Finite Group Databasebeta

← #29572597

Subgroups — #29572597

Order 15616.

794 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 143 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#21446288C212yesmaximal, complemented
2#21446117C211yesmaximal, characteristic
3#21444183C212yesmaximal, complemented
4#21444157C211yesmaximal, characteristic
5C61×#3501C211yesmaximal, characteristic
6C22×#11758194C2211yescharacteristic
7C2×#14519696—22no—
8#16839726C2212yes—
9C2×C22⋊#92418—21no—
10C2×C22⋊#92418C2211yescharacteristic
11#16839531—22no—
12#16839517—21no—
13#16839488—21no—
14#16839488—24no—
15#16839486—21no—
16C61×#554—24no—
17C61×#554—22no—
18D4×Q8×C61—22no—
19C61×#544—21no—
20C61×#543—21no—
21C61×#543C412yes—
22C61×#533—21no—
23C2×C22⋊Q8×C61—21no—
24C2×C22⋊Q8×C61C2212yes—
25C61×#518C412yes—
26C2×C4×D4×C61C2211yescharacteristic
27C2×#11758194—22no—
28C2×#11758194—22no—
29C2×#11758194—42no—
30C22×#92418—21no—
31C22×#92418—21no—
32#14519696—44no—
33C22⋊#92418—22no—
34C22⋊#92418—22no—
35C22⋊#92418—22no—
36C22⋊#92418—42no—
37C22×Q8×C61—21no—
38C22×Q8×C61D412yes—
39C22×D4×C61C2×C411yescharacteristic
40C23×C4×C61D411yescharacteristic
41C23×C4×C61C2311yescharacteristic
42C4⋊Q8×C61—21no—
43C4⋊Q8×C61—21no—
44C4⋊Q8×C61—24no—
45#125×C61—22no—
46#125×C61—24no—
47C22⋊Q8×C61—24no—
48C22⋊Q8×C61—24no—
49C22⋊Q8×C61—22no—
50C22⋊Q8×C61—22no—
51C22⋊Q8×C61—22no—
52C22⋊Q8×C61—22no—
53C22⋊Q8×C61—24no—
54C22⋊Q8×C61—24no—
55C22⋊Q8×C61—22no—
56C22⋊Q8×C61—22no—
57C22⋊Q8×C61—22no—
58C22⋊Q8×C61—22no—
59C4×Q8×C61—22no—
60C4×Q8×C61—22no—
61C4×D4×C61—24no—
62C4×D4×C61—24no—
63C4×D4×C61—22no—
64C4×D4×C61—22no—
65#117×C61—22no—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61—22no—
68C2×C4⋊C4×C61D411yescharacteristic
69C2×C4⋊C4×C61D414yes—
70C2×C4⋊C4×C61D412yes—
71C2×C22⋊C4×C61—22no—
72C2×C22⋊C4×C61C2×C411yescharacteristic
73C2×C22⋊C4×C61C2×C414yes—
74C2×C22⋊C4×C61D411yescharacteristic
75C2×C42×C61D411yescharacteristic
76C24×C61C22⋊C411yescharacteristic
77C24×C61C22×C411yescharacteristic
78C2×Q8×C61—22no—
79C2×Q8×C61—22no—
80C2×Q8×C61—22no—
81C2×Q8×C61—22no—
82C2×Q8×C61—42no—
83C2×Q8×C61—42no—
84C2×Q8×C61—81no—
85C2×Q8×C61—81no—
86C2×D4×C61—22no—
87C2×D4×C61—22no—
88C2×D4×C61—22no—
89C22×C4×C61—22no—
90C22×C4×C61—22no—
91C22×C4×C61—21no—
92C22×C4×C61—21no—
93C22×C4×C61—41no—
94C22×C4×C61—41no—
95C22×C4×C61—41no—
96C22×C4×C61—41no—
97C22×C4×C61C22⋊C411yescharacteristic
98C22×C4×C61C22⋊C411yescharacteristic
99C22×C4×C61C22⋊C414yes—
100C22×C4×C61C22⋊C414yes—
101C22×C4×C61C22⋊C411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D412yes—
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D412yes—
106C4⋊C4×C61—22no—
107C4⋊C4×C61—24no—
108C4⋊C4×C61—22no—
109C4⋊C4×C61—22no—
110C4⋊C4×C61—24no—
111C4⋊C4×C61—22no—
112C4⋊C4×C61—24no—
113C4⋊C4×C61—22no—
114C4⋊C4×C61—22no—
115C4⋊C4×C61—22no—
116C4⋊C4×C61—24no—
117C22⋊C4×C61—24no—
118C22⋊C4×C61—24no—
119C22⋊C4×C61—22no—
120C22⋊C4×C61—24no—
121C22⋊C4×C61—22no—
