Finite Group Databasebeta

← #29580533

Subgroups — #29580533

Order 15616.

650 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 103 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#21456576C211yesmaximal, characteristic
2C2×#16840093C211yesmaximal, characteristic, complemented
3#21450291C211yesmaximal, characteristic
4#21446803C211yesmaximal, characteristic, complemented
5#21445903C211yesmaximal, characteristic, complemented
6#21443665C211yesmaximal, characteristic, complemented
7C61×#1519C211yesmaximal, characteristic
8C2×#14519758—22no—
9C2×#14519756C2211yescharacteristic
10C22×C4⋊Dic61C2211yescharacteristic
11#16840093—22no—
12#16840093—22no—
13#16840092—21no—
14#16840078—21no—
15#16839989—22no—
16#16839913—21no—
17#16839900—21no—
18C2×#14519709—21no—
19C2×#14519702—21no—
20C2×C22⋊#92418C2211yescharacteristic
21#16839698—22no—
22#16839665—21no—
23#16839631—21no—
24C4×C22⋊Dic61C2211yescharacteristic
25#16839528—22no—
26#16839524—22no—
27C2×#121×C61C2211yescharacteristic
28C2×C22⋊C8×C61C2211yescharacteristic
29C61×#380C2211yescharacteristic
30C61×#327—22no—
31C22×Dic122D411yescharacteristic
32C23×Dic61—21no—
33C2×#11758200—22no—
34C22×#92418D412yes—
35#14519758—44no—
36#14519758—44no—
37#14519756—21no—
38#14519756—21no—
39#14519756—21no—
40#14519756—21no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61—21no—
46C2×C4⋊Dic61D412yes—
47C2×C4⋊Dic61D411yescharacteristic
48C2×#11758196—24no—
49C2×#11758196—21no—
50C2×C22⋊Dic61—24no—
51C2×C22⋊Dic61—21no—
52C2×C22⋊Dic61D412yes—
53C2×C4×Dic61—21no—
54#14519709—24no—
55#14519708—22no—
56#14519702—24no—
57#14519700—22no—
58#14519698—22no—
59C22⋊#92418—21no—
60C22⋊#92418—21no—
61#14519684—21no—
62C22×D4×C61—21no—
63C22×D4×C61D411yescharacteristic
64C23×C4×C61C2311yescharacteristic
65C22×C8×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×C61D411yescharacteristic
72C2×C22⋊C4×C61—21no—
73C2×C22⋊C4×C61—21no—
74C22⋊C8×C61—21no—
75C22⋊C8×C61D412yes—
76#95×C61—21no—
77C24×C61—21no—
78C24×C61C2×D411yescharacteristic
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—21no—
87C22×C4×C61—42no—
88C22×C4×C61—41no—
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61#4111yescharacteristic
95C22×Dic61—22no—
96C22×Dic61—22no—
97C22×Dic61—21no—
98C22×Dic61—21no—
99C22×Dic61—41no—
100C22×Dic61—41no—
101C22×Dic61—42no—
102C22×Dic61—42no—
103C2×Dic122—21no—
104C2×Dic122—21no—
105C2×Dic122—42no—
106C2×C8×C61—22no—
107C2×C8×C61—42no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—42no—
111C22⋊C4×C61—41no—
112C22⋊C4×C61—41no—
113C22⋊C4×C61—42no—
114C22⋊Dic61—22no—
115C22⋊Dic61—48no—
116C22⋊Dic61—41no—
117C22⋊Dic61—41no—
118C22⋊Dic61SD1614yescomplemented
119#11758200—82no—
120C4⋊Dic61—21no—
121C4⋊Dic61—22no—
122C4⋊Dic61—21no—
123C4⋊Dic61—22no—
124C4⋊Dic61—41no—
125C4⋊Dic61—41no—
126#11758196—48no—
127#11758196—41no—
128#11758196—41no—
129C4×Dic61—42no—
130C2×#92418—22no—
131C2×#92418—42no—
132C23×C61—41no—
133C23×C61—41no—
134C23×C61—42no—
135C23×C61—42no—
136C23×C61—41no—
