—
25C2×C22⋊C8×C61C2211yescharacteristic
26C61×#336—21no—
27C22×Dic122—21no—
28C22×Dic122D412yes—
29C2×#11758194—22no—
30#14519777—21no—
31#14519777—21no—
32#14519776—24no—
33#14519769—22no—
34#14519768—24no—
35#14519756—22no—
36#14519756—22no—
37#14519756—22no—
38#14519756—22no—
39#14519756—22no—
40#14519756—22no—
41#14519756—22no—
42#14519756—22no—
43#14519754—24no—
44#14519754—24no—
45#14519754—24no—
46#14519754—24no—
47C4×Dic122—22no—
48C4×Dic122—22no—
49D4×Dic61—24no—
50D4×Dic61—24no—
51#14519736—22no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61D412yes—
54C2×#11758196—22no—
55C2×#11758196D414yes—
56C2×C22⋊Dic61—22no—
57C2×C22⋊Dic61C2×C414yes—
58#14519695—44no—
59C22⋊#92418—22no—
60C22⋊#92418—22no—
61C22×D4×C61C2×C411yescharacteristic
62C23×C4×C61D411yescharacteristic
63C23×C4×C61C2311yescharacteristic
64C2×M16×C61—22no—
65C2×M16×C61—42no—
66C22×C8×C61—21no—
67C22×C8×C61—21no—
68C4×D4×C61—22no—
69C4×D4×C61—22no—
70C2×C4⋊C4×C61D411yescharacteristic
71C2×C22⋊C4×C61C2×C411yescharacteristic
72C2×C22⋊C4×C61D411yescharacteristic
73C2×C42×C61D411yescharacteristic
74C22⋊C8×C61—22no—
75C22⋊C8×C61—42no—
76C24×C61C22⋊C411yescharacteristic
77C24×C61C22×C411yescharacteristic
78C2×D4×C61—22no—
79C2×D4×C61—22no—
80C2×D4×C61—22no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—41no—
84C22×C4×C61—41no—
85C22×C4×C61—41no—
86C22×C4×C61—41no—
87C22×C4×C61C22⋊C411yescharacteristic
88C22×C4×C61C22⋊C411yescharacteristic
89C22×C4×C61C22⋊C411yescharacteristic
90C22×C4×C61C2×D412yes—
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D412yes—
94C22×Dic61—22no—
95C22×Dic61—22no—
96C22×Dic61C22⋊C414yes—
97C22×Dic61C22⋊C414yes—
98C2×Dic122—22no—
99C2×Dic122—22no—
100C2×Dic122—22no—
101C2×Dic122—22no—
102C2×Dic122—42no—
103C2×Dic122—42no—
104C2×Dic122—81no—
105C2×Dic122—81no—
106M16×C61—24no—
107M16×C61—42no—
108C2×C8×C61—22no—
109C2×C8×C61—41no—
110C2×C8×C61—41no—
111C2×C8×C61—41no—
112C2×C8×C61—41no—
113C2×C8×C61—41no—
114C4⋊C4×C61—22no—
115C22⋊C4×C61—22no—
116C22⋊C4×C61—22no—
117C42×C61—21no—
118C42×C61—21no—
119C22⋊Dic61—24no—
120C22⋊Dic61—24no—
121C22⋊Dic61—24no—
122C4⋊Dic61—22no—
123C4⋊Dic61—22no—
124C4⋊Dic61—22no—
125C4⋊Dic61—22no—
126C4⋊Dic61—22no—
127C4⋊Dic61—22no—
128#11758196—24no—
129#11758196—24no—
130#11758196—24no—
131#11758196—24no—
132C4×Dic61—24no—
133#11758194—42no—
134#11758194—42no—
135C2×#92418—42no—
136C23×C61—21no—
137C23×C61—21no—
138C23×C61—21no—
139C23×C61—21no—
140C23×C61—41no—
141C23×C61—41no—
142C23×C61—41no—
143C23×C61—41no—
144C23×C61#10112yescomplemented
145C23×C61C2×C22⋊C412yes—
146C23×C61C2×C22⋊C411yescharacteristic
147D4×C61—28no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—22no—
151C2×C4×C61—21no—
152C2×C4×C61—22no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—41no—
