Finite Group Databasebeta

← #29589121

Subgroups — #29589121

Order 15616.

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

normal subgroups only

This group has 594 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
1#21448412C211yesmaximal, characteristic
2C2×#16839488C211yesmaximal, characteristic, complemented
3#21443752C212yesmaximal, complemented
4#21443750C212yesmaximal, complemented
5C2×C61×#324C211yesmaximal, characteristic, complemented
6C22×#11758222C411yescharacteristic
7C2×#14519736C411yescharacteristic
8C2×C22⋊#92418—21no—
9#16839667C411yescharacteristic
10#16839667C2211yescharacteristic
11#16839663C411yescharacteristic
12#16839663C2211yescharacteristic
13#16839641—21no—
14#16839641—21no—
15#16839619—21no—
16#16839619—21no—
17#16839488—21no—
18#16839488C2212yes—
19C22×C4⋊C4×C61C2211yescharacteristic
20C2×C22⋊C8×C61—21no—
21C61×#324—21no—
22C61×#324C2212yes—
23C2×#11758222—22no—
24C2×#11758222—41no—
25C2×#11758222—41no—
26C2×#11758222—42no—
27C2×#11758222C814yescomplemented
28C22×Dic122C2×C411yescharacteristic
29C22×D244C2×C411yescharacteristic
30C22×C22⋊D61—21no—
31C22×C4×D61—21no—
32C22×#92418—41no—
33#14519736—42no—
34C2×C4⋊Dic61—21no—
35C2×C22⋊Dic61—21no—
36C2×#11758198—21no—
37C2×#11758198—21no—
38C2×#11758198—21no—
39C2×#11758198—21no—
40C2×C4×Dic61C2×C411yescharacteristic
41C2×C4×Dic61C2×C411yescharacteristic
42C22⋊#92418—22no—
43C22⋊#92418—41no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48C23×C4×C61C2×C411yescharacteristic
49C23×C4×C61D411yescharacteristic
50C23×C4×C61D411yescharacteristic
51C22×C8×C61—41no—
52C2×C4⋊C4×C61—41no—
53C2×C4⋊C4×C61—41no—
54C2×C4⋊C4×C61C2×C411yescharacteristic
55C2×C4⋊C4×C61D411yescharacteristic
56C2×C4⋊C4×C61D411yescharacteristic
57C2×C4⋊C4×C61C2311yescharacteristic
58C22⋊C8×C61—22no—
59C22⋊C8×C61—41no—
60C24×C61C22⋊C411yescharacteristic
61C23×D61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—22no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—81no—
70C22×C4×C61—81no—
71C22×C4×C61C22⋊C411yescharacteristic
72C22×C4×C61C22⋊C411yescharacteristic
73C22×C4×C61C22⋊C411yescharacteristic
74C22×C4×C61C2×C812yes—
75C22×C4×C61M1612yes—
76C22×C4×C61C22×C411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C2×C22⋊D61—24no—
80C2×C22⋊D61—41no—
81C2×C22⋊D61—41no—
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—21no—
91C2×D244—41no—
92C2×D244—41no—
93C2×D244C2×C812yescomplemented
94C2×C4×D61—24no—
95C2×C4×D61—42no—
96C2×C4×D61—42no—
97C2×Dic122—21no—
98C2×Dic122—41no—
99C2×Dic122—41no—
100C2×Dic122C2×C812yescomplemented
101C2×C8×C61—42no—
102C2×C8×C61—82no—
103C4⋊C4×C61—41no—
104C4⋊C4×C61—41no—
105C4⋊C4×C61—41no—
106C4⋊C4×C61—41no—
107C22⋊Dic61—42no—
108#11758198—81no—
109#11758198—81no—
110#11758198—81no—
111#11758198—81no—
112C4⋊Dic61—42no—
113C4×Dic61—41no—
114C4×Dic61—41no—
115C2×#92418—42no—
116C2×#92418—82no—
117C23×C61—41no—
