Finite Group Databasebeta

← #29653932

Subgroups — #29653932

Order 15616.

1002 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 187 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
1C2×#16840393C211yesmaximal, characteristic, complemented
2#21456577C212yesmaximal
3#21451561C211yesmaximal, characteristic
4#21444829C214yesmaximal, complemented
5#21444761C214yesmaximal, complemented
6#21444392C212yesmaximal, complemented
7C2×C61×#324C211yesmaximal, characteristic, complemented
8#16840393—24no—
9#16840393—22no—
10#16840393C2214yes—
11C2×#14519776—22no—
12C2×#14519768—22no—
13C2×#14519760—21no—
14C2×#14519756—21no—
15C2×#14519756C2212yes—
16C2×#14519751C2211yescharacteristic
17C2×#14519736—21no—
18#16840093C2214yes—
19#16840081—22no—
20#16839991—24no—
21#16839915C2214yes—
22#16839903—22no—
23#16839692C2212yes—
24#16839684—22no—
25#16839679C2212yes—
26#16839641—21no—
27#16839622C2211yescharacteristic
28#16839500C2218yes—
29C22×C4⋊C4×C61C2211yescharacteristic
30C2×C22⋊C8×C61C2212yes—
31C61×#324C2214yes—
32C22×Dic122—21no—
33C22×Dic122D412yes—
34C22×D244C2311yescharacteristic
35C22×C22⋊D61—21no—
36C2×#11758190—24no—
37#14519776—28no—
38#14519776—28no—
39#14519768—28no—
40#14519768—28no—
41#14519760—44no—
42#14519756—24no—
43#14519756—24no—
44#14519756—22no—
45#14519756—22no—
46#14519756D414yes—
47#14519756C2314yes—
48#14519751—22no—
49#14519751—22no—
50#14519736—24no—
51#14519736—24no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61D412yes—
57C2×C4⋊Dic61C2311yescharacteristic
58C2×#11758196—22no—
59C2×#11758196—22no—
60C2×#11758196—22no—
61C2×C22⋊Dic61—22no—
62C2×C22⋊Dic61—21no—
63C2×#11758198—21no—
64C2×#11758198—22no—
65C2×#11758198—22no—
66C2×C4×Dic61—22no—
67#14519709—24no—
68#14519702—24no—
69#14519699—24no—
70#14519684—22no—
71#14519684C2312yes—
72C23×C4×C61D411yescharacteristic
73C23×C4×C61D411yescharacteristic
74C23×C4×C61C2311yescharacteristic
75C22×C8×C61—22no—
76C2×C4⋊C4×C61—22no—
77C2×C4⋊C4×C61—22no—
78C2×C4⋊C4×C61D412yes—
79C2×C4⋊C4×C61C2312yes—
80C22⋊C8×C61—22no—
81C22⋊C8×C61C2314yes—
82C24×C61C2×D411yescharacteristic
83C23×D61—21no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—22no—
87C22×C4×C61—21no—
88C22×C4×C61—21no—
89C22×C4×C61—81no—
90C22×C4×C61—81no—
91C22×C4×C61SD1614yescomplemented
92C22×C4×C61C2×D412yes—
93C22×C4×C61C2×D412yes—
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D412yes—
96C22×C4×C61C2×D412yes—
97C22×C4×C61C2411yescharacteristic
98C2×C22⋊D61—41no—
99C2×C22⋊D61—41no—
100C2×C22⋊D61—42no—
101C2×C22⋊D61—42no—
102C22×Dic61—22no—
103C22×Dic61—22no—
104C22×Dic61—21no—
105C22×Dic61—22no—
106C2×D244—22no—
107C2×D244—81no—
108C2×Dic122—24no—
109C2×Dic122—22no—
110C2×Dic122—82no—
111C2×Dic122—82no—
112C2×Dic122C2×D414yes—
113C2×C8×C61—24no—
114C2×C8×C61—82no—
115C4⋊C4×C61—24no—
116C4⋊C4×C61—24no—
117C22⋊Dic61—24no—
118C22⋊Dic61—24no—
119C22⋊Dic61—42no—
120#11758198—42no—
121#11758198—44no—
122#11758198—82no—
123C4⋊Dic61—24no—
124C4⋊Dic61—24no—
125C4⋊Dic61—24no—
126C4⋊Dic61—22no—
127C4⋊Dic61—22no—
128C4⋊Dic61—81no—
129C4⋊Dic61C2×D414yes—
