Finite Group Databasebeta

← #29574988

Subgroups — #29574988

Order 15616.

782 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 111 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#21454860C211yesmaximal, characteristic, complemented
2#21445763C212yesmaximal, complemented
3C4×#14519692C211yesmaximal, characteristic, complemented
4#21443705C212yesmaximal, complemented
5C61×#2388C211yesmaximal, characteristic
6#16839840—24no—
7#16839815—21no—
8C2×#14519704—24no—
9C2×#14519692—21no—
10#16839726—21no—
11#16839726C2212yes—
12C4×C22⋊Dic61C2212yes—
13#16839529—21no—
14C22×#41×C61C2211yescharacteristic
15C2×C4×D4×C61—21no—
16C2×C4×D4×C61C2211yescharacteristic
17C61×#395C411yescharacteristic, complemented
18C61×#394—21no—
19C61×#394C411yescharacteristic, complemented
20C61×#393C411yescharacteristic, complemented
21C61×#392—21no—
22C61×#392C411yescharacteristic, complemented
23C4×C22⋊C4×C61—21no—
24C4×C22⋊C4×C61C2211yescharacteristic
25C2×#11758194—22no—
26C2×#11758194D414yescomplemented
27C2×C22⋊Dic61—22no—
28C2×C4×Dic61—22no—
29C4⋊#92418—22no—
30#14519704—48no—
31#14519692—24no—
32C22⋊#92418—22no—
33C22⋊#92418—22no—
34C22⋊#92418—22no—
35C4×#92418—22no—
36#14519684—22no—
37C2×#41×C61—22no—
38C2×#41×C61—22no—
39C2×#41×C61—44no—
40C22×Q8×C61C2×C411yescharacteristic
41C22×D4×C61—21no—
42C22×D4×C61—21no—
43C22×D4×C61C2×C411yescharacteristic
44C22×D4×C61D411yescharacteristic
45C23×C4×C61—21no—
46C23×C4×C61D411yescharacteristic
47C23×C4×C61C2311yescharacteristic
48C4×D4×C61—22no—
49C4×D4×C61—22no—
50C4×D4×C61—44no—
51C2×C4⋊C4×C61—21no—
52C2×C4⋊C4×C61C2×C411yescharacteristic
53C2×C22⋊C4×C61—21no—
54C2×C22⋊C4×C61—21no—
55C2×C22⋊C4×C61—21no—
56C2×C22⋊C4×C61—21no—
57C2×C22⋊C4×C61—21no—
58C2×C22⋊C4×C61C2×C411yescharacteristic
59C2×C22⋊C4×C61D411yescharacteristic
60C2×C42×C61—21no—
61C2×C42×C61D411yescharacteristic
62#95×C61—21no—
63#95×C61—21no—
64#95×C61C2×C411yescharacteristic
65#95×C61C2×C411yescharacteristic
66C24×C61—21no—
67C24×C61C22⋊C411yescharacteristic
68C24×C61C2×D411yescharacteristic
69#41×C61—28no—
70#41×C61—44no—
71#41×C61—44no—
72#41×C61—44no—
73C2×Q8×C61—22no—
74C2×Q8×C61—42no—
75C2×D4×C61—22no—
76C2×D4×C61—22no—
77C2×D4×C61—22no—
78C2×D4×C61—41no—
79C2×D4×C61—42no—
80C2×D4×C61—42no—
81C2×D4×C61—42no—
82C2×D4×C61—42no—
83C2×D4×C61—42no—
84C2×D4×C61—42no—
85C2×D4×C61—42no—
86C22×C4×C61—24no—
87C22×C4×C61—21no—
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—42no—
93C22×C4×C61—42no—
94C22×C4×C61—42no—
95C22×C4×C61—41no—
96C22×C4×C61—41no—
97C22×C4×C61C22⋊C411yescharacteristic
98C22×C4×C61C22⋊C411yescharacteristic
99C22×C4×C61C22⋊C411yescharacteristic
100C22×C4×C61C22×C411yescharacteristic
101C22×C4×C61C2×D412yes—
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61#4112yes—
104C22×Dic61—42no—
105C4⋊C4×C61—22no—
106C4⋊C4×C61—42no—
107C22⋊C4×C61—22no—
108C22⋊C4×C61—22no—
109C22⋊C4×C61—42no—
110C22⋊C4×C61—42no—
111C22⋊C4×C61—42no—
112C22⋊C4×C61—42no—
113C22⋊C4×C61—42no—
114C42×C61—21no—
115C42×C61—21no—
116C42×C61—42no—
117C22⋊Dic61—24no—
118C22⋊Dic61—42no—
119C4×Dic61—44no—
120#11758194—44no—