118C23×C61—41no—
119C23×C61C22⋊C812yes—
120C23×C61#9911yescharacteristic
121C23×C61#9911yescharacteristic
122C23×C61#10012yes—
123C23×C61C2×C22⋊C411yescharacteristic
124C22×D61—22no—
125C22×D61—41no—
126C22×D61—41no—
127C22×D61—41no—
128C22×D61—41no—
129C22×D61—41no—
130C22×D61—41no—
131C2×C4×C61—22no—
132C2×C4×C61—22no—
133C2×C4×C61—22no—
134C2×C4×C61—22no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—42no—
144C2×C4×C61—42no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—81no—
150C2×C4×C61—81no—
151C2×C4×C61C22⋊C812yes—
152C2×C4×C61C22×C811yescharacteristic
153C2×C4×C61C2×M1611yescharacteristic
154C22⋊D61—44no—
155C22⋊D61—44no—
156C2×Dic61—22no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165D244—42no—
166D244—42no—
167C4×D61—48no—
168Dic122—42no—
169Dic122—42no—
170C8×C61—82no—
171#92418—82no—
172#11432—611nomaximal, Sylow
173C22×C61—21no—
174C22×C61—21no—
175C22×C61—21no—
176C22×C61—21no—
177C22×C61—21no—
178C22×C61—21no—
179C22×C61—42no—
180C22×C61—42no—
181C22×C61#32312yes—
182C22×C61#32312yes—
183C22×C61C2×C22⋊C811yescharacteristic
184C22×C61C2×#9911yescharacteristic
185C22×C61C2×#10011yescharacteristic
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—21no—
195C4×C61—22no—
196C4×C61—41no—
197C4×C61—41no—
198C4×C61—41no—
199C4×C61—41no—
200Dic61—42no—
201Dic61—81no—
202Dic61—81no—
203#2311—611no—
204C2×#324—611no—
205C2×#324D6111yescharacteristic, complemented
206#1319—612no—
207#1318—612no—
208C2×C61—21no—
209C2×C61—21no—
210C2×C61—22no—
211C2×C61—22no—
212C2×C61C2×#32311yescharacteristic
213C2×C61#212711yescharacteristic
214C2×C61#213311yescharacteristic
215D61—82no—
216C22×#41—611no—
217C2×#117—611no—
218C22×C4⋊C4C2×D6111yescharacteristic
219C2×C22⋊C8—21no—
220C2×C22⋊C8—1221no—
221#391—611no—
222#391—611no—
223#390—611no—
224#390—611no—
225#386—1221no—
226#386—1221no—
227#380—1221no—
228#380—1221no—
229#324—21no—
230#324—612no—
231#324—1221no—
232#324C2×D6112yes—
233C61#1143211yescharacteristic, complemented, Sylow
234C2×#41—614no—
235C2×#41—1222no—
236C2×#41—2441no—
237C2×#41—2441no—
238C2×#41—2442no—
239C22×Q8—611no—
240C22×D4—611no—
241C22×D4—1221no—
242C23×C4—1221no—
243C23×C4C4×D6111yescharacteristic
244C23×C4D24411yescharacteristic
245C23×C4C22⋊D6111yescharacteristic
246C22×C8—41no—
247C22×C8—2441no—
248#117—2442no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—1221no—
252C2×C4⋊C4C4×D6111yescharacteristic
253C2×C4⋊C4D24411yescharacteristic
254C2×C4⋊C4C22⋊D6111yescharacteristic
255C2×C4⋊C4C22×D6111yescharacteristic
256C2×C22⋊C4—1221no—
257C2×C22⋊C4—1221no—
258C2×C22⋊C4—1221no—
259C2×C22⋊C4—1221no—
260C2×C22⋊C4—1221no—
261C2×C42—611no—
262C2×C42—611no—
263C22⋊C8—22no—
264C22⋊C8—41no—
265C22⋊C8—1222no—
266C22⋊C8—2441no—
267#95—1221no—
268#95—1221no—
269#95—1221no—
270#95—1221no—
271C24—1221no—
272C24#1175819811yescharacteristic
273#41—2444no—
274#41—2444no—
275#41—2444no—
