Finite Group Databasebeta

← #29577881

Subgroups — #29577881

Order 15616.

602 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 105 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#21454859C211yesmaximal, characteristic, complemented
2#21450721C211yesmaximal, characteristic
3#21445808C211yesmaximal, characteristic, complemented
4#21444245C211yesmaximal, characteristic, complemented
5#21444243C211yesmaximal, characteristic, complemented
6#21443677C211yesmaximal, characteristic, complemented
7C61×#1286C211yesmaximal, characteristic, complemented
8C2×#14519752—21no—
9C2×C4×D244C2211yescharacteristic
10C2×C4×D244C2211yescharacteristic
11#16839841—22no—
12#16839839—22no—
13#16839815C2211yescharacteristic
14#16839786C2211yescharacteristic
15C2×#14519693—21no—
16#16839724—22no—
17#16839689C412yes—
18#16839672C412yes—
19#16839640—22no—
20#16839619—21no—
21#16839530—21no—
22#16839491—22no—
23#16839489—22no—
24C61×#431C2211yescharacteristic
25C61×#423C2211yescharacteristic
26C61×#382C2211yescharacteristic
27C22×D244—22no—
28C22×D244C2×C411yescharacteristic
29C22×D244D411yescharacteristic
30C22×C4×D61—21no—
31C22×C4×D61D412yes—
32C2×#11758194—21no—
33C2×#11758189—24no—
34#14519752—42no—
35#14519752—42no—
36C4×D244—21no—
37C4×D244—21no—
38C4×D244—21no—
39C4×D244—21no—
40C4×D244—21no—
41C4×D244—21no—
42C4×D244—21no—
43C4×D244—21no—
44C2×C4⋊Dic61C2×C411yescharacteristic
45C2×C4⋊Dic61D411yescharacteristic
46C2×#11758198—22no—
47C2×#11758198—21no—
48C2×#11758198—22no—
49C2×#11758198C2×C412yes—
50C4⋊#92418—21no—
51C4⋊#92418—21no—
52C4⋊#92418—21no—
53#14519693—42no—
54C22⋊#92418—21no—
55#14519689—24no—
56C4×#92418D412yescomplemented
57#14519684—22no—
58C2×M16×C61—21no—
59C2×C4⋊C4×C61—21no—
60C2×C42×C61C2311yescharacteristic
61C4⋊C8×C61—21no—
62C4⋊C8×C61—21no—
63C4⋊C8×C61—21no—
64C22⋊C8×C61—21no—
65C4×C8×C61D412yescomplemented
66#95×C61—21no—
67#95×C61C2×C412yes—
68C23×D61—21no—
69C23×D61C22⋊C412yes—
70C22×C4×C61—22no—
71C22×C4×C61C22×C411yescharacteristic
72C22×C4×C61C2×D411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C22×Dic61—21no—
75C22×Dic61C22⋊C412yes—
76C2×D244—21no—
77C2×D244—21no—
78C2×D244—21no—
79C2×D244—21no—
80C2×D244—21no—
81C2×D244—21no—
82C2×D244—21no—
83C2×D244—21no—
84C2×D244—42no—
85C2×D244—44no—
86C2×D244—44no—
87C2×D244—82no—
88C2×C4×D61—22no—
89C2×C4×D61—22no—
90C2×C4×D61—41no—
91C2×C4×D61—41no—
92C2×C4×D61—41no—
93C2×C4×D61—41no—
94C2×C4×D61—41no—
95C2×C4×D61—41no—
96C2×C4×D61—82no—
97M16×C61—41no—
98M16×C61—41no—
99C2×C8×C61—22no—
100C2×C8×C61—41no—
101C4⋊C4×C61—41no—
102C4⋊C4×C61—41no—
103C42×C61C2×D411yescharacteristic
104C42×C61C2×D411yescharacteristic
105C42×C61#4111yescharacteristic
106C42×C61#4111yescharacteristic
107#11758198—22no—
108#11758198—22no—
109#11758198—44no—
110#11758198—41no—
111#11758198—41no—
112#11758198—82no—
113C4⋊Dic61—21no—
114C4⋊Dic61—21no—
115C4⋊Dic61—21no—
116C4⋊Dic61—21no—
117#11758194—41no—
118#11758194—41no—
119C2×#92418—22no—
120C2×#92418—41no—
121#11758189—84no—
122C23×C61C2×C22⋊C411yescharacteristic
123C22×D61—22no—
124C22×D61—22no—
125C22×D61—22no—
126C22×D61—22no—
127C22×D61—42no—
128C22×D61—42no—
129C22×D61—41no—
