Finite Group Databasebeta

← #29572549

Subgroups — #29572549

Order 15616.

784 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 107 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×#16840367C211yesmaximal, characteristic
2#21444532C212yesmaximal, complemented
3#21444445C212yesmaximal, complemented
4#21444153C211yesmaximal, characteristic, complemented
5C61×#1464C211yesmaximal, characteristic, complemented
6#16840367—24no—
7#16840367—24no—
8C2×#14519774C2211yescharacteristic
9C2×#14519769C2211yescharacteristic
10C2×#14519762—21no—
11C2×#14519762—21no—
12C2×#14519751C411yescharacteristic
13C2×#14519751C411yescharacteristic
14C2×#14519750—21no—
15C2×#14519750—21no—
16C2×C4×D244C411yescharacteristic
17C2×C4×D244C411yescharacteristic
18#16839819C2212yes—
19#16839505—24no—
20#16839503—24no—
21#16839488—21no—
22#16839486—21no—
23C2×#117×C61C2211yescharacteristic
24C61×#432C2212yes—
25C61×#324—21no—
26C61×#323—21no—
27C22×D244—21no—
28C22×D244—21no—
29C22×D244C2×C411yescharacteristic
30C22×D244C2×C411yescharacteristic
31C22×C22⋊D61—21no—
32C22×C22⋊D61—21no—
33C22×C4×D61—21no—
34C22×C4×D61—21no—
35#14519774—22no—
36#14519774D414yes—
37#14519769—22no—
38#14519769D414yes—
39#14519762—28no—
40#14519762—28no—
41#14519751—22no—
42#14519751—22no—
43#14519751—22no—
44#14519751—22no—
45#14519750—28no—
46#14519750—28no—
47C4×D244—22no—
48C4×D244—22no—
49C4×D244—22no—
50C4×D244—22no—
51C2×C4⋊Dic61C2×C411yescharacteristic
52C2×C4⋊Dic61C2×C411yescharacteristic
53C2×#11758196—21no—
54C2×#11758196—21no—
55C2×#11758198—21no—
56C2×#11758198—21no—
57C2×#11758198—21no—
58C2×#11758198—21no—
59C4⋊#92418—22no—
60C22⋊#92418—22no—
61#14519688—22no—
62C23×C4×C61C2×C411yescharacteristic
63#117×C61—22no—
64#117×C61—22no—
65C2×C4⋊C4×C61D411yescharacteristic
66C2×C4⋊C4×C61D411yescharacteristic
67C2×C22⋊C4×C61D411yescharacteristic
68C2×C22⋊C4×C61D411yescharacteristic
69C2×C42×C61C2×C411yescharacteristic
70C2×C42×C61C2311yescharacteristic
71C4⋊C8×C61—22no—
72C22⋊C8×C61—22no—
73#97×C61—22no—
74C24×C61C22⋊C411yescharacteristic
75C23×D61—21no—
76C23×D61—21no—
77C22×C4×C61—21no—
78C22×C4×C61—21no—
79C22×C4×C61—42no—
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C22⋊C411yescharacteristic
83C22×C4×C61C22×C411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C2×C22⋊D61—24no—
87C2×C22⋊D61—24no—
88C2×C22⋊D61—82no—
89C2×C22⋊D61—82no—
90C22×Dic61—21no—
91C22×Dic61—21no—
92C2×D244—21no—
93C2×D244—21no—
94C2×D244—24no—
95C2×D244—24no—
96C2×D244—42no—
97C2×D244—42no—
98C2×D244—82no—
99C2×D244—82no—
100C2×D244C22⋊C412yes—
101C2×D244C22⋊C412yes—
102C2×C4×D61—24no—
103C2×C4×D61—24no—
104C2×C4×D61—82no—
105C2×C4×D61—82no—
106C2×C8×C61—42no—
107C4⋊C4×C61—22no—
108C4⋊C4×C61—22no—
109C22⋊C4×C61—22no—
110C22⋊C4×C61—22no—
111C42×C61—21no—
112C42×C61—21no—
113C42×C61C2×D412yes—
114C42×C61C2×D412yes—
115#11758198—24no—
116#11758198—24no—
117#11758198—24no—
118#11758198—24no—