276#41—2444no—
277C2×Q8—612no—
278C2×Q8—1221no—
279C2×Q8—2441no—
280C2×Q8—2441no—
281C2×D4—612no—
282C2×D4—1224no—
283C2×D4—1221no—
284C2×D4—2441no—
285C2×D4—2441no—
286C2×D4—2441no—
287C2×D4—2441no—
288C2×D4—2442no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—81no—
297C22×C4—81no—
298C22×C4—1221no—
299C22×C4—1224no—
300C22×C4—1221no—
301C22×C4—1221no—
302C22×C4—2442no—
303C22×C4—2442no—
304C22×C4C8×D6112yes—
305C22×C4#1175818912yes—
306C22×C4#1175819811yescharacteristic
307C22×C4#1175819811yescharacteristic
308C22×C4#1175819811yescharacteristic
309C22×C4C2×C4×D6111yescharacteristic
310C22×C4C2×D24411yescharacteristic
311C22×C4C2×C22⋊D6111yescharacteristic
312C2×C8—42no—
313C2×C8—82no—
314C2×C8—4882no—
315C2×C8—2442no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—2442no—
321C22⋊C4—4881no—
322C22⋊C4—4881no—
323C22⋊C4—4881no—
324C22⋊C4—4881no—
325C22⋊C4—2442no—
326C42—2441no—
327C42—2441no—
328C23—41no—
329C23—41no—
330C23—1222no—
331C23—2441no—
332C23—2441no—
333C23—2441no—
334C23—2441no—
335C23—2441no—
336C23—2441no—
337C23#1451968912yes—
338C23#1451969111yescharacteristic
339C23#1451969111yescharacteristic
340C23#1451969312yes—
341C23C2×#1175819811yescharacteristic
342Q8—2442no—
343Q8—2442no—
344D4—2442no—
345D4—2442no—
346D4—2444no—
347D4—2444no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—1222no—
369C2×C4—2448no—
370C2×C4—2441no—
371C2×C4—2441no—
372C2×C4—2441no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4#1451968912yes—
379C2×C4C2×C8×D6111yescharacteristic
380C2×C4C2×#1175818911yescharacteristic
381C8—82no—
382C8—4882no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—42no—
390C22—42no—
391C22—4881no—
392C22—4881no—
393C22—4881no—
394C22—4881no—
395C22—2442no—
396C22—2442no—
397C22—2442no—
398C22#1683948512yes—
399C22#1683948512yes—
400C22C2×#1451968911yescharacteristic
401C22C2×#1451969111yescharacteristic
402C22C2×#1451969311yescharacteristic
403C4—22no—
404C4—21no—
405C4—21no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—4881no—
411C4—4881no—
412C4—2442no—
413C2—21no—
414C2—21no—
415C2—22no—
416C2—22no—
417C2—4882no—
418C2C2×#1683948511yescharacteristic
419C2#2144748611yescharacteristic
420C2#2144750611yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC2×C4×C61C22×C8
FrattiniC2×C4⋊C4C22×D61
FittingC2×C61×#324C2
socleC22×C61C2×#99
prime radicalC22×#11758222C4
cosocleC2×C4⋊C4×C61C23
hypercenterC2×#324D61
hypocenterC61#11432
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. C2×#16839488C2semidirect
3. #21443752C2semidirect
4. #21443750C2semidirect
5. C2×C61×#324C2semidirect
27. C2×#11758222C8semidirect
93. C2×D244C2×C8semidirect
100. C2×Dic122C2×C8semidirect
205. C2×#324D61semidirect
233. C61#11432semidirect

Maximal subgroups

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

subgroupindex
#214484122
C2×#168394882
#214437522
#214437502
C2×C61×#3242
#1143261