Finite Group Databasebeta

← #29580782

Subgroups — #29580782

Order 15616.

548 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 99 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#21455612C211yesmaximal, characteristic
2#21451177C211yesmaximal, characteristic
3#21444383C211yesmaximal, characteristic, complemented
4#21444248C211yesmaximal, characteristic, complemented
5#21444211C211yesmaximal, characteristic
6#21443678C211yesmaximal, characteristic, complemented
7C61×#3104C211yesmaximal, characteristic, complemented
8C2×C4×D244C2211yescharacteristic
9#16840059—22no—
10#16840049—21no—
11#16840039—21no—
12#16839815C2211yescharacteristic
13#16839805—21no—
14#16839786C2211yescharacteristic
15#16839766—21no—
16#16839697—21no—
17#16839673—21no—
18#16839664—21no—
19#16839660—21no—
20#16839633—22no—
21#16839628C2211yescharacteristic
22#16839499—22no—
23Q8⋊#92418—22no—
24#16839491—22no—
25C61×#533C2211yescharacteristic
26C2×C4×Q8×C61C2211yescharacteristic
27C61×#484—21no—
28C61×#477—21no—
29C61×#462—21no—
30C61×#451—21no—
31C4×Q16×C61—22no—
32C61×#431C2211yescharacteristic
33C22×D244D411yescharacteristic
34C22×C4×D61—21no—
35C2×#11758194—21no—
36C4×D244—21no—
37C4×D244—21no—
38C4×D244—21no—
39C4×D244—21no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61D411yescharacteristic
42C2×#11758196—22no—
43C2×#11758198—22no—
44C2×#11758198—21no—
45C2×#11758198—22no—
46#14519710—22no—
47#14519708—22no—
48C4⋊#92418—21no—
49C4⋊#92418—21no—
50C4⋊#92418D412yes—
51#14519697—22no—
52#14519697—22no—
53C22⋊#92418—21no—
54C4×#92418—21no—
55#14519684—22no—
56#14519684—21no—
57#14519684D412yes—
58C22×Q8×C61D411yescharacteristic
59C2×Q16×C61—22no—
60C4⋊Q8×C61D411yescharacteristic
61C4⋊Q8×C61D411yescharacteristic
62#125×C61—21no—
63C22⋊Q8×C61—21no—
64C22⋊Q8×C61—21no—
65C4×Q8×C61—21no—
66C4×Q8×C61—21no—
67C4×Q8×C61—21no—
68C4×Q8×C61—21no—
69C4×Q8×C61—21no—
70C4×Q8×C61—21no—
71#117×C61—21no—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61D411yescharacteristic
74C2×C42×C61—21no—
75C2×C42×C61C2311yescharacteristic
76#107×C61—22no—
77C4⋊C8×C61—21no—
78#103×C61—22no—
79#103×C61—22no—
80C22⋊C8×C61—21no—
81C4×C8×C61D412yes—
82C23×D61—21no—
83C2×Q8×C61—21no—
84C2×Q8×C61—21no—
85C2×Q8×C61—21no—
86C2×Q8×C61—21no—
87C2×Q8×C61—42no—
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×Dic61—21no—
94C22×Dic61—22no—
95C2×D244—21no—
96C2×D244—21no—
97C2×D244—21no—
98C2×D244—21no—
99C2×D244—21no—
100C2×D244Q1612yescomplemented
101C2×C4×D61—41no—
102C2×C4×D61—41no—
103C2×C4×D61—82no—
104Q16×C61—44no—
105C2×C8×C61—22no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—41no—
113C4⋊C4×C61—41no—
114C4⋊C4×C61—41no—
115C4⋊C4×C61—41no—
116C22⋊C4×C61—41no—
117C42×C61—41no—
118C42×C61—41no—
119C42×C61—41no—
120C42×C61C2×D411yescharacteristic
121C42×C61C2×D411yescharacteristic
122C42×C61C2×D411yescharacteristic
123C42×C61#4111yescharacteristic
124#11758198—44no—
125#11758198—41no—
126#11758198—41no—
127#11758198—82no—
128C4⋊Dic61—21no—
