Finite Group Databasebeta

← #29577879

Subgroups — #29577879

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#21450720C211yesmaximal, characteristic
3#21445808C211yesmaximal, characteristic, complemented
4#21444244C211yesmaximal, characteristic, complemented
5#21443679C211yesmaximal, characteristic, complemented
6#21443678C211yesmaximal, characteristic, complemented
7C61×#1486C211yesmaximal, characteristic, complemented
8C2×#14519751—21no—
9C2×C4×D244C2211yescharacteristic
10#16839841—22no—
11#16839839—22no—
12#16839815C2211yescharacteristic
13#16839786C2211yescharacteristic
14C2×#14519693—21no—
15#16839724—22no—
16#16839688C412yes—
17#16839670C412yes—
18#16839638—22no—
19#16839628C2211yescharacteristic
20#16839622—21no—
21#16839530—21no—
22D4⋊#92418—22no—
23C2×C4×D4×C61C2211yescharacteristic
24C61×#431C2211yescharacteristic
25C61×#423C2211yescharacteristic
26C61×D4⋊C8—22no—
27C22×D244D411yescharacteristic
28C22×C22⋊D61—22no—
29C22×C4×D61D412yes—
30C2×#11758194—21no—
31C2×#11758189—24no—
32#14519751—42no—
33#14519751—42no—
34C4×D244—21no—
35C4×D244—21no—
36C4×D244—21no—
37C4×D244—21no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61D411yescharacteristic
40C2×C22⋊Dic61—22no—
41C2×#11758198—22no—
42C2×#11758198C2×C412yes—
43C4⋊#92418—21no—
44C4⋊#92418—21no—
45C4⋊#92418—21no—
46#14519693—42no—
47C22⋊#92418—21no—
48#14519689—24no—
49C4×#92418D412yescomplemented
50#14519684—21no—
51#14519684—22no—
52#14519684C2×C412yes—
53C22×D4×C61C2×C411yescharacteristic
54C23×C4×C61—21no—
55C2×M16×C61—21no—
56C4×D4×C61—21no—
57C4×D4×C61—21no—
58C4×D4×C61—21no—
59C4×D4×C61—21no—
60C2×C4⋊C4×C61C2×C411yescharacteristic
61C2×C22⋊C4×C61—21no—
62C2×C42×C61C2311yescharacteristic
63C4⋊C8×C61—21no—
64C4⋊C8×C61—21no—
65C4⋊C8×C61—21no—
66C22⋊C8×C61—21no—
67C4×C8×C61D412yescomplemented
68C24×C61—21no—
69C23×D61C22⋊C412yes—
70C2×D4×C61—21no—
71C2×D4×C61—21no—
72C2×D4×C61—42no—
73C22×C4×C61—21no—
74C22×C4×C61—41no—
75C22×C4×C61—41no—
76C22×C4×C61—41no—
77C22×C4×C61—41no—
78C22×C4×C61—41no—
79C22×C4×C61—41no—
80C22×C4×C61C22×C411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C2×C22⋊D61—44no—
84C2×C22⋊D61—44no—
85C2×C22⋊D61—82no—
86C22×Dic61—22no—
87C22×Dic61C22⋊C412yes—
88C2×D244—21no—
89C2×D244—21no—
90C2×D244—21no—
91C2×D244—21no—
92C2×D244—21no—
93C2×D244—21no—
94C2×C4×D61—22no—
95C2×C4×D61—22no—
96C2×C4×D61—82no—
97M16×C61—41no—
98M16×C61—41no—
99C2×C8×C61—22no—
100C2×C8×C61—41no—
101C4⋊C4×C61—21no—
102C4⋊C4×C61—21no—
103C22⋊C4×C61—41no—
104C22⋊C4×C61—41no—
105C42×C61C2×D411yescharacteristic
106C42×C61C2×D411yescharacteristic
107C42×C61#4111yescharacteristic
108C42×C61#4111yescharacteristic
109C22⋊Dic61—82no—
110#11758198—22no—
111#11758198—22no—
112#11758198—44no—
113C4⋊Dic61—21no—
114C4⋊Dic61—21no—
115C4⋊Dic61—41no—
116C4⋊Dic61—41no—
117#11758194—41no—
118#11758194—41no—
119C2×#92418—22no—
120C2×#92418—41no—
121#11758189—84no—
122C23×C61—22no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61—41no—
126C23×C61—41no—
127C23×C61—41no—
128C23×C61—41no—
129C23×C61C2×C22⋊C411yescharacteristic
130C22×D61—22no—
131C22×D61—22no—
132C22×D61—22no—
133C22×D61—42no—
134C22×D61—42no—
135D4×C61—22no—
136D4×C61—22no—
137D4×C61—41no—
138D4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—81no—
145C2×C4×C61—81no—
146C2×C4×C61—81no—
147C2×C4×C61—81no—
148C2×C4×C61C4×D411yescharacteristic
149C2×C4×C61C4×D411yescharacteristic
150C2×C4×C61#12011yescharacteristic
151C2×C4×C61#12111yescharacteristic
152C2×C4×C61#12111yescharacteristic
153C2×C4×C61#12311yescharacteristic
154C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C61#13711yescharacteristic
158C22⋊D61—88no—
159C2×Dic61—22no—
160C2×Dic61—42no—
161C2×Dic61—42no—
162C2×Dic61—42no—
163C2×Dic61—42no—
164D244—41no—
165D244—41no—
166D244—41no—
167D244—41no—
168C4×D61—44no—
169C4×D61—82no—
170C8×C61—42no—
171C8×C61—81no—
172#92418—42no—
173#92418—81no—
174#9489—611nomaximal, Sylow
175C22×C61—21no—
176C22×C61—21no—
177C22×C61—42no—
178C22×C61—42no—
179C22×C61—42no—
180C22×C61—81no—
181C22×C61—81no—
182C22×C61—81no—
183C22×C61—81no—
184C22×C61#38611yescharacteristic
185C22×C61#41711yescharacteristic
186C22×C61#41811yescharacteristic
187C2×D61—44no—
188C2×D61—42no—
189C2×D61—42no—
190C2×D61—42no—