119C4⋊Dic61—21no—
120C4⋊Dic61—21no—
121C4⋊Dic61C22⋊C412yes—
122C4⋊Dic61C22⋊C412yes—
123#11758196—24no—
124#11758196—24no—
125C2×#92418—42no—
126C23×C61—21no—
127C23×C61—22no—
128C23×C61—41no—
129C23×C61—41no—
130C23×C61C2×C22⋊C411yescharacteristic
131C22×D61—22no—
132C22×D61—22no—
133C22×D61—42no—
134C22×D61—42no—
135C22×D61—42no—
136C22×D61—42no—
137C22×D61—42no—
138C22×D61—42no—
139C2×C4×C61—22no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—22no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—42no—
147C2×C4×C61—42no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61C2×C22⋊C411yescharacteristic
151C2×C4×C61C2×C22⋊C411yescharacteristic
152C2×C4×C61#12012yes—
153C2×C4×C61#12012yes—
154C22⋊D61—84no—
155C22⋊D61—84no—
156C2×Dic61—22no—
157C2×Dic61—22no—
158C2×Dic61—42no—
159C2×Dic61—42no—
160D244—42no—
161D244—42no—
162D244—42no—
163D244—42no—
164D244—84no—
165D244—84no—
166C4×D61—84no—
167C4×D61—84no—
168C8×C61—82no—
169#92418—82no—
170#8526—611nomaximal, Sylow
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—41no—
174C22×C61—41no—
175C22×C61—41no—
176C22×C61—82no—
177C22×C61#37911yescharacteristic
178C22×C61#41312yes—
179C2×D61—42no—
180C2×D61—42no—
181C2×D61—42no—
182C2×D61—42no—
183C2×D61—42no—
184C2×D61—42no—
185C2×D61—84no—
186C2×D61—84no—
187C4×C61—22no—
188C4×C61—22no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—41no—
192C4×C61—41no—
193C4×C61—41no—
194Dic61—42no—
195Dic61—42no—
196C2×#540—611no—
197#1555—612no—
198#1534—612no—
199#1464—611no—
200#1464D6111yescharacteristic, complemented
201C2×C61—22no—
202C2×C61—81no—
203C2×C61#178511yescharacteristic
204C2×C61#179511yescharacteristic
205C2×C61#179511yescharacteristic
206D61—82no—
207D61—82no—
208#540—1224no—
209#540—1224no—
210C2×#127—611no—
211C2×#125—611no—
212C2×#123—1221no—
213C2×#123—1221no—
214C2×#121—611no—
215C2×#121—611no—
216C2×#121—1221no—
217C2×#121—1221no—
218C2×C4×D4—611no—
219C2×C4×D4—611no—
220C2×#117C2×D6111yescharacteristic
221#432—612no—
222#432C2×D6112yes—
223#331—1224no—
224#330—1224no—
225#324—21no—
226#324—1221no—
227#323—21no—
228#323—1221no—
229C61#852611yescharacteristic, complemented, Sylow
230C22×D4—611no—
231C22×D4—611no—
232C22×D4—1221no—
233C22×D4—1221no—
234C22×D4—1221no—
235C22×D4—1221no—
236C23×C4—1221no—
237C23×C4—1221no—
238C23×C4C4×D6111yescharacteristic
239#127—614no—
240#127—1222no—
241#125—614no—
242#125—1222no—
243#123—1228no—
244#123—1228no—
245#121—1228no—
246#121—1228no—
247#121—1222no—
248#121—1222no—
249#121—1222no—
250#121—1222no—
251C4×D4—1222no—
252C4×D4—1222no—
253C4×D4—1222no—
254C4×D4—1222no—
255#117—22no—
256#117—22no—
257C2×C4⋊C4—611no—
258C2×C4⋊C4—611no—
259C2×C4⋊C4—1221no—
260C2×C4⋊C4—1221no—
261C2×C4⋊C4D24411yescharacteristic
262C2×C4⋊C4C22⋊D6111yescharacteristic
263C2×C22⋊C4—1221no—
264C2×C22⋊C4—1221no—
265C2×C22⋊C4—1221no—
266C2×C22⋊C4—1221no—
267C2×C22⋊C4D24411yescharacteristic
