Finite Group Databasebeta

← #29580458

Subgroups — #29580458

Order 15616.

636 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 101 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#21456567C211yesmaximal, characteristic
2#21450307C211yesmaximal, characteristic, complemented
3#21446656C211yesmaximal, characteristic, complemented
4#21445909C211yesmaximal, characteristic, complemented
5#21444388C211yesmaximal, characteristic, complemented
6#21443665C211yesmaximal, characteristic, complemented
7C2×C61×#481C211yesmaximal, characteristic, complemented
8C2×#14519751C2211yescharacteristic
9#16840075—21no—
10#16840073—21no—
11#16839996—22no—
12#16839896—21no—
13#16839895—21no—
14C2×#14519692—22no—
15C2×C22⋊#92418C2211yescharacteristic
16#16839688—22no—
17#16839663—21no—
18#16839641—22no—
19#16839627—21no—
20#16839622C2211yescharacteristic
21C4×C22⋊Dic61C2211yescharacteristic
22#16839524—22no—
23#16839497—22no—
24C2×#121×C61C2211yescharacteristic
25C22×C4⋊C4×C61C2211yescharacteristic
26C61×#481—22no—
27C61×#481—22no—
28C2×#106×C61—21no—
29C2×#102×C61—21no—
30C2×C22⋊C8×C61C2211yescharacteristic
31C22×D244D411yescharacteristic
32C22×C22⋊D61—21no—
33C2×#11758199—22no—
34C22×#92418D412yes—
35#14519751—21no—
36#14519751—21no—
37#14519751—21no—
38#14519751—21no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61D411yescharacteristic
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61D412yes—
43C2×#11758198—21no—
44C2×#11758198—24no—
45C2×C4×Dic61—21no—
46#14519710—22no—
47#14519698—22no—
48#14519697—22no—
49#14519692—44no—
50C22⋊#92418—21no—
51C22⋊#92418—21no—
52#14519684—24no—
53#14519684—21no—
54#14519684—21no—
55C22×D4×C61—21no—
56C22×D4×C61D411yescharacteristic
57C23×C4×C61—21no—
58C23×C4×C61C2311yescharacteristic
59C22×C8×C61—21no—
60#121×C61—21no—
61#121×C61—21no—
62#121×C61—21no—
63#121×C61—21no—
64C2×C4⋊C4×C61—21no—
65C2×C4⋊C4×C61—21no—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61D412yes—
70C2×C4⋊C4×C61D411yescharacteristic
71C2×C22⋊C4×C61—21no—
72#106×C61—24no—
73#102×C61—24no—
74C22⋊C8×C61—21no—
75C22⋊C8×C61D412yes—
76C24×C61—21no—
77C24×C61C2×D411yescharacteristic
78C23×D61—21no—
79C2×D4×C61—21no—
80C2×D4×C61—21no—
81C2×D4×C61—42no—
82C2×D4×C61—41no—
83C2×D4×C61—41no—
84C2×D4×C61—42no—
85C2×D4×C61—42no—
86C22×C4×C61—22no—
87C22×C4×C61—21no—
88C22×C4×C61—42no—
89C22×C4×C61—41no—
90C22×C4×C61—41no—
91C22×C4×C61—82no—
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61#4111yescharacteristic
98C2×C22⋊D61—41no—
99C2×C22⋊D61—41no—
100C2×C22⋊D61—42no—
101C2×C22⋊D61—42no—
102C22×Dic61—21no—
103C22×Dic61—22no—
104C22×Dic61—41no—
105C2×D244—21no—
106C2×D244—21no—
107C2×D244—42no—
108C2×C8×C61—22no—
109C2×C8×C61—42no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61—22no—
114C4⋊C4×C61—41no—
115C4⋊C4×C61—41no—
116C22⋊C4×C61—41no—
117C22⋊C4×C61—41no—
118C22⋊Dic61—22no—
119C22⋊Dic61—22no—
120C22⋊Dic61—42no—
121#11758199—82no—
122#11758198—41no—
123#11758198—41no—
124#11758198—84no—
125C4⋊Dic61—21no—
126C4⋊Dic61—21no—
127C4⋊Dic61—42no—
128C4×Dic61—42no—
129C2×#92418—22no—
130C2×#92418—42no—
131C23×C61—41no—