191C2×D61—82no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—81no—
197C4×C61#46811yescharacteristic
198C4×C61#46811yescharacteristic
199C4×C61#46911yescharacteristic
200C4×C61#46911yescharacteristic
201Dic61—42no—
202Dic61—82no—
203#2975—611no—
204#2535—611no—
205#1800—611no—
206#1486—611no—
207#1486D6111yescharacteristic, complemented
208#1286—611no—
209#1286—611no—
210C2×C61—21no—
211C2×C61—21no—
212C2×C61—82no—
213C2×C61#189111yescharacteristic
214C2×C61#189511yescharacteristic
215C2×C61#191111yescharacteristic
216D61—82no—
217C2×#121—1221no—
218C2×C4×D4—611no—
219C2×C4×D4C2×D6111yescharacteristic
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#382—611no—
232#380—1221no—
233#343—1221no—
234D4⋊C8—22no—
235D4⋊C8—1222no—
236C61#948911yescharacteristic, complemented, Sylow
237C22×D4—611no—
238C22×D4—1222no—
239C22×D4C4×D6111yescharacteristic
240C23×C4—21no—
241C23×C4—612no—
242C2×M16—21no—
243C2×M16—1224no—
244C2×M16—1221no—
245#121—2442no—
246#121—2442no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—1221no—
252C4×D4—1221no—
253C4×D4—1221no—
254C4×D4—1221no—
255C2×C4⋊C4—611no—
256C2×C4⋊C4—1221no—
257C2×C4⋊C4C4×D6111yescharacteristic
258C2×C22⋊C4—21no—
259C2×C22⋊C4—612no—
260C2×C22⋊C4—1222no—
261C2×C22⋊C4—1222no—
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×C8C22⋊D6112yescomplemented
275#95—612no—
276#95—1222no—
277#95—1221no—
278C24—21no—
279C24—612no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—42no—
283C2×D4—1221no—
284C2×D4—1221no—
285C2×D4—1221no—
286C2×D4—1221no—
287C2×D4—1221no—
288C2×D4—1221no—
289C2×D4—4882no—
290C2×D4—2444no—
291C2×D4—2444no—
292C22×C4—21no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—612no—
300C22×C4—1222no—
301C22×C4—1222no—
302C22×C4—1222no—
303C22×C4—4882no—
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—21no—
317C4⋊C4—21no—
318C4⋊C4—1221no—
319C4⋊C4—1221no—
320C4⋊C4—2441no—
321C4⋊C4—2441no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—1222no—
325C22⋊C4—1222no—
326C22⋊C4—4882no—
327C22⋊C4—2444no—
328C42#1175822211yescharacteristic
329C42D4×D6111yescharacteristic
330C42#1175822411yescharacteristic
331C42C2×C22⋊D6111yescharacteristic
332C23—22no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—1222no—
340C23—1222no—
341C23—1222no—
342C23—2442no—
343C23—2442no—
344C23C22⋊C4×D6111yescharacteristic
345D4—22no—
346D4—22no—
347D4—41no—
348D4—41no—
349D4—4888no—
350D4—2441no—
351D4—2441no—
352D4—2441no—
353D4—2441no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—1222no—
364C2×C4—4882no—
365C2×C4—2444no—
366C2×C4—2442no—
367C2×C4—2442no—
368C2×C4—2442no—
369C2×C4—2442no—
370C2×C4#1451973711yescharacteristic
371C2×C4C4×C22⋊D6111yescharacteristic
372C2×C4#1451974511yescharacteristic
373C2×C4#1451974811yescharacteristic
374C2×C4#1451975011yescharacteristic
375C2×C4#1451975911yescharacteristic
376C2×C4#1451980111yescharacteristic
377C2×C4#1451980111yescharacteristic
378C2×C4#1451980411yescharacteristic
379C2×C4#1451980611yescharacteristic
380C8—42no—
381C8—81no—
382C8—4881no—
383C8—2442no—
384C22—21no—
385C22—21no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—4882no—
394C22—2444no—
395C22—2442no—
396C22—2442no—
397C22—2442no—
398C22#1683963811yescharacteristic
399C22#1683976011yescharacteristic
400C22#1683976511yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—81no—
406C4—4882no—
407C4—2442no—
408C4#1684000511yescharacteristic
409C4#1684000711yescharacteristic
410C4#1684001211yescharacteristic
411C4#1684001411yescharacteristic
412C2—21no—
413C2—21no—
414C2—82no—
415C2—4882no—
416C2#2144623411yescharacteristic
417C2#2144625411yescharacteristic
418C2#2144634811yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1486C2
socleC22×C61#386
prime radical#21450720C2
cosocleC2×C42×C61C23
hypercenter#1486D61
hypocenterC61#9489
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
1. #21454859C2semidirect
3. #21445808C2semidirect
4. #21444244C2semidirect
5. #21443679C2semidirect
6. #21443678C2semidirect
7. C61×#1486C2semidirect
49. C4×#92418D4semidirect
67. C4×C8×C61D4semidirect
207. #1486D61semidirect
236. C61#9489semidirect
274. C4×C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214548592
#214507202
#214458082
#214442442
#214436792
#214436782
C61×#14862
#948961