129C4⋊Dic61—21no—
130C4⋊Dic61—41no—
131C4⋊Dic61—41no—
132#11758196—82no—
133#11758194—41no—
134#11758194—41no—
135C2×#92418—22no—
136C2×#92418—41no—
137C23×C61#12011yescharacteristic
138C22×D61—22no—
139C22×D61—42no—
140C22×D61—42no—
141C22×D61—41no—
142C22×D61—41no—
143Q8×C61—22no—
144Q8×C61—22no—
145Q8×C61—41no—
146Q8×C61—41no—
147Q8×C61—42no—
148Q8×C61—81no—
149C2×C4×C61—22no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61#12011yescharacteristic
160C2×C4×C61#12011yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61#12711yescharacteristic
165C2×C4×C61C2×Q1611yescharacteristic
166C2×C4×C61#13511yescharacteristic
167C2×C4×C61#13711yescharacteristic
168C2×C4×C61#13711yescharacteristic
169C2×Dic61—42no—
170C2×Dic61—42no—
171C2×Dic61—42no—
172C2×Dic61—41no—
173C2×Dic61—42no—
174D244—22no—
175D244—22no—
176D244—41no—
177D244—41no—
178C4×D61—82no—
179C4×D61—82no—
180C8×C61—42no—
181#92418—42no—
182#92418—81no—
183#10002—611nomaximal, Sylow
184C22×C61—21no—
185C22×C61—21no—
186C22×C61#39211yescharacteristic
187C22×C61#45111yescharacteristic
188C22×C61#45211yescharacteristic
189C2×D61—44no—
190C2×D61—42no—
191C2×D61—82no—
192C2×D61—81no—
193C2×D61—81no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—42no—
199C4×C61—81no—
200C4×C61—81no—
201C4×C61—81no—
202C4×C61#45111yescharacteristic
203C4×C61#45211yescharacteristic
204C4×C61#46711yescharacteristic
205C4×C61#47011yescharacteristic
206Dic61—82no—
207Dic61—81no—
208#3104D6111yescharacteristic, complemented
209#3077—611no—
210#2580—611no—
211#1518—611no—
212#1487—611no—
213#1478—611no—
214#1286—611no—
215C2×C61—21no—
216C2×C61—21no—
217C2×C61#200111yescharacteristic
218C2×C61#200811yescharacteristic
219C2×C61#201011yescharacteristic
220D61—82no—
221#533C2×D6111yescharacteristic
222C2×C4×Q8C2×D6111yescharacteristic
223C2×C4×D4—611no—
224#484—21no—
225#478—1222no—
226#477—21no—
227#476—1221no—
228#474—1221no—
229#462—21no—
230#451—21no—
231C4×Q16—22no—
232#431—611no—
233#431C2×D6111yescharacteristic
234#428—1221no—
235#423—611no—
236#418—1221no—
237#399—1221no—
238#393—1221no—
239#390—1221no—
240#389—1221no—
241#385—1222no—
242#382—611no—
243#328—1222no—
244Q8⋊C8—1222no—
245D4⋊C8—1222no—
246C61#1000211yescharacteristic, complemented, Sylow
247C22×Q8C22⋊D6111yescharacteristic
248C22×D4—611no—
249C23×C4—1221no—
250C2×Q16—22no—
251C2×M16—1221no—
252C4⋊Q8C22⋊D6111yescharacteristic
253C4⋊Q8C22⋊D6111yescharacteristic
254#125—21no—
255C22⋊Q8—21no—
256C22⋊Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×D4—1221no—
264C4×D4—1221no—
265C4×D4—1221no—
266C4×D4—1221no—
267#117—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—611no—
270C2×C4⋊C4—1222no—
271C2×C4⋊C4—1221no—
272C2×C4⋊C4C22⋊D6111yescharacteristic
273C2×C22⋊C4—1222no—
274C2×C22⋊C4—1222no—
275C2×C22⋊C4—1221no—
276C2×C42—21no—
277C2×C42C22×D6111yescharacteristic
278#107—22no—
279#107—1222no—
280#106—1222no—
281C4⋊C8—21no—
282C4⋊C8—612no—
283C4⋊C8—1221no—
284C4⋊C8—1221no—
285#103—22no—
286#103—22no—
287#102—1222no—
