Finite Group Databasebeta

← #353×#6923 #30606533

Subgroups — #353×#6923 #30606533

Order 15616.

720 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 96 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#127×#6923C211yesmaximal, characteristic
2D61⋊#353C211yesmaximal, characteristic
3#99×#6923C212yesmaximal, complemented
4#21460912C212yesmaximal, complemented
5#353×D61C211yesmaximal, characteristic
6C2×D4×#6923—22no—
7C2×D4×#6923C411yescharacteristic, complemented
8C2×D4×#6923C2211yescharacteristic
9D61⋊#99C2212yes—
10#16840714—22no—
11#16840714C411yescharacteristic, complemented
12#16840714C2211yescharacteristic
13#16840708C411yescharacteristic, complemented
14C22⋊C4×#6923—22no—
15C42×#6923C411yescharacteristic, complemented
16#16840667C412yescomplemented
17#16840666C412yescomplemented
18#16840639—22no—
19#127×D61C2211yescharacteristic
20#99×D61C2212yes—
21#16839555C411yescharacteristic, complemented
22#16839554C412yescomplemented
23C61×#353C411yescharacteristic, complemented
24C23×#6923—21no—
25C23×#6923D412yes—
26D4×#6923—41no—
27D4×#6923—41no—
28D4×#6923—41no—
29D4×#6923—41no—
30D4×#6923—48no—
31C2×C4⋊#6923—22no—
32C2×C4⋊#6923C2×C411yescharacteristic
33C2×C4⋊#6923C2×C411yescharacteristic
34C2×C22⋊#6923—22no—
35C2×C22⋊#6923—22no—
36C2×C22⋊#6923—21no—
37C2×C22⋊#6923—22no—
38C2×C22⋊#6923D412yes—
39C2×C4×#6923—21no—
40C2×C4×#6923—21no—
41C2×C4×#6923—42no—
42C2×C4×#6923C2×C411yescharacteristic
43C2×C4×#6923C2×C411yescharacteristic
44#14519836C2×C414yescomplemented
45#14519835—22no—
46(C2×#6923)⋊C4—42no—
47C2×D4×D61—22no—
48C2×D4×D61C2×C411yescharacteristic
49C2×D4×D61C2311yescharacteristic
50#14519774C2×C411yescharacteristic
51#14519773—22no—
52#14519773C2×C411yescharacteristic
53#14519773C2×C411yescharacteristic
54C22⋊C4×D61—22no—
55C42×D61C2×C411yescharacteristic
56#14519692C2×C412yescomplemented
57#14519691—22no—
58#127×C61C2×C411yescharacteristic
59#99×C61C2×C412yescomplemented
60C23×D61—21no—
61C23×D61C2×D412yes—
62C22×#6923—21no—
63C22×#6923—22no—
64C22×#6923—22no—
65C22×#6923—21no—
66C22×#6923—44no—
67C22×#6923—44no—
68C22×#6923—42no—
69C22×#6923—41no—
70C22×#6923—41no—
71C22×#6923C22⋊C411yescharacteristic
72C22×#6923C22⋊C411yescharacteristic
73C22×#6923C22⋊C412yes—
74C2×D4×C61—22no—
75C2×D4×C61C4211yescharacteristic, complemented
76C2×D4×C61C22×C411yescharacteristic
77C2×C22⋊D61—22no—
78C2×C22⋊D61—22no—
79C2×C22⋊D61—21no—
80C2×C22⋊D61#4112yes—
81D4×D61—41no—
82D4×D61—41no—
83D4×D61—44no—
84D4×D61—44no—
85D4×D61—41no—
86D4×D61—41no—
87C2×D244—22no—
88C2×D244C4211yescharacteristic, complemented
89C2×D244C22×C411yescharacteristic
90C2×C4×D61—21no—
91C2×C4×D61—42no—
92C2×C4×D61C22×C411yescharacteristic
93C22⋊#6923—24no—
94C22⋊#6923—42no—
95C22⋊#6923—44no—
96C22⋊#6923—44no—
97C22⋊#6923—42no—
98C22⋊#6923—42no—
99C4⋊#6923—41no—
100C4⋊#6923—44no—
101C4⋊#6923—41no—
102C4×#6923—24no—
103C4×#6923—21no—
104C4×#6923—21no—
105C4×#6923—42no—
106C4×#6923—41no—
107C4×#6923—84no—
108C22⋊C4×C61—22no—
109C42×C61C4211yescharacteristic, complemented
110C22⋊Dic61—22no—
111#11758198—42no—
112C4×Dic61—21no—
113C4×Dic61C4211yescharacteristic, complemented
114C23×C61—21no—
115C23×C61C4×D412yes—
116C22×D61—22no—
117C22×D61—22no—
118C22×D61—21no—
