Finite Group Databasebeta

← #29577764

Subgroups — #29577764

Order 15616.

726 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 97 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#21449478C211yesmaximal, characteristic
2#21448404C211yesmaximal, characteristic, complemented
3#21445742C211yesmaximal, characteristic, complemented
4#21444348C211yesmaximal, characteristic
5#21443673C211yesmaximal, characteristic, complemented
6#21443671C211yesmaximal, characteristic, complemented
7C61×#1510C211yesmaximal, characteristic
8C23×Dic122C2211yescharacteristic
9C2×#14519764C412yescomplemented
10C2×#14519733—21no—
11C2×#14519695—22no—
12#16839726C2211yescharacteristic
13#16839674—22no—
14#16839667C412yescomplemented
15#16839666—22no—
16#16839666—22no—
17#16839642—21no—
18#16839642—21no—
19#16839625C2211yescharacteristic
20#16839623—21no—
21C4×C22⋊Dic61C2211yescharacteristic
22#16839496—22no—
23C2×#121×C61C2211yescharacteristic
24C61×#407C2211yescharacteristic
25C61×#381C2211yescharacteristic
26C61×#327—22no—
27C22×Dic122—21no—
28C22×Dic122—21no—
29C22×Dic122—21no—
30C22×Dic122—21no—
31C22×Dic122—44no—
32C22×Dic122C2×C412yes—
33C22×Dic122D412yes—
34C23×Dic61D411yescharacteristic
35C23×Dic61D411yescharacteristic
36C2×#11758194—21no—
37#14519764—44no—
38#14519733—44no—
39C2×#11758196—22no—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61—22no—
43C2×C22⋊Dic61—22no—
44C2×C4×Dic61—22no—
45C2×C4×Dic61C2×C412yes—
46#14519695—44no—
47C22⋊#92418—21no—
48C22⋊#92418—21no—
49#14519684—24no—
50#14519684—21no—
51#14519684—21no—
52#14519684—22no—
53#14519684—22no—
54#14519684D412yes—
55C22×D4×C61—21no—
56C22×D4×C61C2×C411yescharacteristic
57C23×C4×C61C2311yescharacteristic
58C2×M16×C61—21no—
59#121×C61—21no—
60#121×C61—21no—
61#121×C61—21no—
62#121×C61—21no—
63C2×C4⋊C4×C61C2×C411yescharacteristic
64C2×C22⋊C4×C61—21no—
65C2×C22⋊C4×C61—21no—
66C22⋊C8×C61—21no—
67C22⋊C8×C61—21no—
68#95×C61—21no—
69#95×C61D412yes—
70C24×C61—21no—
71C24×C61C2×D411yescharacteristic
72C2×D4×C61—21no—
73C2×D4×C61—21no—
74C2×D4×C61—42no—
75C2×D4×C61—44no—
76C2×D4×C61—41no—
77C2×D4×C61—41no—
78C22×C4×C61—21no—
79C22×C4×C61—22no—
80C22×C4×C61—42no—
81C22×C4×C61C22×C411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61#4111yescharacteristic
86C22×C4×C61#4111yescharacteristic
87C22×Dic61—21no—
88C22×Dic61—21no—
89C22×Dic61—24no—
90C22×Dic61—21no—
91C22×Dic61—21no—
92C22×Dic61—22no—
93C22×Dic61—22no—
94C22×Dic61—42no—
95C22×Dic61C22⋊C412yes—
96C22×Dic61C22⋊C412yes—
97C2×Dic122—22no—
98C2×Dic122—22no—
99C2×Dic122—22no—
100C2×Dic122—22no—
101C2×Dic122—24no—
102C2×Dic122—24no—
103C2×Dic122—42no—
104C2×Dic122—42no—
105C2×Dic122—42no—
106C2×Dic122—42no—
107C2×Dic122—42no—
108C2×Dic122—42no—
109C2×Dic122—44no—
110C2×Dic122—44no—
111M16×C61—42no—
112C2×C8×C61—41no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61—21no—
115C22⋊C4×C61—42no—
116C22⋊C4×C61—41no—
117C22⋊C4×C61—41no—
118C22⋊Dic61—44no—
119C22⋊Dic61—42no—
120C22⋊Dic61—42no—
121C22⋊Dic61—44no—
122#11758196—44no—
123C4×Dic61—44no—
