Finite Group Databasebeta

← #29590335

Subgroups — #29590335

Order 15616.

648 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 87 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#21449034C211yesmaximal, characteristic
2#21444140C211yesmaximal, characteristic, complemented
3#21443839C212yesmaximal, complemented
4#21443815C212yesmaximal, complemented
5C4×#99×C61C211yesmaximal, characteristic, complemented
6C22×#11758222C411yescharacteristic, complemented
7C2×C4×C22⋊D61—21no—
8#16839815—21no—
9#16839682C411yescharacteristic, complemented
10#16839679C411yescharacteristic, complemented
11#16839677—21no—
12#16839674C2211yescharacteristic
13#16839671C2211yescharacteristic
14#16839670—21no—
15C4×#11758198—21no—
16C4×C22⋊Dic61C411yescharacteristic, complemented
17#16839486—21no—
18#16839486C2212yes—
19C2×C4×D4×C61C2211yescharacteristic
20C2×#99×C61—21no—
21C4×C22⋊C4×C61—21no—
22C61×#342C2212yes—
23C2×#11758222—44no—
24C2×#11758222—41no—
25C2×#11758222—41no—
26C22×Dic122C2×C411yescharacteristic
27C22×D244C2×C411yescharacteristic
28C22×C22⋊D61—21no—
29C22×C22⋊D61—21no—
30C22×C4×D61—21no—
31C4×C22⋊D61—44no—
32C2×#11758196—21no—
33C2×C22⋊Dic61—21no—
34C2×C22⋊Dic61—21no—
35C2×#11758198—21no—
36C2×#11758198—21no—
37C2×#11758198—21no—
38C2×C4×Dic61—21no—
39C4⋊#92418—41no—
40C22⋊#92418—22no—
41C4×#92418—41no—
42#14519684—21no—
43#14519684—21no—
44#14519684C2×C411yescharacteristic
45#14519684C2×C411yescharacteristic
46C22×D4×C61D411yescharacteristic
47C23×C4×C61C2×C411yescharacteristic
48C23×C4×C61D411yescharacteristic
49C4×D4×C61—41no—
50C4×D4×C61—41no—
51C2×C4⋊C4×C61C2×C411yescharacteristic
52C2×C22⋊C4×C61—22no—
53C2×C22⋊C4×C61D411yescharacteristic
54C2×C22⋊C4×C61C2311yescharacteristic
55C2×C42×C61—41no—
56C2×C42×C61D411yescharacteristic
57#99×C61—24no—
58#95×C61—41no—
59C24×C61C22⋊C411yescharacteristic
60C24×C61C2×D411yescharacteristic
61C23×D61—21no—
62C2×D4×C61—22no—
63C2×D4×C61—22no—
64C2×D4×C61—41no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—42no—
70C22×C4×C61—42no—
71C22×C4×C61—42no—
72C22×C4×C61C22⋊C411yescharacteristic
73C22×C4×C61C22⋊C411yescharacteristic
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22×C411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C2×C22⋊D61—42no—
78C2×C22⋊D61—42no—
79C2×C22⋊D61—42no—
80C2×C22⋊D61—42no—
81C2×C22⋊D61—42no—
82C2×C22⋊D61—42no—
83C22×Dic61—21no—
84C22×Dic61—21no—
85C22×Dic61—21no—
86#11758222—44no—
87#11758222—44no—
88#11758222—44no—
89#11758222—44no—
90C2×D244—41no—
91C2×D244—42no—
92C2×C4×D61—24no—
93C2×C4×D61—42no—
94C2×C4×D61—41no—
95C2×C4×D61—41no—
96C2×Dic122—41no—
97C2×Dic122—42no—
98C4⋊C4×C61—41no—
99C22⋊C4×C61—24no—
100C22⋊C4×C61—41no—
101C22⋊C4×C61—41no—
102C22⋊C4×C61—42no—
103C42×C61—21no—
104C42×C61—21no—
105C42×C61—82no—
106C22⋊Dic61—42no—
107C22⋊Dic61—42no—
108#11758198—42no—
109#11758198—42no—
110#11758198—42no—
111#11758196—42no—
112C4×Dic61—42no—
113C2×#92418—42no—
114C23×C61—21no—
115C23×C61—21no—
116C23×C61—21no—
