Finite Group Databasebeta

← #29571654

Subgroups — #29571654

Order 15616.

750 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 115 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×#16840452C211yesmaximal, characteristic
2#21444420C212yesmaximal, complemented
3#21444267C212yesmaximal, complemented
4#21444160C211yesmaximal, characteristic
5C4×#100×C61C211yesmaximal, characteristic
6#16840452—24no—
7#16840452—24no—
8C2×#14519770—21no—
9C2×#14519769C2211yescharacteristic
10C2×#14519767—21no—
11C2×#14519758—21no—
12C2×#14519756C411yescharacteristic
13C2×#14519756C411yescharacteristic
14C2×C4×Dic122C2211yescharacteristic
15C2×#14519736C411yescharacteristic
16C2×#14519733—21no—
17C22×C4⋊Dic61C411yescharacteristic
18#16839819C2212yes—
19#16839503—24no—
20#16839494—24no—
21#16839486—21no—
22#16839486—21no—
23C2×C4×D4×C61C2211yescharacteristic
24C2×#100×C61—21no—
25C4×M16×C61C2212yes—
26C61×#343—21no—
27C22×Dic122C2×C411yescharacteristic
28C23×Dic61—21no—
29#14519770—28no—
30#14519769—22no—
31#14519769D414yes—
32#14519767—28no—
33#14519758—28no—
34#14519756—22no—
35#14519756—22no—
36#14519756—22no—
37#14519756—22no—
38C4×Dic122—22no—
39C4×Dic122D414yes—
40#14519736—22no—
41#14519736—22no—
42#14519733—28no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61—22no—
45C2×C4⋊Dic61—22no—
46C2×C4⋊Dic61C2×C411yescharacteristic
47C2×C4⋊Dic61C2×C411yescharacteristic
48C2×C4⋊Dic61C2×C411yescharacteristic
49C2×#11758196—21no—
50C2×#11758196—21no—
51C2×#11758196—21no—
52C2×#11758196—21no—
53C2×#11758196—21no—
54C2×C22⋊Dic61—21no—
55C2×C22⋊Dic61—21no—
56C2×C22⋊Dic61—21no—
57C2×C4×Dic61—21no—
58C2×C4×Dic61—21no—
59C4⋊#92418—22no—
60C22⋊#92418—22no—
61#14519688—22no—
62C22×D4×C61D411yescharacteristic
63C23×C4×C61C2×C411yescharacteristic
64C23×C4×C61C2×C411yescharacteristic
65C2×M16×C61—22no—
66C4×D4×C61—22no—
67C4×D4×C61—22no—
68C2×C4⋊C4×C61D411yescharacteristic
69C2×C22⋊C4×C61D411yescharacteristic
70C2×C22⋊C4×C61D411yescharacteristic
71C2×C42×C61C2311yescharacteristic
72#100×C61—24no—
73#97×C61—22no—
74C4×C8×C61—22no—
75C24×C61C22⋊C411yescharacteristic
76C24×C61C22⋊C411yescharacteristic
77C2×D4×C61—22no—
78C2×D4×C61—22no—
79C2×D4×C61—41no—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—42no—
85C22×C4×C61—42no—
86C22×C4×C61C22⋊C411yescharacteristic
87C22×C4×C61C22⋊C411yescharacteristic
88C22×C4×C61C22×C411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×Dic61—24no—
92C22×Dic61—21no—
93C22×Dic61—21no—
94C22×Dic61—21no—
95C22×Dic61—21no—
96C22×Dic61—42no—
97C22×Dic61—42no—
98C2×Dic122—21no—
99C2×Dic122—42no—
100C2×Dic122C22⋊C412yes—
101M16×C61—24no—
102M16×C61—42no—
103C2×C8×C61—42no—
104C4⋊C4×C61—22no—
105C22⋊C4×C61—22no—
106C22⋊C4×C61—22no—
107C42×C61C2×D412yes—
108C42×C61C2×D412yes—
109C22⋊Dic61—24no—
110C22⋊Dic61—24no—
111C22⋊Dic61—24no—
112C4⋊Dic61—24no—
113C4⋊Dic61—21no—
114C4⋊Dic61—21no—
115C4⋊Dic61—21no—