119C22×D61—42no—
120C22×D61—42no—
121C22×D61—42no—
122C22×D61—42no—
123C22×D61—41no—
124C22×D61—41no—
125C22×D61C2×C22⋊C411yescharacteristic
126C22×D61C4×D412yes—
127C2×#6923—22no—
128C2×#6923—22no—
129C2×#6923—21no—
130C2×#6923—22no—
131C2×#6923—41no—
132C2×#6923—41no—
133C2×#6923—44no—
134C2×#6923—41no—
135C2×#6923—42no—
136C2×#6923—42no—
137C2×#6923—44no—
138C2×#6923—41no—
139C2×#6923—82no—
140C2×#6923—82no—
141C2×#6923—82no—
142C2×#6923—82no—
143C2×#6923—82no—
144C2×#6923#9911yescharacteristic
145C2×#6923#9911yescharacteristic
146D4×C61—41no—
147D4×C61—41no—
148D4×C61—44no—
149C2×C4×C61—21no—
150C2×C4×C61—42no—
151C2×C4×C61C2×C4211yescharacteristic
152C22⋊D61—42no—
153C22⋊D61—44no—
154C22⋊D61—44no—
155C22⋊D61—42no—
156C2×Dic61—21no—
157C2×Dic61—42no—
158C2×Dic61#11711yescharacteristic
159D244—41no—
160D244—41no—
161D244—44no—
162C4×D61—22no—
163C4×D61—22no—
164C4×D61—21no—
165C4×D61—21no—
166C4×D61—82no—
167C4×D61—82no—
168C4×#353—611nomaximal, Sylow
169C22×C61—22no—
170C22×C61—42no—
171C22×C61—42no—
172C22×C61—41no—
173C22×C61C4×C22⋊C411yescharacteristic
174C2×D61—21no—
175C2×D61—21no—
176C2×D61—22no—
177C2×D61—42no—
178C2×D61—42no—
179C2×D61—42no—
180C2×D61—42no—
181C2×D61—41no—
182C2×D61—82no—
183C2×D61—82no—
184C2×D61—81no—
185C2×D61—81no—
186C2×D61C2×#9911yescharacteristic
187C2×D61#41011yescharacteristic
188#6923—21no—
189#6923—42no—
190#6923—42no—
191#6923—41no—
192#6923—84no—
193#6923—81no—
194#6923#35312yesdirect product factor
195C4×C61—22no—
196C4×C61—21no—
197C4×C61—82no—
198Dic61—22no—
199Dic61—21no—
200Dic61—82no—
201C4×#127—611no—
202C2×#353—611no—
203C2×#353—611no—
204C4×#99—612no—
205#1393—612no—
206C2×C61—21no—
207C2×C61—42no—
208C2×C61—81no—
209C2×C61C4×#9911yescharacteristic
210D61—21no—
211D61—42no—
212D61—81no—
213D61C2×#35311yescharacteristic
214D61#212611yescharacteristic
215C2×#127—611no—
216C2×C4×D4—611no—
217C2×C4×D4—611no—
218C2×C4×D4—1222no—
219C2×#99—612no—
220C2×#99—612no—
221#390—611no—
222#390—611no—
223#390—1222no—
224#384—611no—
225C4×C22⋊C4—1222no—
226C43—611no—
227#353—612no—
228#353—612no—
229#353—611no—
230#353—612no—
231#353#692311yescharacteristic, direct product factor
232#342—1222no—
233C61C4×#35311yescharacteristic, complemented, Sylow
234C22×D4—611no—
235C22×D4—611no—
236C22×D4—1222no—
237C23×C4—612no—
238C23×C4—1221no—
239#127—611no—
240#127—611no—
241#127—611no—
242#127—1222no—
243#127C2×#692311yescharacteristic
244C4×D4—2441no—
245C4×D4—2441no—
246C4×D4—2448no—
247C4×D4—2441no—
248C4×D4—2441no—
249C2×C4⋊C4—611no—
250C2×C4⋊C4—611no—
251C2×C4⋊C4—1222no—
252C2×C22⋊C4—612no—
253C2×C22⋊C4—1222no—
254C2×C22⋊C4—1222no—
255C2×C22⋊C4—1221no—
256C2×C22⋊C4—1222no—
257C2×C22⋊C4—1222no—
258C2×C42—611no—
259C2×C42—611no—
260C2×C42—611no—
261C2×C42—1221no—
262C2×C42—1221no—
263C2×C42—2442no—
264#99—614no—
265#99—612no—
266#99—1222no—
267#99—1222no—
268#99C2×#692312yescomplemented
269#95—2442no—
270C24—612no—
271C24—1221no—
272C2×D4—22no—
273C2×D4—612no—
274C2×D4—611no—
275C2×D4—611no—
276C2×D4—1222no—
277C2×D4—1221no—
278C2×D4—1222no—
279C2×D4—1222no—
280C2×D4—2441no—
281C2×D4—2441no—
282C2×D4—2441no—
283C2×D4—2441no—
284C2×D4—2444no—
285C2×D4—2444no—
286C2×D4C4×#692311yescharacteristic, complemented