132C23×C61—41no—
133C23×C61—41no—
134C23×C61—42no—
135C23×C61—81no—
136C23×C61—81no—
137C23×C61#12011yescharacteristic
138C23×C61#12111yescharacteristic
139C23×C61#12711yescharacteristic
140C23×C61#13512yes—
141C23×C61AGL(1,ℤ/8)12yes—
142C22×D61—42no—
143C22×D61—42no—
144C22×D61—41no—
145C22×D61—42no—
146D4×C61—42no—
147D4×C61—42no—
148D4×C61—82no—
149D4×C61—81no—
150D4×C61—81no—
151C2×C4×C61—22no—
152C2×C4×C61—22no—
153C2×C4×C61—22no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—44no—
158C2×C4×C61—41no—
159C2×C4×C61—42no—
160C2×C4×C61—42no—
161C2×C4×C61—82no—
162C2×C4×C61—82no—
163C2×C4×C61#12011yescharacteristic
164C2×C4×C61#12011yescharacteristic
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61#12111yescharacteristic
167C22⋊D61—82no—
168C22⋊D61—81no—
169C22⋊D61—81no—
170C2×Dic61—42no—
171C2×Dic61—42no—
172C2×Dic61—41no—
173C2×Dic61—42no—
174C2×Dic61—42no—
175C2×Dic61—81no—
176C2×Dic61—81no—
177C2×Dic61—81no—
178D244—42no—
179D244—81no—
180C8×C61—42no—
181#92418—42no—
182#9950—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—42no—
190C22×C61—42no—
191C22×C61—81no—
192C22×C61—81no—
193C22×C61—81no—
194C22×C61—162no—
195C22×C61#39211yescharacteristic
196C22×C61#44912yes—
197C22×C61#46312yes—
198C22×C61#49511yescharacteristic
199C22×C61#49611yescharacteristic
200C2×D61—84no—
201C2×D61—81no—
202C2×D61—81no—
203C2×D61—81no—
204C4×C61—21no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—82no—
208C4×C61—81no—
209C4×C61#45812yes—
210Dic61—82no—
211Dic61—81no—
212Dic61—161no—
213#3186—611no—
214C2×#481D6111yescharacteristic, complemented
215#2504—611no—
216#1964—611no—
217#1824—611no—
218#1519—611no—
219#1282—611no—
220C2×C61—22no—
221C2×C61—21no—
222C2×C61—22no—
223C2×C61—21no—
224C2×C61—161no—
225C2×C61#200811yescharacteristic
226C2×C61#201211yescharacteristic
227C2×C61#202211yescharacteristic
228D61—161no—
229C2×#121—611no—
230C2×#121C2×D6111yescharacteristic
231C22×C4⋊C4C2×D6111yescharacteristic
232#481—22no—
233#481—22no—
234#480—1221no—
235#480—1221no—
236#466—1222no—
237#447—1221no—
238#447—1221no—
239C2×#106—21no—
240C2×#102—21no—
241C2×#99—1222no—
242C2×C22⋊C8—611no—
243C2×C22⋊C8C2×D6111yescharacteristic
244#397—1222no—
245#390—1221no—
246#386—1222no—
247#382—1221no—
248#380—611no—
249C4×C22⋊C4—611no—
250#340—1222no—
251#327—1222no—
252C61#995011yescharacteristic, complemented, Sylow
253C22×D4—21no—
254C22×D4—611no—
255C22×D4—1221no—
256C22×D4C22⋊D6111yescharacteristic
257C23×C4—21no—
258C23×C4C22×D6111yescharacteristic
259C2×D8—1222no—
260C22×C8—21no—
261C22×C8—612no—
262#121—21no—
263#121—21no—
264#121—21no—
265#121—21no—
266#121—1221no—
267#121—1221no—
268#121—1221no—
269#121—1221no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—611no—
276C2×C4⋊C4—1221no—
277C2×C4⋊C4C22⋊D6112yes—
278C2×C4⋊C4C22⋊D6111yescharacteristic
279C2×C22⋊C4—21no—
280C2×C22⋊C4—612no—
281C2×C22⋊C4—1224no—
282C2×C22⋊C4—1221no—
283C2×C22⋊C4—1221no—
284C2×C42—1221no—
285#107—1222no—
286#106—24no—
287#102—24no—
288#102—1222no—
289#102—1222no—
290#99—2444no—
291C22⋊C8—21no—
292C22⋊C8—1221no—
293C22⋊C8—1221no—
294C22⋊C8C22⋊D6112yes—