116C4⋊Dic61C22⋊C412yes—
117C4⋊Dic61C22⋊C412yes—
118C4⋊Dic61C22⋊C412yes—
119#11758196—24no—
120#11758196—24no—
121#11758196—24no—
122#11758196—24no—
123#11758196—24no—
124C4×Dic61—24no—
125C4×Dic61—24no—
126C2×#92418—42no—
127C23×C61—21no—
128C23×C61—21no—
129C23×C61—21no—
130C23×C61—21no—
131C23×C61—21no—
132C23×C61—41no—
133C23×C61—41no—
134C23×C61—41no—
135C23×C61—41no—
136C23×C61#10412yes—
137C23×C61C2×C22⋊C411yescharacteristic
138D4×C61—44no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—22no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—41no—
145C2×C4×C61—42no—
146C2×C4×C61—42no—
147C2×C4×C61—42no—
148C2×C4×C61—42no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61C2×C22⋊C411yescharacteristic
152C2×C4×C61C2×C22⋊C411yescharacteristic
153C2×C4×C61#12012yes—
154C2×C4×C61#12012yes—
155C2×Dic61—22no—
156C2×Dic61—22no—
157C2×Dic61—22no—
158C2×Dic61—22no—
159C2×Dic61—42no—
160C2×Dic61—42no—
161C2×Dic61—48no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166Dic122—24no—
167Dic122—42no—
168C8×C61—44no—
169#92418—82no—
170#8371—611nomaximal, Sylow
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—22no—
174C22×C61—42no—
175C22×C61—42no—
176C22×C61—41no—
177C22×C61—41no—
178C22×C61—41no—
179C22×C61—41no—
180C22×C61—41no—
181C22×C61—81no—
182C22×C61—81no—
183C22×C61#37911yescharacteristic
184C22×C61C2×#10411yescharacteristic
185C22×C61#42111yescharacteristic
186C4×C61—22no—
187C4×C61—21no—
188C4×C61—22no—
189C4×C61—41no—
190C4×C61—41no—
191C4×C61#41312yes—
192Dic61—42no—
193Dic61—42no—
194Dic61—42no—
195Dic61—42no—
196C2×#552—611no—
197C4×#100D6111yescharacteristic
198#1528—612no—
199#1491—612no—
200#1466—611no—
201C2×C61—21no—
202C2×C61—21no—
203C2×C61—42no—
204C2×C61—81no—
205C2×C61#178511yescharacteristic
206C2×C61#178911yescharacteristic
207C2×C61#179611yescharacteristic
208#552—1224no—
209#552—1224no—
210C2×#126—1221no—
211C2×#125—611no—
212C2×#125—1221no—
213C2×#123—1221no—
214C2×C22⋊Q8—611no—
215C2×C22⋊Q8—611no—
216C2×C4×Q8—611no—
217C2×C4×D4C2×D6111yescharacteristic
218C2×#117—611no—
219C2×#117—1221no—
220C22×C4⋊C4—611no—
221#432—612no—
222C2×#100—21no—
223C4×M16C2×D6112yes—
224#343—21no—
225#330—1224no—
226Q8⋊C8—1224no—
227#323—1221no—
228#323—1221no—
229C61#837111yescharacteristic, complemented, Sylow
230C22×Q8—611no—
231C22×D4C22⋊D6111yescharacteristic
232C23×C4—1221no—
233C23×C4C4×D6111yescharacteristic
234C23×C4C4×D6111yescharacteristic
235C2×M16—22no—
236#126—1228no—
237#125—614no—
238#125—1228no—
239#125—1222no—
240#123—1228no—
241C22⋊Q8—1222no—
242C22⋊Q8—1222no—
243C22⋊Q8—1222no—
244C22⋊Q8—1222no—
245C4×Q8—614no—
246C4×Q8—1222no—
247C4×D4—22no—
248C4×D4—22no—
249#117—1228no—
250#117—1222no—
251#117—1222no—
252C2×C4⋊C4—611no—
253C2×C4⋊C4—611no—
254C2×C4⋊C4—611no—
255C2×C4⋊C4—1222no—
256C2×C4⋊C4—1222no—
257C2×C4⋊C4—1221no—
258C2×C4⋊C4—1221no—
259C2×C4⋊C4—1221no—
