Finite Group Databasebeta

← #29580667

Subgroups — #29580667

Order 15616.

544 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 81 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, complemented
2#21455755C211yesmaximal, characteristic
3#21445894C211yesmaximal, characteristic, complemented
4#21444352C211yesmaximal, characteristic, complemented
5#21444348C211yesmaximal, characteristic
6#21443727C211yesmaximal, characteristic, complemented
7C61×#2816C211yesmaximal, characteristic
8C2×#14519736C2211yescharacteristic
9#16840100—22no—
10#16840095—21no—
11#16840090—21no—
12#16840083—22no—
13#16840078—21no—
14#16840073—21no—
15#16839802—21no—
16#16839802—21no—
17#16839771—21no—
18#16839767—21no—
19#16839726C2211yescharacteristic
20#16839726C2211yescharacteristic
21#16839625C2211yescharacteristic
22#16839528—22no—
23#16839499—22no—
24#16839496—22no—
25C2×C22⋊Q8×C61C2211yescharacteristic
26C2×#121×C61C2211yescharacteristic
27C61×#399—21no—
28C61×#398—21no—
29C61×#394—21no—
30C61×#394—22no—
31C61×#393—21no—
32C61×#381C2211yescharacteristic
33C2×#11758194—21no—
34C2×#11758194—21no—
35#14519736—21no—
36#14519736—21no—
37#14519736—21no—
38#14519736—21no—
39C2×C4⋊Dic61D411yescharacteristic
40C2×C4⋊Dic61D411yescharacteristic
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61—21no—
43C2×C4×Dic61—21no—
44#14519710—22no—
45#14519710—22no—
46#14519708—22no—
47#14519708—22no—
48#14519701—22no—
49#14519701—22no—
50#14519698—22no—
51#14519698—22no—
52C22⋊#92418—21no—
53C22⋊#92418—21no—
54C22⋊#92418D412yes—
55C22⋊#92418D412yes—
56#14519684—21no—
57#14519684—21no—
58C22×Q8×C61D411yescharacteristic
59C22×D4×C61—21no—
60C22×D4×C61D411yescharacteristic
61C23×C4×C61C2311yescharacteristic
62C22⋊Q8×C61—21no—
63C22⋊Q8×C61—21no—
64C22⋊Q8×C61—21no—
65C22⋊Q8×C61—21no—
66#121×C61—21no—
67#121×C61—21no—
68#121×C61—21no—
69#121×C61—21no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—22no—
72C2×C4⋊C4×C61D411yescharacteristic
73C2×C4⋊C4×C61D411yescharacteristic
74C2×C22⋊C4×C61—21no—
75C2×C22⋊C4×C61—21no—
76C2×C22⋊C4×C61—22no—
77C2×C22⋊C4×C61—22no—
78C2×C22⋊C4×C61D412yes—
79#95×C61—22no—
80#95×C61—21no—
81#95×C61—21no—
82C24×C61—21no—
83C24×C61C2×D411yescharacteristic
84C2×Q8×C61—21no—
85C2×Q8×C61—21no—
86C2×Q8×C61—42no—
87C2×D4×C61—21no—
88C2×D4×C61—21no—
89C2×D4×C61—42no—
90C2×D4×C61—41no—
91C2×D4×C61—41no—
92C2×D4×C61—82no—
93C22×C4×C61—22no—
94C22×C4×C61—21no—
95C22×C4×C61—21no—
96C22×C4×C61—21no—
97C22×C4×C61—42no—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61#4111yescharacteristic
104C22×Dic61—21no—
105C22×Dic61—21no—
106C22×Dic61—41no—
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—82no—
114C22⋊C4×C61—41no—
115C22⋊C4×C61—41no—
116C22⋊C4×C61—41no—
117C22⋊C4×C61—41no—
118C22⋊C4×C61—42no—
119C22⋊C4×C61—82no—
120C22⋊C4×C61—82no—
121C22⋊Dic61—41no—
122C22⋊Dic61—41no—
123C22⋊Dic61—42no—
124C4⋊Dic61—21no—
125C4⋊Dic61—21no—
126C4⋊Dic61—21no—
127C4⋊Dic61—21no—
128C4×Dic61—41no—
129C4×Dic61—41no—
130#11758194—42no—
131#11758194—42no—
132C2×#92418—22no—
133C2×#92418—22no—
134C23×C61—22no—
135C23×C61—41no—
136C23×C61—41no—
137C23×C61—42no—
138C23×C61—42no—
139C23×C61—41no—
140C23×C61—42no—
141C23×C61#12011yescharacteristic
142C23×C61#12111yescharacteristic
143C23×C61#12711yescharacteristic
144Q8×C61—42no—
145Q8×C61—81no—
146D4×C61—42no—
147D4×C61—81no—
148D4×C61—82no—
149D4×C61—82no—
150C2×C4×C61—22no—
151C2×C4×C61—22no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—42no—
158C2×C4×C61—41no—
159C2×C4×C61—42no—
160C2×C4×C61—42no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—42no—
165C2×C4×C61—42no—
166C2×C4×C61#12011yescharacteristic
167C2×C4×C61#12011yescharacteristic
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—81no—
173C2×Dic61—81no—
174C2×Dic61—81no—
175C2×Dic61—81no—
176C2×Dic61—81no—
177#92418—42no—
178#92418—42no—
179#9984—611nomaximal, Sylow
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—82no—
189C22×C61—84no—
190C22×C61—81no—
191C22×C61—81no—
192C22×C61—81no—
193C22×C61#39211yescharacteristic
194C22×C61#45812yes—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—82no—
200C4×C61—81no—
201C4×C61—81no—
202C4×C61—81no—
203Dic61—81no—
204Dic61—81no—
205Dic61—161no—
206#3095—611no—