288#102—1222no—
289C22⋊C8—21no—
290C22⋊C8—1221no—
291C4×C8—1221no—
292C4×C8C22⋊D6112yes—
293#95—612no—
294#95—1222no—
295#95—1221no—
296C24—1221no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—42no—
302C2×D4—612no—
303C2×D4—1221no—
304C2×D4—1221no—
305C2×D4—1221no—
306C2×D4—1221no—
307C2×D4—1221no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—1222no—
311C22×C4—1221no—
312C22×C4—4882no—
313C22×C4—2441no—
314C22×C4—2441no—
315C22×C4D4×D6111yescharacteristic
316C22×C4D4×D6111yescharacteristic
317C22×C4C2×C22⋊D6111yescharacteristic
318Q16—44no—
319M16—2441no—
320M16—2441no—
321C2×C8—22no—
322C2×C8—1222no—
323C2×C8—2441no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—1221no—
335C4⋊C4—1221no—
336C4⋊C4—4882no—
337C4⋊C4—2441no—
338C4⋊C4—2441no—
339C22⋊C4—41no—
340C22⋊C4—4882no—
341C22⋊C4—2444no—
342C22⋊C4—2441no—
343C22⋊C4—2441no—
344C42—41no—
345C42—41no—
346C42—41no—
347C42D4×D6111yescharacteristic
348C42#1175822411yescharacteristic
349C42C2×C22⋊D6111yescharacteristic
350C42C2×C22⋊D6111yescharacteristic
351C23—1222no—
352C23—2442no—
353C23—2442no—
354C23—2441no—
355C23—2441no—
356C23#1451974511yescharacteristic
357Q8—22no—
358Q8—22no—
359Q8—41no—
360Q8—41no—
361Q8—42no—
362Q8—81no—
363D4—1222no—
364D4—1222no—
365D4—2441no—
366D4—2441no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—4882no—
378C2×C4—4882no—
379C2×C4—2442no—
380C2×C4—2442no—
381C2×C4—2442no—
382C2×C4—2442no—
383C2×C4—2441no—
384C2×C4#1451974511yescharacteristic
385C2×C4#1451974611yescharacteristic
386C2×C4#1451974811yescharacteristic
387C2×C4#1451974911yescharacteristic
388C2×C4#1451974911yescharacteristic
389C2×C4#1451977311yescharacteristic
390C2×C4Q16×D6111yescharacteristic
391C2×C4#1451980011yescharacteristic
392C2×C4#1451980911yescharacteristic
393C2×C4#1451980911yescharacteristic
394C8—42no—
395C8—4881no—
396C8—2442no—
397C22—21no—
398C22—21no—
399C22—4882no—
400C22—4881no—
401C22—4881no—
402C22—2444no—
403C22—2442no—
404C22#1683967011yescharacteristic
405C22#1683991611yescharacteristic
406C22#1683992311yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—42no—
412C4—81no—
413C4—81no—
414C4—81no—
415C4—4882no—
416C4—4881no—
417C4#1683991611yescharacteristic
418C4#1683992311yescharacteristic
419C4#1684000011yescharacteristic
420C4#1684001711yescharacteristic
421C2—21no—
422C2—21no—
423C2—4882no—
424C2#2144682611yescharacteristic
425C2#2144686911yescharacteristic
426C2#2144688411yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#3104C2
socleC22×C61#392
prime radicalC22×D244D4
hypercenter#3104D61
hypocenterC61#10002
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
3. #21444383C2semidirect
4. #21444248C2semidirect
6. #21443678C2semidirect
7. C61×#3104C2semidirect
100. C2×D244Q16semidirect
208. #3104D61semidirect
246. C61#10002semidirect

Maximal subgroups

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

subgroupindex
#214556122
#214511772
#214443832
#214442482
#214442112
#214436782
C61×#31042
#1000261