Finite Group Databasebeta

← #29580396

Subgroups — #29580396

Order 15616.

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

normal subgroups only

This group has 600 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#21455755C211yesmaximal, characteristic
2#21450106C211yesmaximal, characteristic, complemented
3#21445919C211yesmaximal, characteristic, complemented
4#21444348C211yesmaximal, characteristic
5#21444149C211yesmaximal, characteristic, complemented
6#21443672C211yesmaximal, characteristic, complemented
7C61×#2591C211yesmaximal, characteristic
8C2×#14519736C2211yescharacteristic
9#16840083—22no—
10#16840078—21no—
11#16840073—21no—
12#16839802—21no—
13#16839767—21no—
14#16839726C2211yescharacteristic
15#16839726C2211yescharacteristic
16#16839701—21no—
17#16839692—21no—
18#16839643—22no—
19#16839635—21no—
20#16839625—21no—
21#16839625C2211yescharacteristic
22#16839522—22no—
23#16839496—22no—
24#16839488—22no—
25C2×#121×C61C2211yescharacteristic
26C22×C4⋊C4×C61C2211yescharacteristic
27C61×#400—21no—
28C61×#394—21no—
29C61×#388—22no—
30C61×#386—21no—
31C61×#386—21no—
32C4×C22⋊C4×C61C2211yescharacteristic
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×#11758196—22no—
42C2×C22⋊Dic61—21no—
43C2×C22⋊Dic61—21no—
44C2×C4×Dic61—21no—
45#14519710—22no—
46#14519708—22no—
47#14519698—22no—
48#14519698—22no—
49C22⋊#92418—21no—
50C22⋊#92418—21no—
51C22⋊#92418—21no—
52C22⋊#92418D412yes—
53#14519684—22no—
54#14519684—22no—
55#14519684—22no—
56#14519684—21no—
57#14519684D412yes—
58C22×D4×C61—21no—
59C22×D4×C61D411yescharacteristic
60C23×C4×C61D411yescharacteristic
61C23×C4×C61D411yescharacteristic
62C23×C4×C61C2311yescharacteristic
63#121×C61—21no—
64#121×C61—21no—
65#121×C61—21no—
66#121×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61D411yescharacteristic
74C2×C22⋊C4×C61—22no—
75C2×C22⋊C4×C61—21no—
76C2×C22⋊C4×C61—21no—
77C2×C22⋊C4×C61—22no—
78C2×C42×C61D412yes—
79#95×C61—22no—
80#95×C61—22no—
81#95×C61—21no—
82C24×C61—21no—
83C24×C61C2×D411yescharacteristic
84C2×D4×C61—21no—
85C2×D4×C61—21no—
86C2×D4×C61—42no—
87C2×D4×C61—44no—
88C2×D4×C61—41no—
89C2×D4×C61—41no—
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—21no—
93C22×C4×C61—21no—
94C22×C4×C61—22no—
95C22×C4×C61—21no—
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61—42no—
99C22×C4×C61—42no—
100C22×C4×C61—42no—
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61#4111yescharacteristic
107C22×Dic61—22no—
108C22×Dic61—21no—
109C22×Dic61—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—44no—
117C22⋊C4×C61—42no—
118C22⋊C4×C61—41no—
119C22⋊C4×C61—41no—
120C22⋊C4×C61—82no—
121C42×C61—42no—
122C22⋊Dic61—42no—
123C22⋊Dic61—41no—
124C22⋊Dic61—41no—
125C4⋊Dic61—21no—
126C4⋊Dic61—21no—
127C4⋊Dic61—21no—
128C4⋊Dic61—21no—
129#11758196—44no—
130C4×Dic61—41no—
131C4×Dic61—41no—
132#11758194—42no—
133#11758194—42no—
134C2×#92418—22no—
135C2×#92418—41no—
136C23×C61—22no—
137C23×C61—22no—
138C23×C61—41no—
139C23×C61—42no—
140C23×C61—42no—
141C23×C61—41no—
142C23×C61—42no—
143C23×C61#12011yescharacteristic
144C23×C61#12011yescharacteristic
145C23×C61#12111yescharacteristic
146D4×C61—42no—
147D4×C61—81no—
148D4×C61—82no—
149D4×C61—82no—
150C2×C4×C61—22no—
151C2×C4×C61—22no—
152C2×C4×C61—22no—
153C2×C4×C61—22no—
154C2×C4×C61—22no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—42no—
164C2×C4×C61—42no—
165C2×C4×C61—42no—
166C2×C4×C61—42no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—42no—
170C2×C4×C61—42no—
171C2×C4×C61—81no—
172C2×C4×C61#12011yescharacteristic
173C2×C4×C61#12111yescharacteristic
174C2×C4×C61#12111yescharacteristic
175C2×C4×C61#12711yescharacteristic
176C2×Dic61—42no—
177C2×Dic61—42no—
178C2×Dic61—42no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61—42no—
182C2×Dic61—42no—
183#92418—42no—
184#92418—81no—
185#9941—611nomaximal, Sylow
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—44no—
194C22×C61—84no—
195C22×C61—81no—
196C22×C61—81no—
197C22×C61—81no—
198C22×C61#39211yescharacteristic
199C22×C61#45412yes—
200C22×C61#45812yes—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61—21no—
204C4×C61—21no—
205C4×C61—42no—
206C4×C61—82no—
207C4×C61—81no—
208C4×C61—81no—
209Dic61—82no—