207#3094—611no—
208#2816D6111yescharacteristic
209#1820—611no—
210#1511—611no—
211#1510—611no—
212#1306—611no—
213C2×C61—22no—
214C2×C61—41no—
215C2×C61—41no—
216C2×C61—161no—
217C2×C61#200811yescharacteristic
218C2×C61#200911yescharacteristic
219C2×C61#202211yescharacteristic
220C2×C22⋊Q8C2×D6111yescharacteristic
221C2×#121C2×D6111yescharacteristic
222C2×#117—611no—
223#485—1222no—
224#484—1221no—
225#483—1221no—
226#482—1222no—
227#481—1221no—
228#480—1221no—
229#428—1221no—
230#428—1221no—
231#419—1221no—
232#418—1221no—
233#407—611no—
234#407—611no—
235#399—21no—
236#398—21no—
237#394—21no—
238#394—22no—
239#393—21no—
240#381—611no—
241#381C2×D6111yescharacteristic
242#342—1222no—
243#328—1222no—
244#327—1222no—
245C61#998411yescharacteristic, complemented, Sylow
246C22×Q8C22⋊D6111yescharacteristic
247C22×D4—21no—
248C22×D4C22⋊D6111yescharacteristic
249C23×C4C22×D6111yescharacteristic
250C2×M16—1221no—
251C2×M16—1221no—
252C22⋊Q8—21no—
253C22⋊Q8—21no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256#121—21no—
257#121—21no—
258#121—21no—
259#121—21no—
260#117—1221no—
261#117—1221no—
262#117—1221no—
263#117—1221no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—611no—
267C2×C4⋊C4—611no—
268C2×C4⋊C4C22⋊D6111yescharacteristic
269C2×C4⋊C4C22⋊D6111yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—1221no—
275C2×C22⋊C4—1221no—
276C2×C22⋊C4C22⋊D6112yes—
277C2×C42—1221no—
278#107—1222no—
279#107—1222no—
280#106—1222no—
281#106—1222no—
282#103—1222no—
283#103—1222no—
284#102—1222no—
285#102—1222no—
286C22⋊C8—612no—
287C22⋊C8—612no—
288C22⋊C8—1221no—
289C22⋊C8—1221no—
290#95—22no—
291#95—21no—
292#95—21no—
293#95—1221no—
294#95—1221no—
295C24—21no—
296C24C2×C22⋊D6111yescharacteristic
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—42no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—82no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—42no—
311C22×C4—1221no—
312C22×C4—1221no—
313C22×C4—2441no—
314C22×C4D4×D6111yescharacteristic
315C22×C4D4×D6111yescharacteristic
316C22×C4D4×D6111yescharacteristic
317C22×C4#1175822411yescharacteristic
318C22×C4C2×C22⋊D6111yescharacteristic
319C22×C4C2×C22⋊D6111yescharacteristic
320M16—2442no—
321M16—2442no—
322C2×C8—1222no—
323C2×C8—1222no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—82no—
331C4⋊C4—1221no—
332C4⋊C4—1221no—
333C4⋊C4—1221no—
334C4⋊C4—1221no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—42no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—2442no—
343C22⋊C4—2441no—
344C22⋊C4—2441no—
345C42—2441no—
346C42—2441no—
347C23—22no—
348C23—41no—
349C23—41no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—41no—
354C23#1451974611yescharacteristic
355C23#1451974811yescharacteristic
356C23#1451977311yescharacteristic
357Q8—42no—
358Q8—81no—
359D4—42no—
360D4—81no—
361D4—82no—
362D4—82no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—4881no—
380C2×C4—4881no—
381C2×C4—4881no—
382C2×C4—4881no—
383C2×C4—4881no—
384C2×C4—2441no—
385C2×C4—2441no—
386C2×C4#1451974511yescharacteristic
387C2×C4#1451974511yescharacteristic
388C2×C4#1451974911yescharacteristic
389C2×C4#1451974911yescharacteristic
390C8—2442no—
391C8—2442no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—82no—
401C22—84no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22#1683967011yescharacteristic
406C22#1683995212yes—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—82no—
412C4—81no—
413C4—81no—
414C4—81no—
415C4—9761no—
416C4—4881no—
417C4—4881no—
418C2—22no—
419C2—41no—
420C2—41no—
421C2—161no—
422C2#2144687211yescharacteristic
423C2#2144687911yescharacteristic
424C2#2144694811yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#2816C2
socleC22×C61#392
prime radicalC2×#121×C61C22
hypercenter#2816D61
hypocenterC61#9984
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2816

Complements

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

NH ≅ G/Ntype
1. #21455763C2semidirect
3. #21445894C2semidirect
4. #21444352C2semidirect
6. #21443727C2semidirect
245. C61#9984semidirect

Maximal subgroups

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

subgroupindex
#214557632
#214557552
#214458942
#214443522
#214443482
#214437272
C61×#28162
#998461