157C2×C4×C61—42no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—81no—
163C2×C4×C61#10112yescomplemented
164C2×C4×C61#12011yescharacteristic
165C2×C4×C61#12012yes—
166C2×C4×C61#12011yescharacteristic
167C2×Dic61—24no—
168C2×Dic61—24no—
169C2×Dic61—44no—
170C2×Dic61—44no—
171C2×Dic61—82no—
172C2×Dic61—82no—
173Dic122—42no—
174Dic122—42no—
175Dic122—82no—
176Dic122—82no—
177Dic122—82no—
178C8×C61—42no—
179C8×C61—81no—
180#92418—82no—
181#8537—611nomaximal, Sylow
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—22no—
185C22×C61—22no—
186C22×C61—22no—
187C22×C61—22no—
188C22×C61—42no—
189C22×C61—41no—
190C22×C61—41no—
191C22×C61—81no—
192C22×C61—81no—
193C22×C61—81no—
194C22×C61#37911yescharacteristic
195C22×C61C2×#10111yescharacteristic
196C22×C61C2×#10111yescharacteristic
197C4×C61—22no—
198C4×C61—22no—
199C4×C61—21no—
200C4×C61—41no—
201C4×C61—41no—
202C4×C61B(4,𝔽2)11yescharacteristic
203C4×C61#45811yescharacteristic
204Dic61—44no—
205Dic61—44no—
206#3501—611no—
207#1901—612no—
208#1870D6111yescharacteristic
209#1472—612no—
210#1465—611no—
211C2×C61—21no—
212C2×C61—21no—
213C2×C61—42no—
214C2×C61—42no—
215C2×C61#178411yescharacteristic
216C2×C61#179011yescharacteristic
217C2×C61#179111yescharacteristic
218C22×M16C2×D6111yescharacteristic
219#554—1222no—
220#554—1224no—
221D4×Q8—1222no—
222#544—1221no—
223#543—612no—
224#543—1221no—
225#533—1221no—
226C2×C22⋊Q8—612no—
227C2×C22⋊Q8—1221no—
228#518—612no—
229C2×C4×D4C2×D6111yescharacteristic
230C2×#101—1222no—
231#407—612no—
232C2×C22⋊C8—21no—
233C2×C22⋊C8C2×D6111yescharacteristic
234#343—1222no—
235#336—21no—
236#324—1224no—
237#324—1221no—
238#323—1221no—
239C61#853711yescharacteristic, complemented, Sylow
240C22×Q8—612no—
241C22×Q8—1221no—
242C22×D4C4×D6111yescharacteristic
243C23×C4C22⋊D6111yescharacteristic
244C23×C4C22×D6111yescharacteristic
245C2×M16—22no—
246C2×M16—42no—
247C2×M16—1222no—
248C22×C8—21no—
249C22×C8—21no—
250C4⋊Q8—1221no—
251C4⋊Q8—1221no—
252C4⋊Q8—1224no—
253#125—1222no—
254#125—1224no—
255C22⋊Q8—1224no—
256C22⋊Q8—1224no—
257C22⋊Q8—1224no—
258C22⋊Q8—1224no—
259C22⋊Q8—1222no—
260C22⋊Q8—1222no—
261C22⋊Q8—1222no—
262C22⋊Q8—1222no—
263C22⋊Q8—1222no—
264C22⋊Q8—1222no—
265C22⋊Q8—1222no—
266C22⋊Q8—1222no—
267C4×Q8—1222no—
268C4×Q8—1222no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—1224no—
272C4×D4—1224no—
273#117—1222no—
274C2×C4⋊C4—614no—
275C2×C4⋊C4—612no—
276C2×C4⋊C4—1222no—
277C2×C4⋊C4—1221no—
278C2×C4⋊C4D24411yescharacteristic
279C2×C22⋊C4—614no—
280C2×C22⋊C4—1222no—
281C2×C22⋊C4C4×D6111yescharacteristic
282C2×C22⋊C4D24411yescharacteristic
283C2×C42C22⋊D6111yescharacteristic
284#101—2444no—
285C22⋊C8—22no—
286C22⋊C8—42no—
287C22⋊C8—1222no—
288C22⋊C8—1222no—