122C42×C61—24no—
123C42×C61—21no—
124C42×C61—21no—
125#11758194—24no—
126#11758194—42no—
127#11758194—42no—
128#11758194—42no—
129C2×#92418—22no—
130C2×#92418—41no—
131C2×#92418—41no—
132C2×#92418—41no—
133C2×#92418—41no—
134C2×#92418—41no—
135C2×#92418—42no—
136C23×C61—21no—
137C23×C61—21no—
138C23×C61—21no—
139C23×C61—21no—
140C23×C61—41no—
141C23×C61—41no—
142C23×C61—41no—
143C23×C61—41no—
144C23×C61#10112yescomplemented
145C23×C61C2×C22⋊C412yes—
146C23×C61C2×C22⋊C411yescharacteristic
147Q8×C61—42no—
148Q8×C61—42no—
149Q8×C61—82no—
150Q8×C61—82no—
151Q8×C61—82no—
152D4×C61—28no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—22no—
157C2×C4×C61—21no—
158C2×C4×C61—22no—
159C2×C4×C61—21no—
160C2×C4×C61—24no—
161C2×C4×C61—24no—
162C2×C4×C61—21no—
163C2×C4×C61—42no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—44no—
169C2×C4×C61—41no—
170C2×C4×C61—44no—
171C2×C4×C61—81no—
172C2×C4×C61—82no—
173C2×C4×C61—82no—
174C2×C4×C61#10112yescomplemented
175C2×C4×C61#12011yescharacteristic
176C2×C4×C61#12012yes—
177C2×C4×C61#12011yescharacteristic
178#92418—42no—
179#92418—81no—
180#92418—82no—
181#8537—611nomaximal, Sylow
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—22no—
185C22×C61—22no—
186C22×C61—22no—
187C22×C61—22no—
188C22×C61—42no—
189C22×C61—41no—
190C22×C61—41no—
191C22×C61—81no—
192C22×C61—81no—
193C22×C61—81no—
194C22×C61#37911yescharacteristic
195C22×C61C2×#10111yescharacteristic
196C22×C61C2×#10111yescharacteristic
197C4×C61—22no—
198C4×C61—21no—
199C4×C61—22no—
200C4×C61—41no—
201C4×C61—44no—
202C4×C61—44no—
203C4×C61—41no—
204C4×C61B(4,𝔽2)11yescharacteristic
205C4×C61#45811yescharacteristic
206#3501D6111yescharacteristic
207#1901—612no—
208#1870—611no—
209#1472—612no—
210#1465—611no—
211C2×C61—21no—
212C2×C61—21no—
213C2×C61—42no—
214C2×C61—42no—
215C2×C61#178411yescharacteristic
216C2×C61#179011yescharacteristic
217C2×C61#179111yescharacteristic
218C22×M16—611no—
219#554—24no—
220#554—22no—
221D4×Q8—22no—
222#544—21no—
223#543—21no—
224#543Dic6112yes—
225#533—21no—
226C2×C22⋊Q8—21no—
227C2×C22⋊Q8C2×D6112yes—
228#518Dic6112yes—
229C2×C4×D4C2×D6111yescharacteristic
230C2×#101—1222no—
231#407—612no—
232C2×C22⋊C8—611no—
233C2×C22⋊C8—1221no—
234#343—1222no—
235#336—1221no—
236#324—1221no—
237#324—1224no—
238#323—1221no—
239C61#853711yescharacteristic, complemented, Sylow
240C22×Q8—21no—
241C22×Q8C22⋊D6112yes—
242C22×D4C2×Dic6111yescharacteristic
243C23×C4C22⋊D6111yescharacteristic
244C23×C4C22×D6111yescharacteristic
245C2×M16—1222no—
246C2×M16—1222no—
247C2×M16—2442no—
248C22×C8—1221no—
249C22×C8—1221no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—24no—
253#125—22no—
254#125—24no—
255C22⋊Q8—22no—
256C22⋊Q8—22no—
257C22⋊Q8—24no—
258C22⋊Q8—24no—
259C22⋊Q8—22no—
260C22⋊Q8—22no—
261C22⋊Q8—24no—
262C22⋊Q8—24no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265C22⋊Q8—22no—
266C22⋊Q8—22no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×D4—24no—