121#11758194—82no—
122C2×#92418—24no—
123C23×C61—21no—
124C23×C61—21no—
125C23×C61—22no—
126C23×C61—21no—
127C23×C61—21no—
128C23×C61—21no—
129C23×C61—41no—
130C23×C61—41no—
131C23×C61—41no—
132C23×C61—41no—
133C23×C61—41no—
134C23×C61—41no—
135C23×C61—41no—
136C23×C61—41no—
137C23×C61—41no—
138C23×C61—41no—
139C23×C61C2×C22⋊C411yescharacteristic
140C23×C61#12112yes—
141Q8×C61—24no—
142Q8×C61—42no—
143D4×C61—24no—
144D4×C61—44no—
145D4×C61—42no—
146D4×C61—48no—
147D4×C61—48no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—24no—
151C2×C4×C61—22no—
152C2×C4×C61—22no—
153C2×C4×C61—22no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—41no—
157C2×C4×C61—42no—
158C2×C4×C61—44no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—42no—
164C2×C4×C61—44no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—81no—
171C2×C4×C61#10412yescomplemented
172C2×C4×C61C4×D412yes—
173C2×C4×C61#12011yescharacteristic
174C2×C4×C61#12311yescharacteristic
175C2×Dic61—44no—
176C2×Dic61—82no—
177C2×Dic61—82no—
178#92418—44no—
179#9010—611nomaximal, Sylow
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—22no—
183C22×C61—22no—
184C22×C61—22no—
185C22×C61—42no—
186C22×C61—42no—
187C22×C61—42no—
188C22×C61—41no—
189C22×C61—41no—
190C22×C61—42no—
191C22×C61—41no—
192C22×C61—81no—
193C22×C61—81no—
194C22×C61—81no—
195C22×C61—81no—
196C22×C61—81no—
197C22×C61—81no—
198C22×C61—81no—
199C22×C61#38611yescharacteristic
200C22×C61C2×#10411yescharacteristic
201C22×C61#42111yescharacteristic
202C4×C61—22no—
203C4×C61—21no—
204C4×C61—22no—
205C4×C61—42no—
206C4×C61—41no—
207C4×C61—42no—
208C4×C61—41no—
209C4×C61—81no—
210C4×C61B(4,𝔽2)12yes—
211Dic61—84no—
212#2975—611no—
213#2388D6111yescharacteristic
214#1789—612no—
215C4×#99—611no—
216#1297—612no—
217C2×C61—21no—
218C2×C61—21no—
219C2×C61—42no—
220C2×C61—81no—
221C2×C61—82no—
222C2×C61#187111yescharacteristic
223C2×C61#188411yescharacteristic
224C2×C61#188911yescharacteristic
225C22×#41C2×D6111yescharacteristic
226C2×C4×D4—21no—
227C2×C4×D4C2×D6111yescharacteristic
228#436—1224no—
229#431—1221no—
230C2×#104—1224no—
231C2×#99—1221no—
232#407—612no—
233#407—1221no—
234#395Dic6111yescharacteristic, complemented
235#394—21no—
236#394Dic6111yescharacteristic, complemented
237#393Dic6111yescharacteristic, complemented
238#392—21no—
239#392Dic6111yescharacteristic, complemented
240C4×C22⋊C4—21no—
241C4×C22⋊C4—612no—
242C4×C22⋊C4C2×D6111yescharacteristic
243#342—1221no—
244C61#901011yescharacteristic, complemented, Sylow
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—44no—
248C22×Q8C2×Dic6111yescharacteristic
249C22×D4—21no—
250C22×D4—21no—
251C22×D4C2×Dic6111yescharacteristic
252C22×D4C22⋊D6111yescharacteristic
253C23×C4—21no—
254C23×C4C22⋊D6111yescharacteristic
255C23×C4C22×D6111yescharacteristic
256C2×M16—614no—
257C2×M16—1222no—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—44no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4C2×Dic6111yescharacteristic
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—1222no—
269C2×C22⋊C4C2×Dic6111yescharacteristic
270C2×C22⋊C4C22⋊D6111yescharacteristic