137C23×C61—42no—
138C23×C61#12011yescharacteristic
139C23×C61#12111yescharacteristic
140C23×C61#12711yescharacteristic
141C23×C61C2×SD1612yes—
142C23×C61AGL(1,ℤ/8)12yes—
143D4×C61—42no—
144D4×C61—81no—
145D4×C61—82no—
146D4×C61—81no—
147D4×C61—81no—
148C2×C4×C61—22no—
149C2×C4×C61—22no—
150C2×C4×C61—22no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—42no—
156C2×C4×C61—81no—
157C2×C4×C61—81no—
158C2×C4×C61—81no—
159C2×C4×C61#12011yescharacteristic
160C2×C4×C61#12011yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×Dic61—24no—
164C2×Dic61—42no—
165C2×Dic61—42no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—42no—
169C2×Dic61—44no—
170C2×Dic61—82no—
171C2×Dic61—81no—
172C2×Dic61—81no—
173C2×Dic61—81no—
174C2×Dic61—81no—
175Dic122—42no—
176Dic122—42no—
177C8×C61—42no—
178#92418—42no—
179#9960—611nomaximal, Sylow
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—42no—
187C22×C61—42no—
188C22×C61—84no—
189C22×C61—81no—
190C22×C61—81no—
191C22×C61—81no—
192C22×C61#39211yescharacteristic
193C22×C61#45012yes—
194C22×C61#46012yes—
195C22×C61#49211yescharacteristic
196C22×C61#49611yescharacteristic
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—81no—
201C4×C61—161no—
202C4×C61#45812yes—
203Dic61—44no—
204Dic61—81no—
205Dic61—82no—
206Dic61—81no—
207#3187—611no—
208C2×#483—611no—
209#2501—611no—
210#1996—611no—
211#1823—611no—
212#1519D6111yescharacteristic
213#1282—611no—
214C2×C61—22no—
215C2×C61—21no—
216C2×C61—22no—
217C2×C61—21no—
218C2×C61—161no—
219C2×C61#200911yescharacteristic
220C2×C61#201411yescharacteristic
221C2×C61#202211yescharacteristic
222C2×#123—1222no—
223C2×C22⋊Q8—611no—
224C2×#121C2×D6111yescharacteristic
225C22×C4⋊C4—611no—
226#483—1222no—
227#483—1221no—
228#483—1222no—
229#481—1221no—
230#465—1222no—
231#450—1221no—
232#448—1221no—
233C2×#106—1221no—
234C2×#103—1221no—
235C2×C22⋊C8—611no—
236C2×C22⋊C8C2×D6111yescharacteristic
237#399—1222no—
238#390—1221no—
239#384—1221no—
240#380C2×D6111yescharacteristic
241C4×C22⋊C4—611no—
242#342—1222no—
243#340—1222no—
244#327—22no—
245C61#996011yescharacteristic, complemented, Sylow
246C22×Q8—611no—
247C22×D4—21no—
248C22×D4D24411yescharacteristic
249C23×C4—1221no—
250C23×C4C22×D6111yescharacteristic
251C2×SD16—1222no—
252C22×C8—21no—
253C22×C8—612no—
254#123—2444no—
255#123—2444no—
256C22⋊Q8—1221no—
257C22⋊Q8—1221no—
258C22⋊Q8—1221no—
259C22⋊Q8—1221no—
260#121—21no—
261#121—21no—
262#121—21no—
263#121—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—612no—
266C2×C4⋊C4—611no—
267C2×C4⋊C4—1221no—
268C2×C4⋊C4—1221no—
269C2×C4⋊C4—1221no—
270C2×C4⋊C4—1221no—
271C2×C4⋊C4—1224no—
272C2×C4⋊C4—1221no—
273C2×C4⋊C4—1221no—
274C2×C4⋊C4D24411yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—612no—
278C2×C22⋊C4—1224no—
279C2×C22⋊C4—1221no—
280C2×C42—1221no—
281#106—1222no—
282#106—1224no—
283#103—1222no—
284#103—1224no—
285#102—1222no—
286C22⋊C8—21no—
287C22⋊C8—1221no—
288C22⋊C8—1221no—
289C22⋊C8C22⋊D6112yes—
290#95—21no—
291#95—1221no—