268C2×C22⋊C4C22⋊D6111yescharacteristic
269C2×C42C4×D6111yescharacteristic
270C2×C42C22×D6111yescharacteristic
271C4⋊C8—22no—
272C4⋊C8—1222no—
273C22⋊C8—22no—
274C22⋊C8—1222no—
275#97—22no—
276#97—1222no—
277C24—1221no—
278C24—1221no—
279C24#1175819811yescharacteristic
280C2×D4—612no—
281C2×D4—612no—
282C2×D4—1224no—
283C2×D4—1224no—
284C2×D4—1221no—
285C2×D4—1221no—
286C2×D4—1224no—
287C2×D4—1224no—
288C2×D4—4882no—
289C2×D4—4882no—
290C2×D4—4882no—
291C2×D4—4882no—
292C2×D4—2442no—
293C2×D4—2442no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—42no—
297C22×C4—1224no—
298C22×C4—1224no—
299C22×C4—1221no—
300C22×C4—1221no—
301C22×C4—4882no—
302C22×C4—4882no—
303C22×C4#1175819811yescharacteristic
304C22×C4#1175819811yescharacteristic
305C22×C4#1175819811yescharacteristic
306C22×C4C2×C4×D6111yescharacteristic
307C22×C4C2×D24411yescharacteristic
308C22×C4C2×C22⋊D6111yescharacteristic
309C2×C8—42no—
310C2×C8—2442no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—612no—
314C4⋊C4—612no—
315C4⋊C4—1221no—
316C4⋊C4—1221no—
317C4⋊C4—1224no—
318C4⋊C4—1224no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—1224no—
322C22⋊C4—1224no—
323C22⋊C4—1224no—
324C22⋊C4—1224no—
325C42—21no—
326C42—21no—
327C42D4×D6112yes—
328C42D4×D6112yes—
329C23—21no—
330C23—22no—
331C23—41no—
332C23—41no—
333C23—1222no—
334C23—1222no—
335C23—2442no—
336C23—2442no—
337C23—2442no—
338C23—2442no—
339C23—2442no—
340C23—2442no—
341C23C2×#1175819811yescharacteristic
342D4—4884no—
343D4—4884no—
344D4—4884no—
345D4—4884no—
346D4—2442no—
347D4—2442no—
348D4—2442no—
349D4—2442no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—1222no—
362C2×C4—1222no—
363C2×C4—4884no—
364C2×C4—4884no—
365C2×C4—2442no—
366C2×C4—2442no—
367C2×C4C22⋊C4×D6111yescharacteristic
368C2×C4C22⋊C4×D6111yescharacteristic
369C2×C4#1451974512yes—
370C2×C4#1451974712yes—
371C8—82no—
372C8—4882no—
373C22—21no—
374C22—21no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—82no—
379C22—4884no—
380C22—4884no—
381C22—2442no—
382C22—2442no—
383C22—2442no—
384C22—2442no—
385C22—2442no—
386C22—2442no—
387C22#1683961711yescharacteristic
388C22#1683974412yes—
389C4—21no—
390C4—21no—
391C4—22no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—2442no—
397C4—2442no—
398C2—22no—
399C2—81no—
400C2—4882no—
401C2—4882no—
402C2#2144574511yescharacteristic
403C2#2144578711yescharacteristic
404C2#2144579011yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1464C2
socleC22×C61#379
prime radicalC2×#16840367C2
cosocleC2×C42×C61C23
hypercenter#1464D61
hypocenterC61#8526
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #21444532C2semidirect
3. #21444445C2semidirect
4. #21444153C2semidirect
5. C61×#1464C2semidirect
200. #1464D61semidirect
229. C61#8526semidirect

Maximal subgroups

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

subgroupindex
C2×#168403672
#214445322
#214444452
#214441532
C61×#14642
#852661