Finite Group Databasebeta

← #29578056

Subgroups — #29578056

Order 15616.

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

normal subgroups only

This group has 472 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#21455742C211yesmaximal, characteristic
2#21454869C211yesmaximal, characteristic
3#21445905C211yesmaximal, characteristic, complemented
4#21444363C211yesmaximal, characteristic
5#21443724C211yesmaximal, characteristic, complemented
6#21443695C211yesmaximal, characteristic, complemented
7C61×#1821C211yesmaximal, characteristic, complemented
8#16840273C2211yescharacteristic
9#16840097C412yescomplemented
10#16840092—21no—
11#16840080C412yescomplemented
12#16840075—21no—
13#16839838—21no—
14#16839831—21no—
15#16839830—21no—
16#16839825—21no—
17#16839820—21no—
18#16839802—21no—
19#16839761—21no—
20#16839727—21no—
21#16839726C2211yescharacteristic
22C2×C22⋊#92418C2211yescharacteristic
23#16839622C2211yescharacteristic
24#16839524—21no—
25#16839524—21no—
26#16839500—21no—
27#16839497—21no—
28C22×C4⋊C4×C61C2211yescharacteristic
29C61×#407C2211yescharacteristic
30C61×#407C2211yescharacteristic
31C61×#343—21no—
32C61×#343—21no—
33C2×#11758222D411yescharacteristic
34C2×#11758194—21no—
35C2×#11758189—21no—
36C22×#92418D411yescharacteristic
37C22×#92418D411yescharacteristic
38C2×C8×D61—21no—
39#14519760—21no—
40#14519756C2×C411yescharacteristic
41#14519756D411yescharacteristic
42#14519751C2×C411yescharacteristic
43#14519751D411yescharacteristic
44#14519746—21no—
45C4×C22⋊D61—21no—
46C4×C22⋊D61—21no—
47#14519736D411yescharacteristic
48C2×C4⋊Dic61—21no—
49C2×C22⋊Dic61—21no—
50#14519710—22no—
51#14519708—22no—
52#14519706—21no—
53#14519706—21no—
54#14519700—22no—
55#14519697—22no—
56C22⋊#92418—21no—
57C22⋊#92418—21no—
58C22⋊#92418—21no—
59C22⋊#92418C2×C412yes—
60#14519689—21no—
61#14519689—21no—
62#14519687—21no—
63C8×Dic61—21no—
64#14519684—21no—
65C23×C4×C61—21no—
66C23×C4×C61C2311yescharacteristic
67C2×M16×C61—21no—
68C2×M16×C61D411yescharacteristic
69C2×M16×C61D411yescharacteristic
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61—21no—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61C2×C412yes—
76C22⋊C8×C61—21no—
77C22⋊C8×C61—21no—
78C22⋊C8×C61—21no—
79C22⋊C8×C61—21no—
80C24×C61#4111yescharacteristic
81C22×C4×C61—22no—
82C22×C4×C61—41no—
83C22×C4×C61—41no—
84C22×C4×C61—41no—
85C22×C4×C61—41no—
86C22×C4×C61—81no—
87C22×C4×C61—81no—
88C22×C4×C61C22×C411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61#4111yescharacteristic
94C2×C22⋊D61—21no—
95C2×C22⋊D61—21no—
96C2×C22⋊D61—21no—
97C22×Dic61—41no—
98#11758222—41no—
99#11758222—41no—
100C2×D244C22⋊C411yescharacteristic
101C2×C4×D61—21no—
102C2×Dic122C22⋊C411yescharacteristic
103M16×C61—41no—
104M16×C61—41no—
105M16×C61—41no—
106M16×C61—41no—
107C2×C8×C61—21no—
108C2×C8×C61—21no—
109C2×C8×C61—41no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61—22no—
112C4⋊C4×C61—41no—
113C4⋊C4×C61—41no—
114C22⋊Dic61—21no—
115C22⋊Dic61—21no—
116C22⋊Dic61—21no—
117C22⋊Dic61—81no—
118#11758198—21no—
