Finite Group Databasebeta

← #29577834

Subgroups — #29577834

Order 15616.

726 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 93 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#21449150C211yesmaximal, characteristic
2#21448419C211yesmaximal, characteristic, complemented
3#21445763C211yesmaximal, characteristic
4#21444148C211yesmaximal, characteristic, complemented
5#21443673C211yesmaximal, characteristic, complemented
6#21443672C211yesmaximal, characteristic, complemented
7C61×#1789C211yesmaximal, characteristic
8C2×#14519758—21no—
9C22×#11758196C2211yescharacteristic
10C2×#14519704—22no—
11#16839726C2211yescharacteristic
12#16839681C412yescomplemented
13#16839676C412yescomplemented
14#16839646—22no—
15#16839644—22no—
16#16839642—21no—
17#16839635—22no—
18#16839625C2211yescharacteristic
19#16839623—21no—
20#16839620—21no—
21C4×C22⋊Dic61C2211yescharacteristic
22#16839487—22no—
23C22×#41×C61C2211yescharacteristic
24C2×#104×C61—22no—
25C61×#407C2211yescharacteristic
26C4×C22⋊C4×C61C2211yescharacteristic
27C23×Dic61D411yescharacteristic
28C23×Dic61D411yescharacteristic
29C2×#11758194—21no—
30#14519758—44no—
31C2×#11758196—21no—
32C2×#11758196—21no—
33C2×#11758196—21no—
34C2×#11758196—21no—
35C2×#11758196—22no—
36C2×#11758196—22no—
37C2×#11758196C2×C412yes—
38C2×#11758196D412yes—
39C2×C22⋊Dic61—21no—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—22no—
42C2×C22⋊Dic61—22no—
43C2×C4×Dic61D412yes—
44#14519704—44no—
45C22⋊#92418—21no—
46C22⋊#92418—21no—
47#14519684—22no—
48#14519684—24no—
49#14519684—21no—
50#14519684—21no—
51#14519684—22no—
52#14519684C2×C412yes—
53C2×#41×C61—21no—
54C2×#41×C61—21no—
55C2×#41×C61—21no—
56C2×#41×C61—21no—
57C2×#41×C61—44no—
58C22×Q8×C61C2×C411yescharacteristic
59C22×D4×C61—21no—
60C22×D4×C61C2×C411yescharacteristic
61C23×C4×C61—21no—
62C23×C4×C61C2311yescharacteristic
63C2×M16×C61—21no—
64C2×C22⋊C4×C61—21no—
65C2×C42×C61D412yes—
66#104×C61—44no—
67C22⋊C8×C61—21no—
68C22⋊C8×C61—21no—
69#95×C61—21no—
70C24×C61—21no—
71C24×C61C2×D411yescharacteristic
72#41×C61—24no—
73#41×C61—24no—
74#41×C61—42no—
75#41×C61—42no—
76#41×C61—42no—
77#41×C61—42no—
78#41×C61—42no—
79#41×C61—42no—
80C2×Q8×C61—21no—
81C2×Q8×C61—21no—
82C2×Q8×C61—42no—
83C2×D4×C61—24no—
84C2×D4×C61—21no—
85C2×D4×C61—21no—
86C2×D4×C61—42no—
87C2×D4×C61—42no—
88C2×D4×C61—41no—
89C2×D4×C61—41no—
90C22×C4×C61—21no—
91C22×C4×C61—22no—
92C22×C4×C61—42no—
93C22×C4×C61—42no—
94C22×C4×C61—42no—
95C22×C4×C61—41no—
96C22×C4×C61—41no—
97C22×C4×C61C22×C411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61#4111yescharacteristic
102C22×C4×C61#4111yescharacteristic
103C22×Dic61—21no—
104C22×Dic61—21no—
105C22×Dic61—21no—
106C22×Dic61—21no—
107C22×Dic61—22no—
108C22×Dic61—22no—
109C22×Dic61—42no—
110C22×Dic61—42no—
111C22×Dic61C22⋊C412yes—
112C22×Dic61C22⋊C412yes—
113M16×C61—42no—
114C2×C8×C61—41no—
115C22⋊C4×C61—42no—
116C42×C61—42no—
117C22⋊Dic61—44no—
118C22⋊Dic61—42no—
119C22⋊Dic61—44no—
120C22⋊Dic61—42no—
121#11758196—22no—
122#11758196—22no—
123#11758196—22no—
124#11758196—22no—