124C4×Dic61—42no—
125#11758194—42no—
126C2×#92418—41no—
127C23×C61—22no—
128C23×C61—22no—
129C23×C61—41no—
130C23×C61—42no—
131C23×C61—42no—
132C23×C61—41no—
133C23×C61—42no—
134C23×C61C2×C22⋊C411yescharacteristic
135C23×C61#12011yescharacteristic
136C23×C61#12311yescharacteristic
137D4×C61—81no—
138D4×C61—81no—
139D4×C61—82no—
140D4×C61—82no—
141C2×C4×C61—22no—
142C2×C4×C61—22no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—42no—
148C2×C4×C61—42no—
149C2×C4×C61—42no—
150C2×C4×C61—41no—
151C2×C4×C61—42no—
152C2×C4×C61C4×D411yescharacteristic
153C2×C4×C61C4×D411yescharacteristic
154C2×C4×C61#12111yescharacteristic
155C2×C4×C61#12111yescharacteristic
156C2×Dic61—22no—
157C2×Dic61—22no—
158C2×Dic61—24no—
159C2×Dic61—22no—
160C2×Dic61—22no—
161C2×Dic61—22no—
162C2×Dic61—22no—
163C2×Dic61—22no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—42no—
167C2×Dic61—42no—
168C2×Dic61—42no—
169C2×Dic61—42no—
170C2×Dic61—42no—
171C2×Dic61—42no—
172C2×Dic61—42no—
173C2×Dic61#9914yes—
174Dic122—44no—
175Dic122—44no—
176Dic122—44no—
177Dic122—44no—
178C8×C61—81no—
179#92418—81no—
180#9466—611nomaximal, Sylow
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—44no—
189C22×C61—84no—
190C22×C61—81no—
191C22×C61—81no—
192C22×C61—81no—
193C22×C61#38611yescharacteristic
194C22×C61C2×#9912yes—
195C22×C61#45412yes—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—82no—
201C4×C61—81no—
202Dic61—24no—
203Dic61—42no—
204Dic61—82no—
205Dic61—82no—
206#2436—611no—
207#2310—611no—
208#1784—611no—
209#1510—611no—
210#1510D6111yescharacteristic
211#1284—611no—
212#1284—611no—
213C2×C61—22no—
214C2×C61—22no—
215C2×C61—41no—
216C2×C61—161no—
217C2×C61#188811yescharacteristic
218C2×C61#189811yescharacteristic
219C2×C61#191211yescharacteristic
220C23×Q8—611no—
221C2×#124—612no—
222C2×#121C2×D6111yescharacteristic
223C2×#117—1221no—
224C2×#101—1222no—
225#407—611no—
226#407C2×D6111yescharacteristic
227#393—1222no—
228#391—612no—
229#391—1222no—
230#391—1222no—
231#386—1221no—
232#386—1221no—
233#381—611no—
234#381—1221no—
235#381C2×D6111yescharacteristic
236C4×C22⋊C4—611no—
237#327—22no—
238#327—1222no—
239C61#946611yescharacteristic, complemented, Sylow
240C22×Q8—612no—
241C22×Q8—612no—
242C22×Q8—1221no—
243C22×Q8—1221no—
244C22×Q8—1221no—
245C22×Q8—1221no—
246C22×Q8—2444no—
247C22×D4—21no—
248C22×D4C4×D6111yescharacteristic
249C23×C4—611no—
250C23×C4—611no—
251C23×C4C22×D6111yescharacteristic
252C2×M16—21no—
253C2×M16—1221no—
254#124—2444no—
255#121—21no—
256#121—21no—
257#121—21no—
258#121—21no—
259#117—2444no—
260C2×C4⋊C4—1222no—
261C2×C4⋊C4C4×D6111yescharacteristic
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—1222no—
265C2×C22⋊C4—1222no—
266C2×C22⋊C4—1221no—
267C2×C22⋊C4—1221no—
268C2×C42—612no—
269C2×C42—1222no—
270#101—2444no—
271C22⋊C8—21no—
272C22⋊C8—21no—
273C22⋊C8—1221no—
274C22⋊C8—1221no—
275#95—21no—
276#95—612no—
277#95—1222no—
278#95—1222no—
279#95—1224no—
280#95—1221no—
281#95—1221no—
282#95C22⋊D6112yes—
283C24—21no—
284C24D4×D6111yescharacteristic
285C2×Q8—1224no—
