Finite Group Databasebeta

← #30598333

Subgroups — #30598333

Order 15616.

750 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C4×#14519837C211yesmaximal, characteristic
2#21461283C212yesmaximal, complemented
3#21461161C212yesmaximal, complemented
4#21461078C211yesmaximal, characteristic
5C2×#16840454C211yesmaximal, characteristic
6#16840783C2212yes—
7C2×#14519837—21no—
8C4×#11758217C2212yes—
9#16840642—21no—
10#16840619—24no—
11Q8⋊#92420—24no—
12#16840606—21no—
13#16840605—21no—
14#16840454—24no—
15#16840454—24no—
16C2×#14519772—21no—
17C2×#14519768—21no—
18C2×#14519768C2211yescharacteristic
19C2×#14519761—21no—
20C2×#14519753C411yescharacteristic
21C2×Q8×Dic61C2211yescharacteristic
22C2×C4×C22⋊D61C2211yescharacteristic
23C2×#14519735C411yescharacteristic
24C2×#14519733—21no—
25C22×C4⋊Dic61C411yescharacteristic
26C2×C22⋊Q8×C61C411yescharacteristic
27C2×#11758217—22no—
28C4⋊#92420—22no—
29#14519837—24no—
30C22⋊#92420—22no—
31#14519832—22no—
32#14519832—22no—
33C4×#92420—22no—
34C22×C22⋊D61D411yescharacteristic
35C23×Dic61—21no—
36C22×C4×D61C2×C411yescharacteristic
37#14519772—28no—
38#14519768—28no—
39#14519768—22no—
40#14519768D414yes—
41#14519761—28no—
42#14519753—22no—
43#14519753—22no—
44Q8×Dic61—22no—
45Q8×Dic61D414yes—
46C4×C22⋊D61—22no—
47C4×C22⋊D61—22no—
48#14519735—22no—
49#14519735—22no—
50#14519733—28no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—22no—
55C2×C4⋊Dic61—22no—
56C2×C4⋊Dic61C2×C411yescharacteristic
57C2×C4⋊Dic61C2×C411yescharacteristic
58C2×#11758196—21no—
59C2×#11758196—21no—
60C2×#11758196D411yescharacteristic
61C2×C22⋊Dic61D411yescharacteristic
62C2×#11758198—21no—
63C2×#11758198—21no—
64C2×#11758198D411yescharacteristic
65C2×C4×Dic61—21no—
66C2×C4×Dic61—21no—
67C2×C4×Dic61C2311yescharacteristic
68C22×Q8×C61C2×C411yescharacteristic
69C23×C4×C61C2×C411yescharacteristic
70C22⋊Q8×C61—22no—
71C22⋊Q8×C61—22no—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61C2×C411yescharacteristic
74C2×C22⋊C4×C61—21no—
75C24×C61C22⋊C411yescharacteristic
76C23×D61C22⋊C411yescharacteristic
77C2×Q8×C61—21no—
78C2×Q8×C61—42no—
79C2×Q8×C61C22⋊C412yes—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—42no—
85C22×C4×C61C22⋊C411yescharacteristic
86C22×C4×C61C22×C411yescharacteristic
87C2×C22⋊D61—22no—
88C2×C22⋊D61—22no—
89C2×C22⋊D61—41no—
90C22×Dic61—24no—
91C22×Dic61—21no—
92C22×Dic61—21no—
93C22×Dic61—42no—
94C22×Dic61—42no—
95C22×Dic61C22⋊C411yescharacteristic
96C22×Dic61C2×D411yescharacteristic
97C22×Dic61C2×D411yescharacteristic
98C2×C4×D61—21no—
99C2×C4×D61—21no—
100C2×C4×D61—42no—
101#11758217—24no—
102#11758217—42no—
103C2×#92420—42no—
104C2×#92420—42no—
105C4⋊C4×C61—24no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61C22⋊C412yes—
108C22⋊C4×C61—24no—
109C22⋊Dic61—22no—
110#11758198—24no—