130C22×D61—41no—
131C22×D61—41no—
132C22×D61—41no—
133C22×D61—41no—
134C22×D61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—42no—
138C2×C4×C61—42no—
139C2×C4×C61—42no—
140C2×C4×C61C4×D411yescharacteristic
141C2×C4×C61C4×D411yescharacteristic
142C2×C4×C61#12011yescharacteristic
143C2×C4×C61#12111yescharacteristic
144C2×C4×C61#12111yescharacteristic
145C2×C4×C61#12311yescharacteristic
146C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C61#13711yescharacteristic
150C2×Dic61—22no—
151C2×Dic61—42no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155D244—22no—
156D244—22no—
157D244—41no—
158D244—41no—
159D244—41no—
160D244—41no—
161D244—41no—
162D244—41no—
163D244—88no—
164C4×D61—44no—
165C4×D61—82no—
166C4×D61—81no—
167C4×D61—81no—
168C4×D61—81no—
169C4×D61—81no—
170C8×C61—42no—
171C8×C61—81no—
172#92418—42no—
173#92418—81no—
174#9489—611nomaximal, Sylow
175C22×C61—21no—
176C22×C61—21no—
177C22×C61#38611yescharacteristic
178C22×C61#41711yescharacteristic
179C22×C61#41811yescharacteristic
180C2×D61—42no—
181C2×D61—44no—
182C2×D61—42no—
183C2×D61—42no—
184C2×D61—42no—
185C2×D61—42no—
186C2×D61—42no—
187C2×D61—81no—
188C2×D61—81no—
189C2×D61—81no—
190C2×D61—81no—
191C2×D61—82no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—82no—
197C4×C61#46811yescharacteristic
198C4×C61#46811yescharacteristic
199C4×C61#46911yescharacteristic
200C4×C61#46911yescharacteristic
201Dic61—42no—
202Dic61—81no—
203#2975—611no—
204#2535—611no—
205#1800—611no—
206#1486—611no—
207#1486—611no—
208#1286—611no—
209#1286D6111yescharacteristic, complemented
210C2×C61—21no—
211C2×C61—21no—
212C2×C61#189111yescharacteristic
213C2×C61#189511yescharacteristic
214C2×C61#191111yescharacteristic
215D61—82no—
216D61—82no—
217C2×#121—1221no—
218C2×C4×D4—611no—
219C2×C4×D4—611no—
220#436—1222no—
221#436—1222no—
222#431—611no—
223#431C2×D6111yescharacteristic
224#423—611no—
225#423C2×D6111yescharacteristic
226C2×#100—1221no—
227#407—1222no—
228#397—612no—
229#392—612no—
230#386—1222no—
231#382C2×D6111yescharacteristic
232#380—1221no—
233#343—1221no—
234D4⋊C8—1222no—
235D4⋊C8—1222no—
236C61#948911yescharacteristic, complemented, Sylow
237C22×D4—611no—
238C22×D4—611no—
239C22×D4—1222no—
240C23×C4—612no—
241C23×C4—1221no—
242C2×M16—21no—
243C2×M16—1224no—
244C2×M16—1221no—
245#121—2442no—
246#121—2442no—
247C4×D4—1221no—
248C4×D4—1221no—
249C4×D4—1221no—
250C4×D4—1221no—
251C4×D4—1221no—
252C4×D4—1221no—
253C4×D4—1221no—
254C4×D4—1221no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—611no—
257C2×C4⋊C4—611no—
258C2×C22⋊C4—612no—
259C2×C22⋊C4—1222no—
260C2×C22⋊C4—1222no—
261C2×C22⋊C4—1221no—
262C2×C42C22×D6111yescharacteristic
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—1221no—
267C4⋊C8—1221no—
268C4⋊C8—1221no—
269#100—2442no—
270C22⋊C8—21no—
271C22⋊C8—1224no—
272C22⋊C8—1221no—
273C4×C8—612no—
274C4×C8D24412yescomplemented
275#95—21no—
276#95—1222no—
277#95C4×D6112yes—
278C24—612no—
279C24—1221no—
280C2×D4—1221no—
281C2×D4—1221no—
282C2×D4—1221no—
283C2×D4—1221no—
284C2×D4—1221no—
285C2×D4—1221no—
286C2×D4—1221no—
287C2×D4—1221no—
288C2×D4—4882no—
289C2×D4—2442no—
290C2×D4—2444no—