210Dic61—81no—
211Dic61—81no—
212#3094—611no—
213#2591D6111yescharacteristic
214#2484—611no—
215#1826—611no—
216#1510—611no—
217#1463—611no—
218#1284—611no—
219C2×C61—22no—
220C2×C61—22no—
221C2×C61—41no—
222C2×C61—161no—
223C2×C61#200311yescharacteristic
224C2×C61#201011yescharacteristic
225C2×C61#202211yescharacteristic
226C2×#121C2×D6111yescharacteristic
227C2×#117—611no—
228C22×C4⋊C4C2×D6111yescharacteristic
229#482—1222no—
230#481—1221no—
231#480—1221no—
232#428—1221no—
233#418—1221no—
234#407—611no—
235#407—611no—
236#400—21no—
237#400—1221no—
238#398—1221no—
239#394—21no—
240#388—22no—
241#387—1222no—
242#386—21no—
243#386—21no—
244#385—1221no—
245#381—611no—
246#381—1221no—
247C4×C22⋊C4C2×D6111yescharacteristic
248#339—1222no—
249#327—1222no—
250#324—1222no—
251C61#994111yescharacteristic, complemented, Sylow
252C22×D4—21no—
253C22×D4C22⋊D6111yescharacteristic
254C23×C4C22⋊D6111yescharacteristic
255C23×C4C22⋊D6111yescharacteristic
256C23×C4C22×D6111yescharacteristic
257C2×M16—1221no—
258C2×M16—1221no—
259#121—21no—
260#121—21no—
261#121—21no—
262#121—21no—
263#117—1221no—
264#117—1221no—
265#117—1221no—
266#117—1221no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—611no—
274C2×C4⋊C4—611no—
275C2×C4⋊C4—1222no—
276C2×C4⋊C4C22⋊D6111yescharacteristic
277C2×C22⋊C4—22no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—1221no—
282C2×C22⋊C4—1221no—
283C2×C42—1221no—
284C2×C42C22⋊D6112yes—
285#107—1222no—
286#106—1222no—
287#102—1222no—
288#102—1222no—
289C22⋊C8—612no—
290C22⋊C8—1221no—
291C22⋊C8—1221no—
292C22⋊C8—1221no—
293#95—22no—
294#95—22no—
295#95—21no—
296#95—612no—
297#95—1222no—
298#95—1222no—
299#95—1222no—
300#95—1221no—
301C24—21no—
302C24C2×C22⋊D6111yescharacteristic
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—42no—
306C2×D4—44no—
307C2×D4—41no—
308C2×D4—41no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—1222no—
321C22×C4—1221no—
322C22×C4—1221no—
323C22×C4D4×D6111yescharacteristic
324C22×C4D4×D6111yescharacteristic
325C22×C4D4×D6111yescharacteristic
326C22×C4#1175822411yescharacteristic
327C22×C4C2×C22⋊D6111yescharacteristic
328C22×C4C2×C22⋊D6111yescharacteristic
329M16—2442no—
330M16—2442no—
331C2×C8—1222no—
332C2×C8—2441no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—1221no—
340C4⋊C4—1221no—
341C4⋊C4—1221no—
342C4⋊C4—1221no—
343C4⋊C4—2444no—
344C22⋊C4—44no—
345C22⋊C4—42no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—82no—
349C22⋊C4—2442no—
350C22⋊C4—2441no—
351C22⋊C4—2441no—
352C42—42no—
353C42—2441no—
354C42—2441no—
355C23—22no—
356C23—22no—
357C23—41no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—41no—
362C23#1451974511yescharacteristic
363C23#1451974611yescharacteristic
364C23#1451974911yescharacteristic
365D4—42no—
366D4—81no—
367D4—82no—
368D4—82no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—81no—
391C2×C4—2442no—
392C2×C4—2442no—
393C2×C4—2442no—
394C2×C4—2441no—
395C2×C4—2441no—
396C2×C4—2442no—
397C2×C4—2442no—
398C2×C4#1451974511yescharacteristic
399C2×C4#1451974811yescharacteristic
400C2×C4#1451974911yescharacteristic
401C2×C4#1451977311yescharacteristic
402C8—4881no—
403C8—2442no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—44no—
412C22—84no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22#1683967011yescharacteristic
417C22#1683993412yes—
418C22#1683995212yes—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—42no—
424C4—82no—
425C4—81no—
426C4—81no—
427C4—4882no—
428C4—4881no—
429C4—4881no—
430C2—22no—
431C2—22no—
432C2—41no—
433C2—161no—
434C2#2144684111yescharacteristic
435C2#2144688411yescharacteristic
436C2#2144694811yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#2591C2
socleC22×C61#392
prime radicalC2×#121×C61C22
hypercenter#2591D61
hypocenterC61#9941
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2591

Complements

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

NH ≅ G/Ntype
2. #21450106C2semidirect
3. #21445919C2semidirect
5. #21444149C2semidirect
6. #21443672C2semidirect
251. C61#9941semidirect

Maximal subgroups

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

subgroupindex
#214557552
#214501062
#214459192
#214443482
#214441492
#214436722
C61×#25912
#994161