111#11758198—24no—
112#11758198—22no—
113C4⋊Dic61—24no—
114C4⋊Dic61—21no—
115C4⋊Dic61—21no—
116C4⋊Dic61—24no—
117C4⋊Dic61—24no—
118C4⋊Dic61C22⋊C412yes—
119C4⋊Dic61C22⋊C412yes—
120#11758196—24no—
121#11758196—24no—
122#11758196—22no—
123C4×Dic61—24no—
124C4×Dic61—24no—
125C4×Dic61C2×D412yes—
126C4×Dic61C2×D412yes—
127C23×C61—21no—
128C23×C61—21no—
129C23×C61—41no—
130C23×C61—41no—
131C23×C61#10412yes—
132C23×C61C2×C22⋊C411yescharacteristic
133C22×D61—21no—
134C22×D61—21no—
135C22×D61—21no—
136C22×D61—41no—
137C22×D61—41no—
138Q8×C61—24no—
139Q8×C61—42no—
140C2×C4×C61—22no—
141C2×C4×C61—22no—
142C2×C4×C61—22no—
143C2×C4×C61—21no—
144C2×C4×C61—42no—
145C2×C4×C61—42no—
146C2×C4×C61—42no—
147C2×C4×C61—42no—
148C2×C4×C61—41no—
149C2×C4×C61C2×C22⋊C411yescharacteristic
150C2×C4×C61C2×C22⋊C411yescharacteristic
151C22⋊D61—44no—
152C2×Dic61—21no—
153C2×Dic61—21no—
154C2×Dic61—21no—
155C2×Dic61—22no—
156C2×Dic61—22no—
157C2×Dic61—41no—
158C2×Dic61—48no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61#12012yes—
165C2×Dic61#12012yes—
166C4×D61—42no—
167C4×D61—42no—
168#92420—44no—
169#92420—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#37911yescharacteristic
179C22×C61C2×#10411yescharacteristic
180C22×C61#42111yescharacteristic
181C2×D61—41no—
182C2×D61—41no—
183C2×D61—41no—
184C2×D61—81no—
185C2×D61—81no—
186C4×C61—21no—
187C4×C61—42no—
188C4×C61—41no—
189C4×C61—42no—
190C4×C61#41312yes—
191Dic61—22no—
192Dic61—22no—
193Dic61—41no—
194Dic61—42no—
195Dic61—42no—
196C2×#552—611no—
197C4×#100—611no—
198#1528—612no—
199#1491—612no—
200#1466—611no—
201C2×C61—21no—
202C2×C61—21no—
203C2×C61—42no—
204C2×C61#178511yescharacteristic
205C2×C61#178911yescharacteristic
206C2×C61#179611yescharacteristic
207D61—81no—
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#692311yescharacteristic
216C2×C4×Q8—611no—
217C2×C4×D4—611no—
218C2×#117—611no—
219C2×#117—1221no—
220C22×C4⋊C4—611no—
221#432—612no—
222C2×#100—1221no—
223C4×M16—612no—
224#343—1221no—
225#330—1224no—
226Q8⋊C8—1224no—
227#323—1221no—
228#323—1221no—
229C61#837111yescharacteristic, complemented, Sylow
230C22×Q8C2×#692311yescharacteristic
231C22×D4—611no—
232C23×C4—611no—
233C23×C4—1221no—
234C23×C4C2×#692311yescharacteristic
235C2×M16—1222no—
236#126—1228no—
237#125—614no—
238#125—1222no—
239#125—1228no—
240#123—1228no—
241C22⋊Q8—22no—
242C22⋊Q8—22no—
243C22⋊Q8—1222no—
244C22⋊Q8—1222no—
245C4×Q8—614no—
246C4×Q8—1222no—
247C4×D4—1222no—
248C4×D4—1222no—
249#117—1228no—
250#117—1222no—
251#117—1222no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—611no—
254C2×C4⋊C4—611no—
255C2×C4⋊C4—611no—
256C2×C4⋊C4—1222no—
257C2×C4⋊C4—1222no—
258C2×C4⋊C4—1221no—
259C2×C4⋊C4—1221no—
260C2×C4⋊C4—1221no—
261C2×C4⋊C4—1221no—
262C2×C4⋊C4—1221no—
263C2×C4⋊C4C2×#692311yescharacteristic
264C2×C22⋊C4—21no—