130#11758196—28no—
131#11758196—24no—
132#11758196—24no—
133#11758196—44no—
134C4×Dic61—28no—
135#11758190—84no—
136C23×C61—41no—
137C23×C61—41no—
138C23×C61#12011yescharacteristic
139C23×C61#12011yescharacteristic
140C23×C61C2×SD1612yes—
141C23×C61#13712yes—
142C23×C61C22×D411yescharacteristic
143C22×D61—42no—
144C22×D61—42no—
145C22×D61—41no—
146C22×D61—41no—
147C22×D61—41no—
148C2×C4×C61—22no—
149C2×C4×C61—22no—
150C2×C4×C61—24no—
151C2×C4×C61—24no—
152C2×C4×C61—22no—
153C2×C4×C61—41no—
154C2×C4×C61—42no—
155C2×C4×C61—42no—
156C2×C4×C61—41no—
157C2×C4×C61—42no—
158C2×C4×C61—42no—
159C2×C4×C61—81no—
160C2×C4×C61#12012yes—
161C2×C4×C61C2×SD1614yes—
162C2×C4×C61C22×D412yes—
163C22⋊D61—82no—
164C22⋊D61—82no—
165C2×Dic61—24no—
166C2×Dic61—24no—
167C2×Dic61—44no—
168C2×Dic61—44no—
169C2×Dic61—42no—
170C2×Dic61—42no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—42no—
174C2×Dic61—41no—
175D244—44no—
176Dic122—44no—
177Dic122—82no—
178Dic122—84no—
179C8×C61—44no—
180#18294—611nomaximal, Sylow
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—42no—
188C22×C61—42no—
189C22×C61#44812yes—
190C22×C61#44812yes—
191C22×C61C2×#12011yescharacteristic
192C22×C61C22×SD1611yescharacteristic
193C22×C61C2×#13711yescharacteristic
194C2×D61—82no—
195C2×D61—82no—
196C2×D61—81no—
197C2×D61—81no—
198C2×D61—81no—
199C4×C61—24no—
200C4×C61—22no—
201C4×C61—22no—
202C4×C61—42no—
203Dic61—44no—
204Dic61—44no—
205Dic61—82no—
206Dic61—81no—
207C2×#544—611no—
208#3187—612no—
209#2618—611no—
210#1611—614no—
211#1599—614no—
212#1520—612no—
213C2×#324D6111yescharacteristic, complemented
214C2×C61—21no—
215C2×C61—21no—
216C2×C61—22no—
217C2×C61—22no—
218C2×C61C2×#44811yescharacteristic
219C2×C61#302011yescharacteristic
220C2×C61#302611yescharacteristic
221D61—161no—
222#544—614no—
223#544—1224no—
224#544—1222no—
225C2×C4⋊Q8—1222no—
226C2×#125—1222no—
227C2×#123—1221no—
228C2×C22⋊Q8—612no—
229C2×C22⋊Q8—1221no—
230C2×#121—611no—
231C2×#117—1221no—
232C22×C4⋊C4C2×D6111yescharacteristic
233#483—614no—
234#481—1222no—
235#465—1224no—
236#450—614no—
237#448—1222no—
238C2×C22⋊C8C2×D6112yes—
239#398—612no—
240#396—1222no—
241#394—612no—
242#386—1221no—
243#380—611no—
244#328—618no—
245#324C2×D6114yes—
246C61#1829411yescharacteristic, complemented, Sylow
247C22×Q8—612no—
248C22×Q8—1221no—
249C22×D4—611no—
250C22×D4—1221no—
251C23×C4D24411yescharacteristic
252C23×C4D24411yescharacteristic
253C23×C4C22×D6111yescharacteristic
254C2×SD16—1224no—
255C22×C8—22no—
256C4⋊Q8—1228no—
257C4⋊Q8—1228no—
258#125—1228no—
259#125—1228no—
260#123—2444no—
261C22⋊Q8—614no—
262C22⋊Q8—614no—
263C22⋊Q8—1224no—
264C22⋊Q8—1222no—
265C22⋊Q8—1224no—
266C22⋊Q8—1222no—
267#121—1222no—
268#121—1222no—
269#117—1224no—
270#117—1224no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—612no—
274C2×C4⋊C4—611no—
275C2×C4⋊C4—1221no—
276C2×C4⋊C4—1221no—
277C2×C4⋊C4—1221no—
278C2×C4⋊C4—1222no—
279C2×C4⋊C4—1222no—
280C2×C4⋊C4—1221no—
281C2×C4⋊C4—1222no—
282C2×C4⋊C4D24412yes—
283C2×C4⋊C4C22×D6112yes—
284C2×C22⋊C4—1222no—
285C2×C22⋊C4—1222no—