295#95—1224no—
296#95—1221no—
297#95—1221no—
298C24—21no—
299C24—1221no—
300C24D4×D6111yescharacteristic
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—42no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—1221no—
309C2×D4—1221no—
310C2×D4—2442no—
311C2×D4—2441no—
312C2×D4—2441no—
313C2×D4—2442no—
314C2×D4—2442no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—82no—
321C22×C4—1221no—
322C22×C4—1222no—
323C22×C4—2441no—
324C22×C4D4×D6111yescharacteristic
325C22×C4D4×D6111yescharacteristic
326C22×C4#1175822411yescharacteristic
327C22×C4C2×C22⋊D6111yescharacteristic
328C22×C4C2×C22⋊D6111yescharacteristic
329C22×C4C2×C22⋊D6111yescharacteristic
330D8—4882no—
331C2×C8—22no—
332C2×C8—42no—
333C2×C8—1222no—
334C2×C8—2442no—
335C4⋊C4—21no—
336C4⋊C4—22no—
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—1221no—
342C4⋊C4—1221no—
343C4⋊C4—2442no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—1222no—
347C22⋊C4—1222no—
348C22⋊C4—4884no—
349C22⋊C4—2442no—
350C22⋊C4—2441no—
351C22⋊C4—2441no—
352C42—2442no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—81no—
358C23—81no—
359C23—2442no—
360C23—2442no—
361C23—2442no—
362C23—2441no—
363C23#1451974511yescharacteristic
364C23#1451974911yescharacteristic
365C23#1451977311yescharacteristic
366C23#1451979812yes—
367C23#1451980412yes—
368D4—42no—
369D4—42no—
370D4—81no—
371D4—81no—
372D4—82no—
373D4—4881no—
374D4—4882no—
375D4—4881no—
376D4—4881no—
377D4—2442no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—4881no—
391C2×C4—4881no—
392C2×C4—4881no—
393C2×C4—2442no—
394C2×C4—2442no—
395C2×C4—2442no—
396C2×C4—2442no—
397C2×C4—2441no—
398C2×C4#1451974511yescharacteristic
399C2×C4#1451974611yescharacteristic
400C2×C4#1451974811yescharacteristic
401C2×C4#1451974911yescharacteristic
402C8—42no—
403C8—2442no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—42no—
411C22—42no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—162no—
416C22—4884no—
417C22—4881no—
418C22—4881no—
419C22—4881no—
420C22#1683967011yescharacteristic
421C22#1683990612yes—
422C22#1683997912yes—
423C22#1684014311yescharacteristic
424C22#1684014811yescharacteristic
425C4—21no—
426C4—21no—
427C4—21no—
428C4—82no—
429C4—81no—
430C4—9761no—
431C4—4882no—
432C4—4881no—
433C4#1683995112yes—
434C2—21no—
435C2—22no—
436C2—22no—
437C2—21no—
438C2—161no—
439C2—9761no—
440C2#2144687011yescharacteristic
441C2#2144689311yescharacteristic
442C2#2144694711yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC2×C61×#481C2
socleC22×C61#392
prime radical#21456567C2
hypercenterC2×#481D61
hypocenterC61#9950
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#481

Complements

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

NH ≅ G/Ntype
2. #21450307C2semidirect
3. #21446656C2semidirect
4. #21445909C2semidirect
5. #21444388C2semidirect
6. #21443665C2semidirect
7. C2×C61×#481C2semidirect
214. C2×#481D61semidirect
252. C61#9950semidirect

Maximal subgroups

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

subgroupindex
#214565672
#214503072
#214466562
#214459092
#214443882
#214436652
C2×C61×#4812
#995061