Finite Group Databasebeta

← #29577674

Subgroups — #29577674

Order 15616.

486 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 91 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#21454903C211yesmaximal, characteristic, complemented
2#21450800C211yesmaximal, characteristic
3#21444443C211yesmaximal, characteristic
4#21444162C211yesmaximal, characteristic, complemented
5#21443676C211yesmaximal, characteristic, complemented
6#21443674C211yesmaximal, characteristic, complemented
7C61×#1533C211yesmaximal, characteristic
8C2×#14519775C411yescharacteristic
9C2×#14519764C411yescharacteristic
10C2×C4×Dic122C2211yescharacteristic
11#16839891—22no—
12#16839890—22no—
13#16839819C2211yescharacteristic
14#16839819C2211yescharacteristic
15#16839816—21no—
16#16839812—21no—
17#16839685—21no—
18#16839685—21no—
19#16839669C411yescharacteristic
20#16839665C411yescharacteristic
21#16839650—21no—
22#16839650—21no—
23C4×C4⋊Dic61C2211yescharacteristic
24#16839502—22no—
25#16839487—21no—
26#16839487—21no—
27C2×#124×C61C2211yescharacteristic
28C61×#432C2211yescharacteristic
29C61×#432C2211yescharacteristic
30C61×#329—22no—
31C22×Dic122C2×C411yescharacteristic
32#14519775—42no—
33#14519764—42no—
34C4×Dic122—21no—
35C4×Dic122—21no—
36C4×Dic122—21no—
37C4×Dic122—21no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61C2×C411yescharacteristic
40C2×#11758196—21no—
41C2×#11758196—21no—
42C2×#11758196D411yescharacteristic
43C2×#11758196D411yescharacteristic
44C2×C22⋊Dic61—21no—
45C2×C22⋊Dic61—21no—
46C2×C22⋊Dic61—21no—
47C2×C22⋊Dic61—21no—
48C2×C4×Dic61C2×C411yescharacteristic
49C2×C4×Dic61C2×C411yescharacteristic
50C2×C4×Dic61D411yescharacteristic
51C2×C4×Dic61D411yescharacteristic
52C4⋊#92418—21no—
53C4⋊#92418—21no—
54#14519706—22no—
55#14519706—22no—
56C22⋊#92418—21no—
57C22⋊#92418—21no—
58#14519688—21no—
59#14519688D412yes—
60#14519687—22no—
61C8×Dic61—22no—
62#14519684—21no—
63#14519684—21no—
64#14519684—21no—
65C22×Q8×C61C2×C411yescharacteristic
66C22×D4×C61C2×C411yescharacteristic
67#124×C61—21no—
68#124×C61—21no—
69#124×C61—21no—
70#124×C61—21no—
71C2×C22⋊C4×C61—21no—
72C2×C22⋊C4×C61—21no—
73C2×C42×C61C2311yescharacteristic
74C4⋊C8×C61—21no—
75C4⋊C8×C61—21no—
76C22⋊C8×C61—21no—
77C22⋊C8×C61—21no—
78#97×C61—21no—
79#97×C61D412yes—
80C24×C61—21no—
81C2×Q8×C61—21no—
82C2×Q8×C61—21no—
83C2×Q8×C61—41no—
84C2×Q8×C61—41no—
85C2×D4×C61—21no—
86C2×D4×C61—21no—
87C2×D4×C61—41no—
88C2×D4×C61—41no—
89C22×C4×C61—21no—
90C22×C4×C61C22×C411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×Dic61—21no—
94C22×Dic61—21no—
95C22×Dic61C22⋊C411yescharacteristic
96C22×Dic61C22⋊C411yescharacteristic
97C22×Dic61C22⋊C411yescharacteristic
98C22×Dic61C22⋊C411yescharacteristic
99C2×Dic122—21no—
100C2×Dic122—21no—
101C2×Dic122—22no—
102C2×Dic122—41no—
103C2×C8×C61—22no—
104C2×C8×C61—41no—
105C22⋊C4×C61—41no—
106C22⋊C4×C61—41no—
107C22⋊C4×C61—41no—
108C22⋊C4×C61—41no—
109C42×C61C2×D411yescharacteristic
110C42×C61C2×D411yescharacteristic
111C42×C61#4111yescharacteristic
112C42×C61#4111yescharacteristic