117C23×C61—21no—
118C23×C61—21no—
119C23×C61—41no—
120C23×C61—41no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61C2×C22⋊C411yescharacteristic
124C23×C61#13512yes—
125C22×D61—22no—
126C22×D61—41no—
127C22×D61—41no—
128C22×D61—41no—
129C22×D61—41no—
130C22×D61—41no—
131C22×D61—41no—
132D4×C61—44no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—22no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—42no—
140C2×C4×C61—42no—
141C2×C4×C61—42no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—44no—
146C2×C4×C61—81no—
147C2×C4×C61—82no—
148C2×C4×C61—82no—
149C2×C4×C61#9911yescharacteristic
150C2×C4×C61#9911yescharacteristic
151C2×C4×C61#10412yes—
152C22⋊D61—48no—
153C22⋊D61—48no—
154C2×Dic61—22no—
155C2×Dic61—22no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163D244—42no—
164D244—42no—
165C4×D61—44no—
166C4×D61—44no—
167Dic122—42no—
168Dic122—42no—
169#92418—82no—
170#11725—611nomaximal, Sylow
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—22no—
174C22×C61—42no—
175C22×C61—41no—
176C22×C61—41no—
177C22×C61—41no—
178C22×C61—41no—
179C22×C61—41no—
180C22×C61—81no—
181C22×C61—81no—
182C22×C61—81no—
183C22×C61C2×#9911yescharacteristic
184C22×C61#41611yescharacteristic
185C22×C61C2×#10411yescharacteristic
186C2×D61—42no—
187C2×D61—42no—
188C2×D61—42no—
189C2×D61—81no—
190C2×D61—81no—
191C2×D61—81no—
192C2×D61—81no—
193C4×C61—21no—
194C4×C61—22no—
195C4×C61—41no—
196C4×C61—41no—
197C4×C61—41no—
198C4×C61—84no—
199C4×C61AΣL(1,𝔽16)12yes—
200Dic61—42no—
201Dic61—42no—
202Dic61—81no—
203#2388—611no—
204C4×#99D6111yescharacteristic, complemented
205#1460—611no—
206#1351—612no—
207#1343—612no—
208C2×C61—21no—
209C2×C61—21no—
210C2×C61—42no—
211C2×C61—81no—
212C2×C61#150111yescharacteristic
213C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
214C2×C61#212611yescharacteristic
215D61—82no—
216C22×#41—611no—
217C2×C4×D4—1221no—
218C2×C4×D4C2×D6111yescharacteristic
219#431—1221no—
220C2×#99—21no—
221#395—611no—
222#394—611no—
223#394—1221no—
224#393—611no—
225#392—611no—
226#392—1221no—
227C4×C22⋊C4—21no—
228C4×C22⋊C4—611no—
229C4×C22⋊C4—1221no—
230#342C2×D6112yes—
231#323—612no—
232#323—1221no—
233C61#1172511yescharacteristic, complemented, Sylow
234C2×#41—2441no—
235C2×#41—2441no—
236C2×#41—2444no—
237C22×Q8—611no—
238C22×D4—611no—
239C22×D4—1221no—
240C22×D4—1221no—
241C22×D4C22⋊D6111yescharacteristic
242C23×C4—1221no—
243C23×C4C4×D6111yescharacteristic
244C23×C4C22⋊D6111yescharacteristic
245C4×D4—41no—
246C4×D4—41no—
247C4×D4—2444no—
248C2×C4⋊C4—1221no—
249C2×C4⋊C4C4×D6111yescharacteristic
250C2×C22⋊C4—22no—
251C2×C22⋊C4—1221no—
252C2×C22⋊C4—1221no—
253C2×C22⋊C4—1221no—
254C2×C22⋊C4—1221no—
255C2×C22⋊C4—1221no—
256C2×C22⋊C4D24411yescharacteristic
257C2×C22⋊C4C22×D6111yescharacteristic
258C2×C42—41no—
259C2×C42—1221no—
260C2×C42D24411yescharacteristic
261C4⋊C8—2441no—
262#99—24no—
263C22⋊C8—1222no—
264C4×C8—2441no—
265#95—41no—
266#95—611no—
267#95—611no—
268#95—1221no—
269#95—1221no—