287C2×D4C22×#692311yescharacteristic
288C22×C4—611no—
289C22×C4—611no—
290C22×C4—611no—
291C22×C4—612no—
292C22×C4—1221no—
293C22×C4—1222no—
294C22×C4—1222no—
295C22×C4—1221no—
296C22×C4—1221no—
297C22×C4—2442no—
298C22×C4—2441no—
299C22×C4—2441no—
300C22×C4—2444no—
301C22×C4—2444no—
302C22×C4—2442no—
303C4⋊C4—2441no—
304C4⋊C4—2441no—
305C4⋊C4—2444no—
306C22⋊C4—22no—
307C22⋊C4—1224no—
308C22⋊C4—1222no—
309C22⋊C4—2442no—
310C22⋊C4—2442no—
311C22⋊C4—2442no—
312C22⋊C4—2444no—
313C22⋊C4—2444no—
314C22⋊C4—2442no—
315C42—611no—
316C42—1221no—
317C42—1221no—
318C42—1221no—
319C42—1224no—
320C42—4884no—
321C42—2441no—
322C42—2442no—
323C42C4×#692311yescharacteristic, complemented
324C23—21no—
325C23—611no—
326C23—612no—
327C23—1222no—
328C23—1222no—
329C23—1221no—
330C23—2441no—
331C23—2441no—
332C23—2442no—
333C23—2442no—
334C23—2442no—
335C23—2442no—
336C23D4×#692312yes—
337D4—41no—
338D4—41no—
339D4—44no—
340D4—2441no—
341D4—2441no—
342D4—2444no—
343D4—2442no—
344D4—2444no—
345D4—2444no—
346D4—2442no—
347C2×C4—21no—
348C2×C4—42no—
349C2×C4—611no—
350C2×C4—611no—
351C2×C4—611no—
352C2×C4—1221no—
353C2×C4—1221no—
354C2×C4—1222no—
355C2×C4—1222no—
356C2×C4—1222no—
357C2×C4—1221no—
358C2×C4—1221no—
359C2×C4—1222no—
360C2×C4—1222no—
361C2×C4—4882no—
362C2×C4—4882no—
363C2×C4—4882no—
364C2×C4—4882no—
365C2×C4—4882no—
366C2×C4—4882no—
367C2×C4—4882no—
368C2×C4—2444no—
369C2×C4—2442no—
370C2×C4—2441no—
371C2×C4—2444no—
372C2×C4—2442no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4—2442no—
377C2×C4C2×C4×#692311yescharacteristic
378C22—22no—
379C22—42no—
380C22—42no—
381C22—41no—
382C22—611no—
383C22—611no—
384C22—1221no—
385C22—1221no—
386C22—1222no—
387C22—4882no—
388C22—4881no—
389C22—4881no—
390C22—4882no—
391C22—2442no—
392C22—2442no—
393C22—2442no—
394C22—2442no—
395C22—2441no—
396C22C22⋊C4×#692311yescharacteristic
397C4—22no—
398C4—21no—
399C4—82no—
400C4—612no—
401C4—1222no—
402C4—1221no—
403C4—1221no—
404C4—4882no—
405C4—4881no—
406C4—4884no—
407C4—2442no—
408C4—2441no—
409C4—2442no—
410C2—21no—
411C2—42no—
412C2—81no—
413C2—611no—
414C2—611no—
415C2—1221no—
416C2—4881no—
417C2—2442no—
418C2#99×#692311yescharacteristic
subgroupG/H
centerC2#99×#6923
commutatorC2×C4×C61C2×C42
FrattiniC2×D4C22×#6923
FittingC61×#353C4
socleC2×C61C4×#99
prime radical#127×D61C22
cosocleC2×D4×D61C23
hypercenter#353#6923
hypocenterC61C4×#353
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
3. #99×#6923C2semidirect
4. #21460912C2semidirect
7. C2×D4×#6923C4semidirect
11. #16840714C4semidirect
13. #16840708C4semidirect
15. C42×#6923C4semidirect
16. #16840667C4semidirect
17. #16840666C4semidirect
21. #16839555C4semidirect
22. #16839554C4semidirect
23. C61×#353C4semidirect
44. #14519836C2×C4semidirect
56. #14519692C2×C4semidirect
59. #99×C61C2×C4semidirect
75. C2×D4×C61C42semidirect
88. C2×D244C42semidirect
109. C42×C61C42semidirect
113. C4×Dic61C42semidirect
194. #6923#353direct (N × H)
231. #353#6923direct (N × H)
233. C61C4×#353semidirect
268. #99C2×#6923semidirect
286. C2×D4C4×#6923semidirect
323. C42C4×#6923semidirect

Maximal subgroups

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

subgroupindex
#127×#69232
D61⋊#3532
#99×#69232
#214609122
#353×D612
C4×#35361