Finite Group Databasebeta

← #29580368

Subgroups — #29580368

Order 15616.

526 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 85 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#21455763C211yesmaximal, characteristic
2#21451617C211yesmaximal, characteristic
3#21446805C211yesmaximal, characteristic
4#21445916C211yesmaximal, characteristic
5#21444339C211yesmaximal, characteristic
6#21443724C211yesmaximal, characteristic
7C61×#3195C211yesmaximal, characteristic
8C2×#14519736C2211yescharacteristic
9#16840100—22no—
10#16840095—21no—
11#16840090—21no—
12#16839802—21no—
13#16839771—21no—
14#16839726C2211yescharacteristic
15#16839701—21no—
16#16839692—21no—
17#16839690—22no—
18#16839669—21no—
19#16839658—21no—
20#16839622C2211yescharacteristic
21C4×C22⋊Dic61C2211yescharacteristic
22#16839529—22no—
23#16839523—22no—
24#16839499—22no—
25C2×C22⋊Q8×C61C2211yescharacteristic
26C2×C22⋊Q8×C61C2211yescharacteristic
27C61×#483—21no—
28C61×#483—21no—
29C61×#470—22no—
30C61×#451—21no—
31C61×#451—21no—
32C61×#407C2211yescharacteristic
33C2×#11758194—21no—
34#14519736—21no—
35#14519736—21no—
36#14519736—21no—
37#14519736—21no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61D411yescharacteristic
40C2×C4⋊Dic61D411yescharacteristic
41C2×#11758196—22no—
42C2×C22⋊Dic61—21no—
43C2×C22⋊Dic61—21no—
44C2×C22⋊Dic61D412yes—
45C2×C4×Dic61—21no—
46C2×C4×Dic61—21no—
47#14519710—22no—
48#14519708—22no—
49#14519701—22no—
50#14519701—22no—
51C22⋊#92418—21no—
52C22⋊#92418D412yes—
53#14519684—22no—
54#14519684—22no—
55#14519684—22no—
56#14519684—21no—
57#14519684—21no—
58C22×Q8×C61D411yescharacteristic
59C22×Q8×C61D411yescharacteristic
60C23×C4×C61C2311yescharacteristic
61C2×Q16×C61—22no—
62C2×M16×C61D412yes—
63C22⋊Q8×C61—21no—
64C22⋊Q8×C61—21no—
65C22⋊Q8×C61—21no—
66C22⋊Q8×C61—21no—
67C22⋊Q8×C61—21no—
68C22⋊Q8×C61—21no—
69C22⋊Q8×C61—21no—
70C22⋊Q8×C61—21no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61D411yescharacteristic
74C2×C4⋊C4×C61D411yescharacteristic
75C2×C22⋊C4×C61—21no—
76C2×C22⋊C4×C61—21no—
77#106×C61—22no—
78#103×C61—22no—
79#103×C61—22no—
80C22⋊C8×C61—21no—
81C22⋊C8×C61—21no—
82C24×C61C2×D411yescharacteristic
83C2×Q8×C61—21no—
84C2×Q8×C61—21no—
85C2×Q8×C61—21no—
86C2×Q8×C61—21no—
87C2×Q8×C61—42no—
88C2×Q8×C61—42no—
89C22×C4×C61—21no—
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—21no—
93C22×C4×C61—42no—
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61#4111yescharacteristic
100C22×Dic61—21no—
101C22×Dic61—22no—
102C22×Dic61—21no—
103C22×Dic61—41no—
104Q16×C61—82no—
105M16×C61—42no—
106C2×C8×C61—22no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—41no—
112C4⋊C4×C61—41no—
113C4⋊C4×C61—41no—
114C4⋊C4×C61—41no—
115C22⋊C4×C61—41no—
116C22⋊C4×C61—41no—
117C22⋊C4×C61—41no—
118C22⋊C4×C61—41no—
119C22⋊Dic61—22no—
120C22⋊Dic61—41no—
121C22⋊Dic61—41no—
122C22⋊Dic61—41no—
123C22⋊Dic61—42no—
124C4⋊Dic61—21no—
125C4⋊Dic61—21no—
126C4⋊Dic61—21no—
127C4⋊Dic61—21no—
128C4⋊Dic61—42no—
129#11758196—82no—
130C4×Dic61—41no—
131C4×Dic61—41no—
132C4×Dic61—42no—
133#11758194—42no—
134C2×#92418—22no—
135C23×C61—21no—
136C23×C61—21no—
137C23×C61—41no—
138C23×C61—41no—
139C23×C61#12011yescharacteristic
140C23×C61#12111yescharacteristic
141C23×C61#12711yescharacteristic
142Q8×C61—42no—
143Q8×C61—42no—
144Q8×C61—42no—
145Q8×C61—81no—
146C2×C4×C61—22no—
147C2×C4×C61—22no—
148C2×C4×C61—22no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—42no—
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—81no—
160C2×C4×C61—81no—
161C2×C4×C61#12011yescharacteristic
162C2×C4×C61#12011yescharacteristic
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61#12111yescharacteristic
165C2×Dic61—42no—
166C2×Dic61—42no—
167C2×Dic61—42no—
168C2×Dic61—42no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—42no—
173C2×Dic61—41no—
174C2×Dic61—81no—
175C2×Dic61—81no—
176C8×C61—42no—
177#92418—42no—
178#9937—611nomaximal, Sylow
179C22×C61—21no—
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—81no—
188C22×C61—81no—
189C22×C61#39211yescharacteristic
190C22×C61#45511yescharacteristic
191C22×C61#45611yescharacteristic
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—81no—
196C4×C61—81no—
197C4×C61—81no—
198C4×C61—81no—
199C4×C61#45812yes—
200Dic61—82no—