292C24—21no—
293C24D4×D6111yescharacteristic
294C2×Q8—1221no—
295C2×Q8—1221no—
296C2×Q8—2442no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—42no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—42no—
304C22×C4—21no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—1222no—
308C22×C4—1221no—
309C22×C4—1221no—
310C22×C4—1222no—
311C22×C4—2441no—
312C22×C4—2441no—
313C22×C4—2442no—
314C22×C4—2442no—
315C22×C4C2×D24411yescharacteristic
316C22×C4#1175822211yescharacteristic
317C22×C4D4×D6111yescharacteristic
318C22×C4D4×D6111yescharacteristic
319C22×C4D4×D6111yescharacteristic
320C22×C4C2×C22⋊D6111yescharacteristic
321SD16—4882no—
322C2×C8—22no—
323C2×C8—42no—
324C2×C8—1222no—
325C2×C8—2442no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—42no—
329C4⋊C4—1222no—
330C4⋊C4—1221no—
331C4⋊C4—1221no—
332C4⋊C4—1222no—
333C4⋊C4—2448no—
334C4⋊C4—2441no—
335C4⋊C4—2441no—
336C4⋊C4—2441no—
337C4⋊C4—2441no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—42no—
341C22⋊C4—614no—
342C22⋊C4—1222no—
343C22⋊C4—2448no—
344C22⋊C4—2441no—
345C22⋊C4—2441no—
346C42—2442no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23#1451974711yescharacteristic
354C23#1451975011yescharacteristic
355C23#1451977311yescharacteristic
356C23SD16×D6112yes—
357C23#1451980112yes—
358Q8—2442no—
359Q8—2442no—
360D4—42no—
361D4—81no—
362D4—82no—
363D4—81no—
364D4—81no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—1224no—
377C2×C4—4882no—
378C2×C4—4881no—
379C2×C4—4881no—
380C2×C4—4881no—
381C2×C4—4881no—
382C2×C4—2444no—
383C2×C4—2441no—
384C2×C4—2441no—
385C2×C4—2442no—
386C2×C4—2442no—
387C2×C4—2442no—
388C2×C4#1451974511yescharacteristic
389C2×C4#1451974711yescharacteristic
390C2×C4#1451975011yescharacteristic
391C2×C4#1451975111yescharacteristic
392C8—42no—
393C8—2442no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—42no—
401C22—42no—
402C22—84no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22#1683967111yescharacteristic
407C22#1683991412yes—
408C22#1683996112yes—
409C22#1684013311yescharacteristic
410C22#1684015011yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—81no—
415C4—161no—
416C4—4881no—
417C4—4881no—
418C4—4882no—
419C4—2444no—
420C4#1683995312yes—
421C2—21no—
422C2—22no—
423C2—22no—
424C2—21no—
425C2—161no—
426C2#2144688011yescharacteristic
427C2#2144690811yescharacteristic
428C2#2144694911yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1519C2
socleC22×C61#392
prime radicalC2×#121×C61C22
hypercenter#1519D61
hypocenterC61#9960
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1519

Complements

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

NH ≅ G/Ntype
2. C2×#16840093C2semidirect
4. #21446803C2semidirect
5. #21445903C2semidirect
6. #21443665C2semidirect
118. C22⋊Dic61SD16semidirect
245. C61#9960semidirect

Maximal subgroups

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

subgroupindex
#214565762
C2×#168400932
#214502912
#214468032
#214459032
#214436652
C61×#15192
#996061