260C2×C4⋊C4—1221no—
261C2×C4⋊C4—1221no—
262C2×C4⋊C4—1221no—
263C2×C4⋊C4C22⋊D6111yescharacteristic
264C2×C22⋊C4—1221no—
265C2×C22⋊C4—1221no—
266C2×C22⋊C4—1221no—
267C2×C22⋊C4D24411yescharacteristic
268C2×C22⋊C4D24411yescharacteristic
269C2×C42—1221no—
270C2×C42—1221no—
271C2×C42C22×D6111yescharacteristic
272C4⋊C8—1222no—
273#100—24no—
274C22⋊C8—1222no—
275#97—22no—
276#97—1222no—
277C4×C8—22no—
278C24#1175819811yescharacteristic
279C24#1175819811yescharacteristic
280C2×Q8—612no—
281C2×Q8—1221no—
282C2×Q8—2442no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—41no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—1224no—
293C22×C4—1221no—
294C22×C4—1221no—
295C22×C4—1221no—
296C22×C4—1221no—
297C22×C4—2442no—
298C22×C4—2442no—
299C22×C4#1175819811yescharacteristic
300C22×C4#1175819811yescharacteristic
301C22×C4C2×C4×D6111yescharacteristic
302C22×C4C2×D24411yescharacteristic
303C22×C4C2×C22⋊D6111yescharacteristic
304M16—24no—
305M16—42no—
306C2×C8—42no—
307C2×C8—2442no—
308C4⋊C4—22no—
309C4⋊C4—612no—
310C4⋊C4—612no—
311C4⋊C4—612no—
312C4⋊C4—1224no—
313C4⋊C4—1224no—
314C4⋊C4—1224no—
315C4⋊C4—1221no—
316C4⋊C4—1221no—
317C4⋊C4—1221no—
318C4⋊C4—1224no—
319C4⋊C4—1224no—
320C4⋊C4—1224no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—1224no—
324C22⋊C4—1224no—
325C22⋊C4—1224no—
326C42—1224no—
327C42—1224no—
328C42D4×D6112yes—
329C42D4×D6112yes—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23#1451970512yes—
340C23C2×#1175819811yescharacteristic
341Q8—1224no—
342Q8—2442no—
343D4—44no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—1222no—
357C2×C4—1222no—
358C2×C4—1222no—
359C2×C4—1222no—
360C2×C4—2442no—
361C2×C4—2448no—
362C2×C4—2442no—
363C2×C4—2441no—
364C2×C4—2441no—
365C2×C4—2441no—
366C2×C4—2441no—
367C2×C4C22⋊C4×D6111yescharacteristic
368C2×C4C22⋊C4×D6111yescharacteristic
369C2×C4#1451974512yes—
370C2×C4#1451974712yes—
371C8—44no—
372C8—4882no—
373C22—21no—
374C22—21no—
375C22—22no—
376C22—42no—
377C22—42no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—81no—
384C22—81no—
385C22#1683961711yescharacteristic
386C22C2×#1451970511yescharacteristic
387C22#1683978011yescharacteristic
388C4—21no—
389C4—22no—
390C4—22no—
391C4—41no—
392C4—41no—
393C4—2442no—
394C4—2442no—
395C4—2442no—
396C4—2442no—
397C4#1683974312yes—
398C2—21no—
399C2—21no—
400C2—42no—
401C2—81no—
402C2#2144574411yescharacteristic
403C2#2144576411yescharacteristic
404C2#2144579511yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC4×#100×C61C2
socleC22×C61#379
prime radicalC22×D4×C61D4
cosocleC2×C42×C61C23
hypercenterC4×#100D61
hypocenterC61#8371
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
2. #21444420C2semidirect
3. #21444267C2semidirect
229. C61#8371semidirect

Maximal subgroups

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

subgroupindex
C2×#168404522
#214444202
#214442672
#214441602
C4×#100×C612
#837161