119#11758198—21no—
120C4⋊Dic61—41no—
121C4⋊Dic61—41no—
122C4⋊Dic61C22⋊C411yescharacteristic
123C4⋊Dic61C22⋊C411yescharacteristic
124#11758196—21no—
125#11758196—21no—
126C4×Dic61—21no—
127#11758194—41no—
128#11758194—41no—
129C2×#92418—22no—
130C2×#92418—21no—
131C2×#92418—21no—
132C2×#92418—41no—
133C2×#92418—41no—
134#11758189—81no—
135C8×D61—81no—
136C23×C61—21no—
137C23×C61—21no—
138C23×C61—41no—
139C23×C61—41no—
140C23×C61C4×D411yescharacteristic
141C23×C61#12111yescharacteristic
142C23×C61#12311yescharacteristic
143C22×D61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—42no—
154C2×C4×C61—82no—
155C2×C4×C61—81no—
156C2×C4×C61—81no—
157C2×C4×C61—82no—
158C2×C4×C61C2×C22⋊C411yescharacteristic
159C2×C4×C61C4×D411yescharacteristic
160C2×C4×C61#12011yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C22⋊D61—41no—
163C22⋊D61—41no—
164C22⋊D61—81no—
165C22⋊D61—81no—
166C2×Dic61—21no—
167C2×Dic61—21no—
168C2×Dic61—21no—
169C2×Dic61—81no—
170C2×Dic61—81no—
171C2×Dic61—81no—
172D244—41no—
173C4×D61—41no—
174C4×D61—41no—
175Dic122—41no—
176C8×C61—41no—
177C8×C61—41no—
178C8×C61—81no—
179#92418—42no—
180#92418—41no—
181#92418—41no—
182#9522—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—42no—
192C22×C61—42no—
193C22×C61#38611yescharacteristic
194C22×C61#47111yescharacteristic
195C22×C61#47211yescharacteristic
196C2×D61—41no—
197C2×D61—41no—
198C2×D61—41no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—82no—
203C4×C61#45811yescharacteristic
204C4×C61#45811yescharacteristic
205Dic61—41no—
206Dic61—41no—
207Dic61—41no—
208Dic61—161no—
209#3092—611no—
210#2976—611no—
211#1823—611no—
212#1821D6111yescharacteristic, complemented
213#1513—611no—
214#1305—611no—
215#1293—611no—
216C2×C61—21no—
217C2×C61—21no—
218C2×C61—41no—
219C2×C61—41no—
220C2×C61#189011yescharacteristic
221C2×C61#190211yescharacteristic
222C2×C61#190711yescharacteristic
223D61—81no—
224#525—611no—
225C22×C4⋊C4C2×D6111yescharacteristic
226#484—612no—
227#483—1221no—
228#481—612no—
229#480—1221no—
230#436—1221no—
231#435—1221no—
232#435—1221no—
233C8×D4—1221no—
234#433—1221no—
235#428—1221no—
236#417—1221no—
237#408—1221no—
238#407—611no—
239#407C2×D6111yescharacteristic
240#407C2×D6111yescharacteristic
241C2×C22⋊C8—611no—
242#380—611no—
243#343—21no—
244#343—21no—
245#340—1221no—
246#340—1221no—
247#328—1221no—
248#327—1221no—
249C61#952211yescharacteristic, complemented, Sylow
250C2×#41—611no—
251C23×C4—21no—
252C23×C4C22×D6111yescharacteristic
253C2×M16—21no—
254C2×M16—1221no—
255C2×M16—1221no—
256C2×M16C22⋊D6111yescharacteristic
257C2×M16C22⋊D6111yescharacteristic
258C22×C8—611no—
259C22×C8—611no—
260C22×C8—1221no—
261#123—1221no—
262C22⋊Q8—611no—
263C22⋊Q8—611no—
264#121—611no—
265#121—611no—
266#120—1221no—
267C4×D4—1221no—
268C4×D4—1221no—
269#117—611no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—1221no—
276C2×C4⋊C4C4×D6112yes—
277C2×C22⋊C4—1221no—
278#107—1222no—
279#106—1222no—
280C4⋊C8—1221no—
281C4⋊C8—1221no—
282#103—1222no—