270C4×D4—24no—
271C4×D4—22no—
272C4×D4—22no—
273#117—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4C22⋊D6111yescharacteristic
277C2×C4⋊C4C22⋊D6114yes—
278C2×C4⋊C4C22⋊D6112yes—
279C2×C22⋊C4—22no—
280C2×C22⋊C4C2×Dic6111yescharacteristic
281C2×C22⋊C4C2×Dic6114yes—
282C2×C22⋊C4C22⋊D6111yescharacteristic
283C2×C42C22⋊D6111yescharacteristic
284#101—2444no—
285C22⋊C8—1222no—
286C22⋊C8—1222no—
287C22⋊C8—1222no—
288C22⋊C8—2442no—
289C24C22⋊Dic6111yescharacteristic
290C24C22×Dic6111yescharacteristic
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—81no—
298C2×Q8—81no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4C22⋊Dic6111yescharacteristic
311C22×C4C22⋊Dic6111yescharacteristic
312C22×C4C22⋊Dic6114yes—
313C22×C4C22⋊Dic6114yes—
314C22×C4C22⋊Dic6111yescharacteristic
315C22×C4C2×C22⋊D6111yescharacteristic
316C22×C4C2×C22⋊D6112yes—
317C22×C4C2×C22⋊D6111yescharacteristic
318C22×C4C2×C22⋊D6112yes—
319M16—1224no—
320M16—2442no—
321M16—2442no—
322M16—2442no—
323C2×C8—1222no—
324C2×C8—2441no—
325C2×C8—2441no—
326C2×C8—2441no—
327C2×C8—2442no—
328C2×C8—2441no—
329C2×C8—2441no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—22no—
344C22⋊C4—24no—
345C22⋊C4—22no—
346C42—24no—
347C42—21no—
348C42—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23#1451969612yescomplemented
358C23C2×C22⋊Dic6112yes—
359C23C2×C22⋊Dic6111yescharacteristic
360Q8—42no—
361Q8—42no—
362Q8—82no—
363Q8—82no—
364Q8—82no—
365D4—28no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—44no—
382C2×C4—41no—
383C2×C4—44no—
384C2×C4—81no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4#1451969612yescomplemented
388C2×C4#1451974611yescharacteristic
389C2×C4#1451974612yes—
390C2×C4#1451974611yescharacteristic
391C8—4881no—
392C8—4882no—
393C8—2442no—
394C22—21no—
395C22—21no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—42no—
401C22—41no—
402C22—41no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22#1683961811yescharacteristic
407C22C2×#1451969611yescharacteristic
408C22C2×#1451969611yescharacteristic
409C4—22no—
410C4—21no—
411C4—22no—
412C4—41no—
413C4—44no—
414C4—44no—
415C4—41no—
416C4#1683994911yescharacteristic
417C4#1683995211yescharacteristic
418C2—21no—
419C2—21no—
420C2—42no—
421C2—42no—
422C2#2144574311yescharacteristic
423C2#2144576711yescharacteristic
424C2#2144577011yescharacteristic
subgroupG/H
centerC22#16839618
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#3501C2
socleC22×C61#379
prime radicalC22×D4×C61C2×C4
cosocleC23×C4×C61C23
hypercenter#3501D61
hypocenterC61#8537
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #21446288C2semidirect
3. #21444183C2semidirect
144. C23×C61#101semidirect
174. C2×C4×C61#101semidirect
239. C61#8537semidirect
357. C23#14519696semidirect
387. C2×C4#14519696semidirect

Maximal subgroups

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

subgroupindex
#214462882
#214461172
#214441832
#214441572
C61×#35012
#853761