201Dic61—82no—
202Dic61—81no—
203Dic61—81no—
204#3195D6111yescharacteristic
205#3095—611no—
206#2625—611no—
207#1996—611no—
208#1826—611no—
209#1508—611no—
210#1305—611no—
211C2×C61—21no—
212C2×C61—21no—
213C2×C61—41no—
214C2×C61—41no—
215C2×C61#200711yescharacteristic
216C2×C61#200911yescharacteristic
217C2×C61#202211yescharacteristic
218C2×C22⋊Q8C2×D6111yescharacteristic
219C2×C22⋊Q8C2×D6111yescharacteristic
220C2×#117—611no—
221#485—1222no—
222#484—1221no—
223#483—21no—
224#483—21no—
225#483—1221no—
226#470—22no—
227#451—21no—
228#451—21no—
229#428—1221no—
230#419—1221no—
231#407—611no—
232#407C2×D6111yescharacteristic
233#400—1221no—
234#398—1221no—
235#397—1222no—
236#391—1221no—
237#389—1221no—
238#380—611no—
239C4×C22⋊C4—611no—
240#342—1222no—
241#339—1222no—
242#328—1222no—
243C61#993711yescharacteristic, complemented, Sylow
244C22×Q8C22⋊D6111yescharacteristic
245C22×Q8C22⋊D6111yescharacteristic
246C23×C4C22×D6111yescharacteristic
247C2×Q16—22no—
248C2×M16—1221no—
249C2×M16C22⋊D6112yes—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C22⋊Q8—21no—
253C22⋊Q8—21no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256C22⋊Q8—21no—
257C22⋊Q8—21no—
258#117—1221no—
259#117—1221no—
260#117—1221no—
261#117—1221no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—611no—
265C2×C4⋊C4—611no—
266C2×C4⋊C4—1222no—
267C2×C4⋊C4—1221no—
268C2×C4⋊C4C22⋊D6111yescharacteristic
269C2×C4⋊C4C22⋊D6111yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—612no—
273C2×C22⋊C4—1221no—
274C2×C22⋊C4—1221no—
275C2×C42—1221no—
276C2×C42—1221no—
277#107—1222no—
278#106—22no—
279#106—1222no—
280#103—22no—
281#103—22no—
282#103—1222no—
283#103—1222no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8—612no—
287C22⋊C8—1221no—
288#95—1222no—
289#95—1222no—
290#95—1222no—
291#95—1221no—
292#95—1221no—
293C24D4×D6111yescharacteristic
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—42no—
299C2×Q8—42no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—42no—
305C22×C4—1221no—
306C22×C4—1222no—
307C22×C4—1221no—
308C22×C4—2441no—
309C22×C4D4×D6111yescharacteristic
310C22×C4D4×D6111yescharacteristic
311C22×C4#1175822411yescharacteristic
312C22×C4C2×C22⋊D6111yescharacteristic
313C22×C4C2×C22⋊D6111yescharacteristic
314C22×C4C2×C22⋊D6111yescharacteristic
315Q16—82no—
316M16—42no—
317M16—2442no—
318C2×C8—22no—
319C2×C8—1222no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—1221no—
329C4⋊C4—1221no—
330C4⋊C4—1221no—
331C4⋊C4—1221no—
332C4⋊C4—4882no—
333C4⋊C4—2442no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—1222no—
339C22⋊C4—2441no—
340C22⋊C4—2441no—
341C22⋊C4—2441no—
342C22⋊C4—2442no—
343C42—2442no—
344C42—2441no—
345C42—2441no—
346C23—21no—
347C23—21no—
348C23—41no—
349C23—41no—
350C23#1451974511yescharacteristic
351C23#1451974911yescharacteristic
352C23#1451977311yescharacteristic
353Q8—42no—
354Q8—42no—
355Q8—42no—
356Q8—81no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—4881no—
373C2×C4—4881no—
374C2×C4—2442no—
375C2×C4—2442no—
376C2×C4—2442no—
377C2×C4—2442no—
378C2×C4—2442no—
379C2×C4—2441no—
380C2×C4—2441no—
381C2×C4—2441no—
382C2×C4—2441no—
383C2×C4#1451974511yescharacteristic
384C2×C4#1451974611yescharacteristic
385C2×C4#1451974811yescharacteristic
386C2×C4#1451974911yescharacteristic
387C8—42no—
388C8—2442no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—81no—
398C22—81no—
399C22#1683967011yescharacteristic
400C22#1683993811yescharacteristic
401C22#1683994311yescharacteristic
402C4—21no—
403C4—21no—
404C4—21no—
405C4—81no—
406C4—81no—
407C4—81no—
408C4—81no—
409C4—4882no—
410C4—4882no—
411C4—4881no—
412C4—4881no—
413C4#1683995212yes—
414C2—21no—
415C2—21no—
416C2—41no—
417C2—41no—
418C2#2144686411yescharacteristic
419C2#2144687711yescharacteristic
420C2#2144694811yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#3195C2
socleC22×C61#392
prime radicalC24×C61C2×D4
hypercenter#3195D61
hypocenterC61#9937
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3195

Complements

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

NH ≅ G/Ntype
243. C61#9937semidirect

Maximal subgroups

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

subgroupindex
#214557632
#214516172
#214468052
#214459162
#214443392
#214437242
C61×#31952
#993761