283#102—1222no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—612no—
289C22⋊C8—1221no—
290C22⋊C8—1221no—
291C22⋊C8—1221no—
292C22⋊C8—1221no—
293C22⋊C8—1221no—
294#97—1221no—
295C4×C8—1221no—
296#95—1221no—
297C24#1175822211yescharacteristic
298#41—2441no—
299#41—2441no—
300C2×Q8—611no—
301C2×D4—611no—
302C2×D4—1221no—
303C2×D4—1221no—
304C2×D4—1221no—
305C22×C4—22no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—81no—
311C22×C4—81no—
312C22×C4—1221no—
313C22×C4—2441no—
314C22×C4C2×C4×D6111yescharacteristic
315C22×C4D4×D6111yescharacteristic
316C22×C4D4×D6111yescharacteristic
317C22×C4D4×D6111yescharacteristic
318C22×C4#1175822411yescharacteristic
319C22×C4C2×C22⋊D6111yescharacteristic
320M16—41no—
321M16—41no—
322M16—41no—
323M16—41no—
324M16—4881no—
325M16—2441no—
326M16—2441no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—41no—
330C2×C8—1222no—
331C2×C8—1221no—
332C2×C8—1221no—
333C2×C8—4881no—
334C2×C8—2441no—
335C2×C8—2441no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—611no—
341C4⋊C4—611no—
342C4⋊C4—1221no—
343C4⋊C4—1221no—
344C4⋊C4—2441no—
345C4⋊C4—2441no—
346C22⋊C4—1221no—
347C22⋊C4—1221no—
348C22⋊C4—1221no—
349C22⋊C4—1221no—
350C22⋊C4—1221no—
351C22⋊C4—4881no—
352C42—1221no—
353C23—21no—
354C23—21no—
355C23—41no—
356C23—41no—
357C23—1221no—
358C23C4×C22⋊D6111yescharacteristic
359C23#1451975011yescharacteristic
360C23#1451975911yescharacteristic
361Q8—2441no—
362D4—4881no—
363D4—4881no—
364D4—2441no—
365D4—2441no—
366D4—2441no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—82no—
381C2×C4—1221no—
382C2×C4—1221no—
383C2×C4—1221no—
384C2×C4—4881no—
385C2×C4—4881no—
386C2×C4—4881no—
387C2×C4—2441no—
388C2×C4—2441no—
389C2×C4C22⋊C4×D6111yescharacteristic
390C2×C4#1451973711yescharacteristic
391C2×C4#1451974511yescharacteristic
392C2×C4#1451974811yescharacteristic
393C8—41no—
394C8—41no—
395C8—81no—
396C8—2442no—
397C8—2441no—
398C8—2441no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—42no—
408C22—42no—
409C22—2441no—
410C22—2441no—
411C22—2441no—
412C22#1683963811yescharacteristic
413C22#1684002411yescharacteristic
414C22#1684003111yescharacteristic
415C4—21no—
416C4—21no—
417C4—21no—
418C4—82no—
419C4—9761no—
420C4—2441no—
421C4—2441no—
422C4—2441no—
423C4#1683995111yescharacteristic
424C4#1683995111yescharacteristic
425C2—21no—
426C2—21no—
427C2—41no—
428C2—41no—
429C2—4881no—
430C2#2144622911yescharacteristic
431C2#2144629511yescharacteristic
432C2#2144632411yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1821C2
socleC22×C61#386
prime radical#16840273C22
cosocleC23×C4×C61C23
hypercenter#1821D61
hypocenterC61#9522
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
3. #21445905C2semidirect
5. #21443724C2semidirect
6. #21443695C2semidirect
7. C61×#1821C2semidirect
9. #16840097C4semidirect
11. #16840080C4semidirect
212. #1821D61semidirect
249. C61#9522semidirect

Maximal subgroups

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

subgroupindex
#214557422
#214548692
#214459052
#214443632
#214437242
#214436952
C61×#18212
#952261