286C2×C22⋊C4—1221no—
287C2×C22⋊C4—1221no—
288C2×C22⋊C4—1222no—
289C2×C42—1222no—
290#106—1224no—
291#103—1224no—
292#102—1224no—
293C22⋊C8—22no—
294C22⋊C8C22×D6114yes—
295#95—612no—
296#95—1222no—
297C24—1221no—
298C24C2×D24411yescharacteristic
299C2×Q8—614no—
300C2×Q8—1224no—
301C2×Q8—1222no—
302C2×Q8—4882no—
303C2×Q8—4882no—
304C2×D4—1222no—
305C2×D4—4881no—
306C2×D4—2442no—
307C2×D4—2441no—
308C2×D4—2441no—
309C2×D4—2442no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—81no—
316C22×C4—81no—
317C22×C4—1222no—
318C22×C4—1222no—
319C22×C4—1222no—
320C22×C4—1221no—
321C22×C4#1175819014yescomplemented
322C22×C4C2×D24412yes—
323C22×C4C2×D24412yes—
324C22×C4C2×D24411yescharacteristic
325C22×C4D4×D6112yes—
326C22×C4D4×D6112yes—
327C22×C4C23×D6111yescharacteristic
328SD16—4884no—
329C2×C8—24no—
330C2×C8—82no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—614no—
334C4⋊C4—1228no—
335C4⋊C4—1224no—
336C4⋊C4—1224no—
337C4⋊C4—1224no—
338C4⋊C4—1224no—
339C4⋊C4—1222no—
340C4⋊C4—1224no—
341C4⋊C4—1222no—
342C4⋊C4—4881no—
343C4⋊C4—2444no—
344C22⋊C4—1224no—
345C22⋊C4—1224no—
346C22⋊C4—4882no—
347C22⋊C4—2442no—
348C22⋊C4—2442no—
349C22⋊C4—2444no—
350C42—1228no—
351C23—41no—
352C23—41no—
353C23—2442no—
354C23—2442no—
355C23—2441no—
356C23—2441no—
357C23—2441no—
358C23#1451974711yescharacteristic
359C23#1451974711yescharacteristic
360C23C2×#1175819012yes—
361C23#1451981012yes—
362C23C22×D24411yescharacteristic
363Q8—4882no—
364Q8—4884no—
365Q8—2444no—
366D4—4882no—
367D4—4882no—
368D4—2444no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—81no—
381C2×C4—1224no—
382C2×C4—1224no—
383C2×C4—2442no—
384C2×C4—2442no—
385C2×C4—2441no—
386C2×C4—2442no—
387C2×C4—2441no—
388C2×C4—2441no—
389C2×C4—2444no—
390C2×C4—2444no—
391C2×C4#1451974712yes—
392C2×C4C2×#1175819014yes—
393C2×C4C2×D4×D6112yes—
394C8—44no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—4882no—
404C22—4882no—
405C22—4881no—
406C22—4881no—
407C22—4881no—
408C22#1683990312yes—
409C22#1683990312yes—
410C22C2×#1451974711yescharacteristic
411C22C22×#1175819011yescharacteristic
412C22C2×#1451981011yescharacteristic
413C4—22no—
414C4—22no—
415C4—24no—
416C4—42no—
417C4—4882no—
418C4—4881no—
419C4—2444no—
420C4—2444no—
421C2—21no—
422C2—21no—
423C2—22no—
424C2—22no—
425C2—9761no—
426C2C2×#1683990311yescharacteristic
427C2#2145518611yescharacteristic
428C2#2145521811yescharacteristic
subgroupG/H
centerC22C2×#14519747
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#324C2
socleC22×C61C2×#120
prime radicalC2×#14519751C22
hypercenterC2×#324D61
hypocenterC61#18294
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
1. C2×#16840393C2semidirect
4. #21444829C2semidirect
5. #21444761C2semidirect
6. #21444392C2semidirect
7. C2×C61×#324C2semidirect
91. C22×C4×C61SD16semidirect
213. C2×#324D61semidirect
246. C61#18294semidirect
321. C22×C4#11758190semidirect

Maximal subgroups

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

subgroupindex
C2×#168403932
#214565772
#214515612
#214448292
#214447612
#214443922
C2×C61×#3242
#1829461