125#11758196—44no—
126#11758196—82no—
127C4×Dic61—42no—
128#11758194—42no—
129C2×#92418—41no—
130C23×C61—22no—
131C23×C61—22no—
132C23×C61—22no—
133C23×C61—41no—
134C23×C61—41no—
135C23×C61—42no—
136C23×C61—41no—
137C23×C61—42no—
138C23×C61C2×C22⋊C411yescharacteristic
139C23×C61#12011yescharacteristic
140C23×C61#12311yescharacteristic
141Q8×C61—22no—
142Q8×C61—22no—
143Q8×C61—41no—
144Q8×C61—41no—
145D4×C61—22no—
146D4×C61—22no—
147D4×C61—41no—
148D4×C61—41no—
149D4×C61—44no—
150D4×C61—44no—
151C2×C4×C61—22no—
152C2×C4×C61—22no—
153C2×C4×C61—22no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—44no—
159C2×C4×C61—44no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—42no—
163C2×C4×C61—42no—
164C2×C4×C61—42no—
165C2×C4×C61C4×D411yescharacteristic
166C2×C4×C61C4×D411yescharacteristic
167C2×C4×C61#12111yescharacteristic
168C2×C4×C61#12111yescharacteristic
169C2×Dic61—22no—
170C2×Dic61—22no—
171C2×Dic61—22no—
172C2×Dic61—22no—
173C2×Dic61—42no—
174C2×Dic61—42no—
175C2×Dic61—42no—
176C2×Dic61—42no—
177C2×Dic61—42no—
178C2×Dic61—42no—
179C2×Dic61—42no—
180C2×Dic61—42no—
181C2×Dic61—42no—
182C2×Dic61—42no—
183C2×Dic61—81no—
184C8×C61—81no—
185#92418—81no—
186#9480—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—44no—
195C22×C61—44no—
196C22×C61—42no—
197C22×C61—82no—
198C22×C61—81no—
199C22×C61—81no—
200C22×C61#38611yescharacteristic
201C22×C61#42112yes—
202C22×C61B(4,𝔽2)12yes—
203C4×C61—21no—
204C4×C61—21no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—42no—
208C4×C61—82no—
209Dic61—42no—
210Dic61—42no—
211Dic61—82no—
212Dic61—82no—
213#2406—611no—
214#2313—611no—
215#1789—611no—
216#1789D6111yescharacteristic
217#1463—611no—
218#1284—611no—
219#1284—611no—
220C2×C61—22no—
221C2×C61—22no—
222C2×C61—41no—
223C2×C61—82no—
224C2×C61#188911yescharacteristic
225C2×C61#189711yescharacteristic
226C2×C61#191211yescharacteristic
227C22×#41C2×D6111yescharacteristic
228C2×#123—1221no—
229C22×C4⋊C4—611no—
230C2×#104—22no—
231C2×#104—1222no—
232#407—611no—
233#407C2×D6111yescharacteristic
234#395—612no—
235#394—612no—
236#387—1222no—
237#387—1222no—
238#386—1221no—
239#385—1222no—
240#381—611no—
241#381—1221no—
242#380—1221no—
243C4×C22⋊C4—611no—
244C4×C22⋊C4C2×D6111yescharacteristic
245#324—1222no—
246C61#948011yescharacteristic, complemented, Sylow
247C2×#41—21no—
248C2×#41—21no—
249C2×#41—21no—
250C2×#41—21no—
251C2×#41—44no—
252C22×Q8C4×D6111yescharacteristic
253C22×D4—21no—
254C22×D4C4×D6111yescharacteristic
255C23×C4—21no—
256C23×C4—611no—
257C23×C4—611no—
258C23×C4C22×D6111yescharacteristic
259C2×M16—21no—
260C2×M16—1221no—
261#123—2444no—
262C2×C4⋊C4—612no—
263C2×C4⋊C4—612no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4—1221no—
266C2×C4⋊C4—1221no—
267C2×C4⋊C4—1221no—
268C2×C4⋊C4—1222no—
269C2×C4⋊C4—1222no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—1222no—
272C2×C22⋊C4—1222no—
273C2×C22⋊C4—1221no—
274C2×C22⋊C4—1221no—
275C2×C42—612no—
276C2×C42C22⋊D6112yes—
277#104—44no—
278#104—2444no—
279C22⋊C8—21no—
280C22⋊C8—21no—
281C22⋊C8—1221no—
282C22⋊C8—1221no—
283#95—21no—