113C22⋊Dic61—42no—
114C22⋊Dic61—42no—
115C22⋊Dic61—42no—
116C22⋊Dic61—42no—
117C4⋊Dic61—21no—
118C4⋊Dic61—21no—
119C4⋊Dic61—42no—
120#11758196—21no—
121#11758196—21no—
122#11758196—21no—
123#11758196—21no—
124#11758196—42no—
125#11758196—42no—
126C4×Dic61—21no—
127C4×Dic61—21no—
128C4×Dic61—21no—
129C4×Dic61—21no—
130C4×Dic61—41no—
131C4×Dic61—41no—
132C2×#92418—22no—
133C2×#92418—41no—
134C23×C61—41no—
135C23×C61—41no—
136C23×C61—41no—
137C23×C61—41no—
138C23×C61—41no—
139C23×C61—41no—
140C23×C61—41no—
141C23×C61C2×C22⋊C411yescharacteristic
142Q8×C61—81no—
143Q8×C61—81no—
144D4×C61—81no—
145D4×C61—81no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61C4×D411yescharacteristic
154C2×C4×C61C4×D411yescharacteristic
155C2×C4×C61#12011yescharacteristic
156C2×C4×C61#12111yescharacteristic
157C2×C4×C61#12111yescharacteristic
158C2×C4×C61#12311yescharacteristic
159C2×Dic61—21no—
160C2×Dic61—21no—
161C2×Dic61—22no—
162C2×Dic61—21no—
163C2×Dic61—21no—
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61—21no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61#10012yes—
175Dic122—42no—
176Dic122—42no—
177C8×C61—42no—
178C8×C61—81no—
179#92418—42no—
180#92418—81no—
181#9448—611nomaximal, Sylow
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—81no—
185C22×C61—81no—
186C22×C61—81no—
187C22×C61—81no—
188C22×C61—81no—
189C22×C61—81no—
190C22×C61—81no—
191C22×C61#38611yescharacteristic
192C22×C61C2×#10011yescharacteristic
193C22×C61#41311yescharacteristic
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—81no—
201Dic61—22no—
202Dic61—42no—
203Dic61—41no—
204Dic61—41no—
205Dic61—41no—
206Dic61—81no—
207#2983—611no—
208#2545—611no—
209#1533—611no—
210#1533D6111yescharacteristic
211#1467—611no—
212#1285—611no—
213#1285—611no—
214C2×C61—21no—
215C2×C61—21no—
216C2×C61—161no—
217C2×C61#188411yescharacteristic
218C2×C61#190011yescharacteristic
219C2×C61#191211yescharacteristic
220C2×C4⋊Q8—611no—
221C2×#124—611no—
222C2×#124C2×D6111yescharacteristic
223C2×C4×Q8—611no—
224#446—1222no—
225#446—1222no—
226#432—611no—
227#432—611no—
228#432—1221no—
229#432C2×D6111yescharacteristic
230#432C2×D6111yescharacteristic
231#431—1221no—
232#396—1221no—
233#396—1221no—
234#391—611no—
235#390—611no—
236#388—1221no—
237#388—1221no—
238C4×C4⋊C4—611no—
239#329—22no—
240#329—1222no—
241#324—1221no—
242#324—1221no—
243C61#944811yescharacteristic, complemented, Sylow
244C22×Q8—611no—
245C22×Q8C4×D6111yescharacteristic
246C22×D4C4×D6111yescharacteristic
247C4⋊Q8—2442no—
248#124—21no—
249#124—21no—
250#124—21no—
251#124—21no—
252#124—2442no—
253C4×Q8—1221no—
254C4×Q8—1221no—
255C4×Q8—1221no—
256C4×Q8—1221no—
257C2×C4⋊C4—611no—
258C2×C4⋊C4—611no—
259C2×C4⋊C4—611no—
260C2×C4⋊C4—1221no—
261C2×C4⋊C4—1221no—
262C2×C4⋊C4—1221no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—1221no—
266C2×C22⋊C4—1221no—
267C2×C22⋊C4—1221no—
268C2×C22⋊C4—1221no—
269C2×C42—611no—
270C2×C42—611no—
271C2×C42—611no—
272C2×C42—611no—
273C2×C42C22×D6111yescharacteristic
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—1222no—