286C2×Q8—1224no—
287C2×Q8—1222no—
288C2×Q8—1222no—
289C2×Q8—1222no—
290C2×Q8—1222no—
291C2×Q8—2442no—
292C2×Q8—2442no—
293C2×Q8—2444no—
294C2×Q8—2442no—
295C2×Q8—2442no—
296C2×Q8—2444no—
297C2×Q8—2442no—
298C2×Q8—2442no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—42no—
302C2×D4—44no—
303C2×D4—41no—
304C2×D4—41no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—42no—
308C22×C4—612no—
309C22×C4—612no—
310C22×C4—1224no—
311C22×C4—1221no—
312C22×C4—1221no—
313C22×C4—1221no—
314C22×C4—1221no—
315C22×C4—1222no—
316C22×C4—1222no—
317C22×C4—2442no—
318C22×C4C2×C4×D6111yescharacteristic
319C22×C4#1175822211yescharacteristic
320C22×C4D4×D6111yescharacteristic
321C22×C4D4×D6111yescharacteristic
322C22×C4#1175822411yescharacteristic
323C22×C4C2×C22⋊D6111yescharacteristic
324M16—42no—
325M16—2442no—
326C2×C8—41no—
327C2×C8—2441no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—2444no—
331C22⋊C4—42no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—2444no—
335C22⋊C4—2444no—
336C22⋊C4—2442no—
337C22⋊C4—2442no—
338C42—2442no—
339C42—2444no—
340C23—22no—
341C23—22no—
342C23—41no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—41no—
347C23C22⋊C4×D6111yescharacteristic
348C23#1451974511yescharacteristic
349C23#1451975911yescharacteristic
350Q8—2444no—
351Q8—2444no—
352Q8—2444no—
353Q8—2444no—
354D4—81no—
355D4—81no—
356D4—82no—
357D4—82no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—614no—
370C2×C4—1222no—
371C2×C4—1222no—
372C2×C4—1224no—
373C2×C4—1222no—
374C2×C4—1222no—
375C2×C4—1222no—
376C2×C4—1222no—
377C2×C4—1222no—
378C2×C4—2442no—
379C2×C4—2442no—
380C2×C4—2442no—
381C2×C4—2442no—
382C2×C4—2442no—
383C2×C4—2442no—
384C2×C4—2441no—
385C2×C4—2441no—
386C2×C4—2442no—
387C2×C4#1451973711yescharacteristic
388C2×C4C4×C22⋊D6111yescharacteristic
389C2×C4#1451974811yescharacteristic
390C2×C4#1451975011yescharacteristic
391C8—81no—
392C8—4881no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—44no—
401C22—84no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22#1683963811yescharacteristic
406C22#99×D6112yes—
407C22#1683993312yes—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—82no—
413C4—81no—
414C4—1224no—
415C4—4882no—
416C4—4882no—
417C4—2442no—
418C2—22no—
419C2—22no—
420C2—41no—
421C2—161no—
422C2#2144621911yescharacteristic
423C2#2144626911yescharacteristic
424C2#2144635311yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1510C2
socleC22×C61#386
prime radicalC2×#121×C61C22
cosocleC23×C4×C61C23
hypercenter#1510D61
hypocenterC61#9466
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1510

Complements

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

NH ≅ G/Ntype
2. #21448404C2semidirect
3. #21445742C2semidirect
5. #21443673C2semidirect
6. #21443671C2semidirect
9. C2×#14519764C4semidirect
14. #16839667C4semidirect
239. C61#9466semidirect

Maximal subgroups

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

subgroupindex
#214494782
#214484042
#214457422
#214443482
#214436732
#214436712
C61×#15102
#946661