271C2×C42—21no—
272C2×C42—1222no—
273C2×C42C22⋊D6111yescharacteristic
274C4⋊C8—1222no—
275#104—2448no—
276#99—1224no—
277C22⋊C8—1222no—
278C22⋊C8—1222no—
279C22⋊C8—1222no—
280C4×C8—1222no—
281#95—21no—
282#95—21no—
283#95—1222no—
284#95C2×Dic6111yescharacteristic
285#95C2×Dic6111yescharacteristic
286C24—21no—
287C24C22⋊Dic6111yescharacteristic
288C24C2×C22⋊D6111yescharacteristic
289#41—28no—
290#41—44no—
291#41—44no—
292#41—44no—
293C2×Q8—22no—
294C2×Q8—42no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—41no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—2442no—
318C22×C4C22⋊Dic6111yescharacteristic
319C22×C4C22⋊Dic6111yescharacteristic
320C22×C4C22⋊Dic6111yescharacteristic
321C22×C4D4×D6112yes—
322C22×C4#1175822412yes—
323C22×C4C22×Dic6111yescharacteristic
324C22×C4C2×C22⋊D6111yescharacteristic
325M16—4882no—
326M16—2444no—
327C2×C8—1224no—
328C4⋊C4—22no—
329C4⋊C4—42no—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—1224no—
338C22⋊C4—2442no—
339C42—21no—
340C42—21no—
341C42—42no—
342C42—2444no—
343C23—21no—
344C23—21no—
345C23—22no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23C2×C22⋊Dic6111yescharacteristic
360C23#1451974912yes—
361Q8—24no—
362Q8—42no—
363D4—24no—
364D4—44no—
365D4—42no—
366D4—48no—
367D4—48no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—44no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—81no—
391C2×C4—4882no—
392C2×C4—4882no—
393C2×C4—2444no—
394C2×C4#1451970412yescomplemented
395C2×C4D4×Dic6112yes—
396C2×C4#1451974511yescharacteristic
397C2×C4#1451975811yescharacteristic
398C8—2444no—
399C22—21no—
400C22—21no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—41no—
408C22—42no—
409C22—41no—
410C22—41no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22#1683964211yescharacteristic
419C22C2×#1451970411yescharacteristic
420C22#1683977911yescharacteristic
421C4—21no—
422C4—22no—
423C4—22no—
424C4—42no—
425C4—41no—
426C4—42no—
427C4—41no—
428C4—81no—
429C4—4884no—
430C4#1683994812yes—
431C2—21no—
432C2—21no—
433C2—42no—
434C2—81no—
435C2—82no—
436C2#2144612211yescharacteristic
437C2#2144620311yescharacteristic
438C2#2144622811yescharacteristic
subgroupG/H
centerC2×C4#14519745
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
Fitting, prime radicalC61×#2388C2
socleC22×C61#386
cosocleC23×C4×C61C23
hypercenter#2388D61
hypocenterC61#9010
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2388

Complements

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

NH ≅ G/Ntype
1. #21454860C2semidirect
2. #21445763C2semidirect
3. C4×#14519692C2semidirect
4. #21443705C2semidirect
17. C61×#395C4semidirect
19. C61×#394C4semidirect
20. C61×#393C4semidirect
22. C61×#392C4semidirect
26. C2×#11758194D4semidirect
171. C2×C4×C61#104semidirect
234. #395Dic61semidirect
236. #394Dic61semidirect
237. #393Dic61semidirect
239. #392Dic61semidirect
244. C61#9010semidirect
394. C2×C4#14519704semidirect

Maximal subgroups

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

subgroupindex
#214548602
#214457632
C4×#145196922
#214437052
C61×#23882
#901061