289C24#1175819811yescharacteristic
290C24C2×C4×D6111yescharacteristic
291C2×Q8—1222no—
292C2×Q8—1222no—
293C2×Q8—1222no—
294C2×Q8—1222no—
295C2×Q8—4881no—
296C2×Q8—4881no—
297C2×Q8—2442no—
298C2×Q8—2442no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—614no—
309C22×C4—614no—
310C22×C4—1222no—
311C22×C4—1222no—
312C22×C4#1175819811yescharacteristic
313C22×C4#1175819811yescharacteristic
314C22×C4#1175819811yescharacteristic
315C22×C4C2×D24411yescharacteristic
316C22×C4D4×D6112yes—
317C22×C4D4×D6112yes—
318C22×C4C2×C22⋊D6111yescharacteristic
319M16—24no—
320M16—42no—
321M16—2442no—
322M16—2442no—
323C2×C8—22no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—2442no—
330C4⋊C4—22no—
331C4⋊C4—1224no—
332C4⋊C4—1222no—
333C4⋊C4—1224no—
334C4⋊C4—1222no—
335C4⋊C4—1222no—
336C4⋊C4—1222no—
337C4⋊C4—1224no—
338C4⋊C4—1222no—
339C4⋊C4—1224no—
340C4⋊C4—1222no—
341C22⋊C4—22no—
342C22⋊C4—22no—
343C22⋊C4—1224no—
344C22⋊C4—1224no—
345C22⋊C4—1224no—
346C42—21no—
347C42—21no—
348C42—1224no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23#1451969512yescomplemented
358C23C22⋊C4×D6112yes—
359C23C2×#1175819811yescharacteristic
360Q8—4882no—
361Q8—4882no—
362Q8—4882no—
363Q8—2442no—
364Q8—2442no—
365D4—28no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—81no—
381C2×C4—1224no—
382C2×C4—1224no—
383C2×C4—4882no—
384C2×C4—4882no—
385C2×C4—2444no—
386C2×C4—2444no—
387C2×C4#1451969512yescomplemented
388C2×C4#1451974511yescharacteristic
389C2×C4#1451974511yescharacteristic
390C2×C4#1451974712yes—
391C8—42no—
392C8—81no—
393C8—4882no—
394C22—21no—
395C22—21no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—42no—
401C22—41no—
402C22—41no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22#1683961711yescharacteristic
407C22C2×#1451969511yescharacteristic
408C22C2×#1451969511yescharacteristic
409C4—22no—
410C4—21no—
411C4—22no—
412C4—41no—
413C4—41no—
414C4—2444no—
415C4—2444no—
416C4#1683994811yescharacteristic
417C4#1683995111yescharacteristic
418C2—21no—
419C2—21no—
420C2—42no—
421C2—42no—
422C2#2144574211yescharacteristic
423C2#2144576611yescharacteristic
424C2#2144576911yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1870C2
socleC22×C61#379
prime radicalC22×D4×C61C2×C4
cosocleC23×C4×C61C23
hypercenter#1870D61
hypocenterC61#8537
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1870

Complements

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

NH ≅ G/Ntype
2. #21446287C2semidirect
3. #21444182C2semidirect
144. C23×C61#101semidirect
163. C2×C4×C61#101semidirect
239. C61#8537semidirect
357. C23#14519695semidirect
387. C2×C4#14519695semidirect

Maximal subgroups

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

subgroupindex
#214596222
#214462872
#214441822
#214441572
C61×#18702
#853761