291C2×D4—2444no—
292C22×C4—22no—
293C22×C4—612no—
294C22×C4—1222no—
295C22×C4—1222no—
296C22×C4—1221no—
297C22×C4—4882no—
298C22×C4—2441no—
299C22×C4—2441no—
300C22×C4—2441no—
301C22×C4—2441no—
302C22×C4—2441no—
303C22×C4—2441no—
304C22×C4C2×C4×D6111yescharacteristic
305C22×C4D4×D6111yescharacteristic
306C22×C4D4×D6111yescharacteristic
307M16—41no—
308M16—41no—
309M16—4884no—
310M16—2441no—
311M16—2441no—
312C2×C8—22no—
313C2×C8—41no—
314C2×C8—1222no—
315C2×C8—2441no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—1221no—
319C4⋊C4—1221no—
320C4⋊C4—1221no—
321C4⋊C4—1221no—
322C22⋊C4—1222no—
323C22⋊C4—1222no—
324C22⋊C4—4882no—
325C22⋊C4—2444no—
326C22⋊C4—2441no—
327C22⋊C4—2441no—
328C42C2×D24411yescharacteristic
329C42#1175822211yescharacteristic
330C42D4×D6111yescharacteristic
331C42#1175822611yescharacteristic
332C23—1222no—
333C23—1222no—
334C23—1222no—
335C23—1222no—
336C23—2441no—
337C23—2441no—
338C23—2441no—
339C23—2442no—
340C23—2442no—
341C23—2441no—
342C23—2441no—
343C23—2441no—
344C23C22⋊C4×D6111yescharacteristic
345D4—1222no—
346D4—1222no—
347D4—4888no—
348D4—2441no—
349D4—2441no—
350D4—2441no—
351D4—2441no—
352D4—2441no—
353D4—2441no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—1222no—
360C2×C4—4882no—
361C2×C4—4881no—
362C2×C4—4881no—
363C2×C4—4881no—
364C2×C4—4881no—
365C2×C4—2444no—
366C2×C4—2441no—
367C2×C4—2441no—
368C2×C4—2441no—
369C2×C4—2442no—
370C2×C4#1451973911yescharacteristic
371C2×C4C4×D24411yescharacteristic
372C2×C4#1451974711yescharacteristic
373C2×C4#1451975011yescharacteristic
374C2×C4#1451975211yescharacteristic
375C2×C4#1451976211yescharacteristic
376C2×C4#1451980111yescharacteristic
377C2×C4#1451980411yescharacteristic
378C2×C4#1451980411yescharacteristic
379C2×C4#1451980911yescharacteristic
380C8—42no—
381C8—81no—
382C8—4881no—
383C8—2442no—
384C22—21no—
385C22—21no—
386C22—4882no—
387C22—4881no—
388C22—4881no—
389C22—4881no—
390C22—4881no—
391C22—2442no—
392C22—2444no—
393C22—2442no—
394C22—2442no—
395C22—2442no—
396C22—2442no—
397C22—2442no—
398C22#1683964011yescharacteristic
399C22#1683976211yescharacteristic
400C22#1683976511yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—82no—
406C4—4881no—
407C4—2442no—
408C4#1684000911yescharacteristic
409C4#1684001111yescharacteristic
410C4#1684001411yescharacteristic
411C4#1684001611yescharacteristic
412C2—21no—
413C2—21no—
414C2—4882no—
415C2—4882no—
416C2#2144623611yescharacteristic
417C2#2144625611yescharacteristic
418C2#2144635011yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1286C2
socleC22×C61#386
prime radical#21450721C2
cosocleC2×C42×C61C23
hypercenter#1286D61
hypocenterC61#9489
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
1. #21454859C2semidirect
3. #21445808C2semidirect
4. #21444245C2semidirect
5. #21444243C2semidirect
6. #21443677C2semidirect
7. C61×#1286C2semidirect
56. C4×#92418D4semidirect
65. C4×C8×C61D4semidirect
209. #1286D61semidirect
236. C61#9489semidirect
274. C4×C8D244semidirect

Maximal subgroups

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

subgroupindex
#214548592
#214507212
#214458082
#214442452
#214442432
#214436772
C61×#12862
#948961