284#95—612no—
285#95—1222no—
286#95—1222no—
287#95—1224no—
288#95—1221no—
289#95—1221no—
290C24—21no—
291C24D4×D6111yescharacteristic
292#41—24no—
293#41—24no—
294#41—42no—
295#41—42no—
296#41—42no—
297#41—42no—
298#41—42no—
299#41—42no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—42no—
303C2×D4—24no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—41no—
309C2×D4—41no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—612no—
318C22×C4—612no—
319C22×C4—1221no—
320C22×C4—1221no—
321C22×C4—1221no—
322C22×C4—1221no—
323C22×C4—1222no—
324C22×C4—1222no—
325C22×C4—2442no—
326C22×C4—2442no—
327C22×C4C2×C4×D6111yescharacteristic
328C22×C4#1175822211yescharacteristic
329C22×C4D4×D6111yescharacteristic
330C22×C4D4×D6111yescharacteristic
331C22×C4#1175822411yescharacteristic
332C22×C4C2×C22⋊D6111yescharacteristic
333M16—42no—
334M16—2442no—
335C2×C8—41no—
336C2×C8—2441no—
337C4⋊C4—1222no—
338C4⋊C4—1222no—
339C4⋊C4—1222no—
340C4⋊C4—1222no—
341C4⋊C4—4882no—
342C4⋊C4—2444no—
343C22⋊C4—42no—
344C22⋊C4—2444no—
345C22⋊C4—2442no—
346C22⋊C4—2444no—
347C22⋊C4—2442no—
348C42—42no—
349C42—2442no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—41no—
354C23—41no—
355C23—42no—
356C23—42no—
357C23—41no—
358C23C22⋊C4×D6111yescharacteristic
359C23#1451974511yescharacteristic
360C23#1451975911yescharacteristic
361Q8—22no—
362Q8—22no—
363Q8—41no—
364Q8—41no—
365D4—22no—
366D4—22no—
367D4—41no—
368D4—41no—
369D4—44no—
370D4—44no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—1222no—
386C2×C4—1222no—
387C2×C4—1222no—
388C2×C4—1222no—
389C2×C4—4881no—
390C2×C4—2442no—
391C2×C4—2442no—
392C2×C4—2442no—
393C2×C4—2442no—
394C2×C4—2442no—
395C2×C4—2442no—
396C2×C4—2442no—
397C2×C4—2442no—
398C2×C4—2442no—
399C2×C4—2442no—
400C2×C4#1451973711yescharacteristic
401C2×C4C4×C22⋊D6111yescharacteristic
402C2×C4#1451974811yescharacteristic
403C2×C4#1451975011yescharacteristic
404C8—81no—
405C8—4881no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—44no—
414C22—44no—
415C22—42no—
416C22—82no—
417C22—81no—
418C22—81no—
419C22#1683963811yescharacteristic
420C22#1683977712yes—
421C22#1683994812yes—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—42no—
427C4—82no—
428C4—4882no—
429C4—4882no—
430C4—2442no—
431C4—2442no—
432C2—22no—
433C2—22no—
434C2—41no—
435C2—82no—
436C2#2144622411yescharacteristic
437C2#2144626411yescharacteristic
438C2#2144635311yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1789C2
socleC22×C61#386
prime radicalC22×#41×C61C22
cosocleC23×C4×C61C23
hypercenter#1789D61
hypocenterC61#9480
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1789

Complements

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

NH ≅ G/Ntype
2. #21448419C2semidirect
4. #21444148C2semidirect
5. #21443673C2semidirect
6. #21443672C2semidirect
12. #16839681C4semidirect
13. #16839676C4semidirect
246. C61#9480semidirect

Maximal subgroups

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

subgroupindex
#214491502
#214484192
#214457632
#214441482
#214436732
#214436722
C61×#17892
#948061