277C4⋊C8—1222no—
278C4⋊C8—1221no—
279C4⋊C8—1221no—
280C22⋊C8—21no—
281C22⋊C8—21no—
282C22⋊C8—1221no—
283C22⋊C8—1221no—
284#97—21no—
285#97—612no—
286#97—1222no—
287#97—1221no—
288#97C22⋊D6112yes—
289C4×C8—1222no—
290#95—1221no—
291#95—1221no—
292#95—1221no—
293C24—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—1222no—
299C2×Q8—1221no—
300C2×Q8—1221no—
301C2×Q8—2441no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—41no—
306C22×C4—21no—
307C22×C4—611no—
308C22×C4—611no—
309C22×C4—611no—
310C22×C4—611no—
311C22×C4—1221no—
312C22×C4—1221no—
313C22×C4C2×C4×D6111yescharacteristic
314C22×C4D4×D6111yescharacteristic
315C22×C4D4×D6111yescharacteristic
316C2×C8—22no—
317C2×C8—41no—
318C2×C8—1222no—
319C2×C8—2441no—
320C4⋊C4—1221no—
321C4⋊C4—1221no—
322C4⋊C4—1221no—
323C4⋊C4—1221no—
324C4⋊C4—1221no—
325C4⋊C4—1221no—
326C4⋊C4—2442no—
327C4⋊C4—2442no—
328C4⋊C4—2442no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—2442no—
334C22⋊C4—2442no—
335C22⋊C4—2442no—
336C22⋊C4—2442no—
337C42—1221no—
338C42—1221no—
339C42—1221no—
340C42—1221no—
341C42—2441no—
342C42—2441no—
343C42#1175822211yescharacteristic
344C42D4×D6111yescharacteristic
345C42#1175822411yescharacteristic
346C42C2×C22⋊D6111yescharacteristic
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23C22⋊C4×D6111yescharacteristic
355Q8—81no—
356Q8—81no—
357Q8—2442no—
358Q8—2442no—
359D4—81no—
360D4—81no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—612no—
369C2×C4—1221no—
370C2×C4—1221no—
371C2×C4—1222no—
372C2×C4—1221no—
373C2×C4—1221no—
374C2×C4—1221no—
375C2×C4—1221no—
376C2×C4—1221no—
377C2×C4—2441no—
378C2×C4—2441no—
379C2×C4—2441no—
380C2×C4—2441no—
381C2×C4—2441no—
382C2×C4—2441no—
383C2×C4—2441no—
384C2×C4#1451973711yescharacteristic
385C2×C4C4×C22⋊D6111yescharacteristic
386C2×C4#1451974511yescharacteristic
387C2×C4#1451974811yescharacteristic
388C2×C4#1451975011yescharacteristic
389C2×C4#1451975911yescharacteristic
390C8—42no—
391C8—81no—
392C8—4881no—
393C8—2442no—
394C22—21no—
395C22—21no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22#1683963811yescharacteristic
404C22#100×D6111yescharacteristic
405C22#1683974311yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—81no—
413C4—1222no—
414C4—4881no—
415C4—2442no—
416C4—2441no—
417C4—2441no—
418C4—2441no—
419C2—21no—
420C2—21no—
421C2—161no—
422C2#2144619911yescharacteristic
423C2#2144627911yescharacteristic
424C2#2144635311yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1533C2
socleC22×C61#386
prime radicalC22×D4×C61C2×C4
cosocleC2×C42×C61C23
hypercenter#1533D61
hypocenterC61#9448
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1533

Complements

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

NH ≅ G/Ntype
1. #21454903C2semidirect
4. #21444162C2semidirect
5. #21443676C2semidirect
6. #21443674C2semidirect
243. C61#9448semidirect

Maximal subgroups

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

subgroupindex
#214549032
#214508002
#214444432
#214441622
#214436762
#214436742
C61×#15332
#944861