270C24—1221no—
271C24#1175819811yescharacteristic
272C24C2×C22⋊D6111yescharacteristic
273#41—2444no—
274#41—2444no—
275#41—2444no—
276#41—2444no—
277C2×Q8—2441no—
278C2×Q8—2442no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—41no—
282C2×D4—2441no—
283C2×D4—2442no—
284C2×D4—2442no—
285C2×D4—2442no—
286C2×D4—2442no—
287C2×D4—2442no—
288C2×D4—2442no—
289C2×D4—2442no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—1224no—
298C22×C4—1221no—
299C22×C4—1221no—
300C22×C4—1221no—
301C22×C4—2442no—
302C22×C4—2441no—
303C22×C4—2441no—
304C22×C4#1175819811yescharacteristic
305C22×C4#1175819811yescharacteristic
306C22×C4#1175819811yescharacteristic
307C22×C4C2×C4×D6111yescharacteristic
308C22×C4C2×D24411yescharacteristic
309C2×C8—2442no—
310C4⋊C4—41no—
311C4⋊C4—2442no—
312C22⋊C4—24no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—42no—
316C22⋊C4—2442no—
317C22⋊C4—2442no—
318C22⋊C4—2442no—
319C22⋊C4—2442no—
320C22⋊C4—2442no—
321C42—21no—
322C42—21no—
323C42—82no—
324C42—2442no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—1222no—
335C23—2441no—
336C23—2441no—
337C23—2441no—
338C23—2441no—
339C23—2441no—
340C23—2441no—
341C23C2×#1175819811yescharacteristic
342C23#1451979712yes—
343Q8—2442no—
344Q8—2442no—
345D4—44no—
346D4—2442no—
347D4—2442no—
348D4—2448no—
349D4—2448no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—44no—
363C2×C4—81no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—1222no—
367C2×C4—1222no—
368C2×C4—2444no—
369C2×C4—2441no—
370C2×C4—2441no—
371C2×C4—2441no—
372C2×C4—2441no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4—2444no—
377C2×C4#1451969111yescharacteristic
378C2×C4#1451969111yescharacteristic
379C2×C4#1451970512yes—
380C8—4882no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—4881no—
394C22—4881no—
395C22—4881no—
396C22—4881no—
397C22—2442no—
398C22—2442no—
399C22—2442no—
400C22C2×#1451969111yescharacteristic
401C22#1683975611yescharacteristic
402C22C2×#1451970511yescharacteristic
403C4—21no—
404C4—22no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—84no—
409C4—4881no—
410C4—2442no—
411C4—2442no—
412C4#1683954812yes—
413C2—21no—
414C2—21no—
415C2—42no—
416C2—81no—
417C2—4882no—
418C2#2144430811yescharacteristic
419C2C2×#1683954811yescharacteristic
420C2#2144748211yescharacteristic
subgroupG/H
centerC2×C4#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC4×#99×C61C2
socleC22×C61C2×#99
prime radical#21449034C2
cosocleC2×C22⋊C4×C61C23
hypercenterC4×#99D61
hypocenterC61#11725
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
2. #21444140C2semidirect
3. #21443839C2semidirect
4. #21443815C2semidirect
5. C4×#99×C61C2semidirect
6. C22×#11758222C4semidirect
9. #16839682C4semidirect
10. #16839679C4semidirect
16. C4×C22⋊Dic61C4semidirect
204. C4×#99D61semidirect
233. C61#11725semidirect

Maximal subgroups

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

subgroupindex
#214490342
#214441402
#214438392
#214438152
C4×#99×C612
#1172561