265C2×C22⋊C4—611no—
266C2×C22⋊C4—611no—
267C2×C22⋊C4—1221no—
268C2×C22⋊C4—1221no—
269C2×C42—611no—
270C2×C42—1221no—
271C2×C42—1221no—
272C4⋊C8—1222no—
273#100—1224no—
274C22⋊C8—1222no—
275#97—1222no—
276#97—1222no—
277C4×C8—1222no—
278C24—611no—
279C24C22⋊#692311yescharacteristic
280C2×Q8—21no—
281C2×Q8—42no—
282C2×Q8C22⋊#692312yes—
283C2×D4—1222no—
284C2×D4—1222no—
285C2×D4—2441no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—42no—
291C22×C4—611no—
292C22×C4—611no—
293C22×C4—611no—
294C22×C4—1224no—
295C22×C4—1221no—
296C22×C4—1221no—
297C22×C4—1221no—
298C22×C4—1221no—
299C22×C4—2442no—
300C22×C4—2442no—
301C22×C4—2442no—
302C22×C4C22⋊#692311yescharacteristic
303C22×C4C22×#692311yescharacteristic
304M16—1224no—
305M16—2442no—
306C2×C8—2442no—
307C2×C8—2442no—
308C4⋊C4—24no—
309C4⋊C4—21no—
310C4⋊C4—612no—
311C4⋊C4—612no—
312C4⋊C4—1222no—
313C4⋊C4—1224no—
314C4⋊C4—1224no—
315C4⋊C4—1224no—
316C4⋊C4—1224no—
317C4⋊C4—1224no—
318C4⋊C4—1221no—
319C4⋊C4—1221no—
320C4⋊C4C22⋊#692312yes—
321C22⋊C4—24no—
322C22⋊C4—1222no—
323C22⋊C4—1224no—
324C22⋊C4—1224no—
325C22⋊C4—1222no—
326C42—612no—
327C42—612no—
328C42—1224no—
329C42—1224no—
330C23—21no—
331C23—21no—
332C23—41no—
333C23—41no—
334C23—1221no—
335C23—1221no—
336C23—1221no—
337C23—2441no—
338C23—2441no—
339C23#1451984212yes—
340C23C2×C22⋊#692311yescharacteristic
341Q8—24no—
342Q8—42no—
343D4—2444no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—612no—
354C2×C4—612no—
355C2×C4—1221no—
356C2×C4—1221no—
357C2×C4—1222no—
358C2×C4—1221no—
359C2×C4—1222no—
360C2×C4—2442no—
361C2×C4—2441no—
362C2×C4—2448no—
363C2×C4—2441no—
364C2×C4—2441no—
365C2×C4—2441no—
366C2×C4—2441no—
367C2×C4—2442no—
368C2×C4—2441no—
369C2×C4C2×C22⋊#692311yescharacteristic
370C2×C4C2×C22⋊#692311yescharacteristic
371C8—4882no—
372C8—2444no—
373C22—21no—
374C22—21no—
375C22—22no—
376C22—42no—
377C22—42no—
378C22—41no—
379C22—41no—
380C22—4881no—
381C22—4881no—
382C22—2441no—
383C22—2441no—
384C22—2441no—
385C22#1684070311yescharacteristic
386C22C2×#1451984211yescharacteristic
387C22#1684076111yescharacteristic
388C4—21no—
389C4—41no—
390C4—42no—
391C4—42no—
392C4—1222no—
393C4—1222no—
394C4—2441no—
395C4—2442no—
396C4—2442no—
397C4#1684074912yes—
398C2—21no—
399C2—21no—
400C2—42no—
401C2—4881no—
402C2#2146165611yescharacteristic
403C2#2146166211yescharacteristic
404C2#2146167211yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61#379
prime radicalC22×C22⋊D61D4
cosocleC2×C4×Dic61C23
hypercenterC2×C22⋊Q8#6923
hypocenterC61#8371
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #21461283C2semidirect
3. #21461161C2semidirect
229. C61#8371semidirect

Maximal subgroups

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

subgroupindex
C4×#145198372
#214612832
#214611612
#214610782
C2×#168404542
#837161