Finite Group Databasebeta

← #29595993

Subgroups — #29595993

Order 15616.

680 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 107 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×#14519773C211yesmaximal, characteristic
2C2×#16839554C212yesmaximal
3#21445807C211yesmaximal, characteristic, complemented
4#21443943C212yesmaximal, complemented
5C61×#1799C211yesmaximal, characteristic, complemented
6C2×#14519773C2211yescharacteristic
7C2×C4×D244C411yescharacteristic, complemented
8C2×C4×C22⋊D61—21no—
9C2×C4×C22⋊D61—21no—
10C2×#14519692C2212yes—
11#16839665C2211yescharacteristic
12#16839663C411yescharacteristic, complemented
13#16839662—21no—
14#16839662—21no—
15#16839631C411yescharacteristic, complemented
16#16839622—21no—
17C42×Dic61C411yescharacteristic, complemented
18#16839554—24no—
19#16839529—21no—
20C2×C4×D4×C61C2211yescharacteristic
21C2×#99×C61C2212yes—
22C61×#380—21no—
23C61×#342—21no—
24C22×D244C2×C411yescharacteristic
25C22×C22⋊D61—21no—
26C22×C22⋊D61—21no—
27C22×C4×D61—21no—
28#14519773—21no—
29#14519773—21no—
30#14519773D412yes—
31#14519773Q812yes—
32C4×D244—41no—
33C4×D244—41no—
34C4×C22⋊D61—44no—
35C4×C22⋊D61—44no—
36C2×C4⋊Dic61—41no—
37C2×C4⋊Dic61C2×C411yescharacteristic
38C2×#11758196—21no—
39C2×#11758196—21no—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61—22no—
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×#11758198—21no—
46C2×C4×Dic61—21no—
47C2×C4×Dic61—21no—
48C2×C4×Dic61C2×C411yescharacteristic
49C2×C4×Dic61C2×C411yescharacteristic
50#14519692—22no—
51#14519692—22no—
52#14519684—41no—
53C22×D4×C61C2311yescharacteristic
54C23×C4×C61D411yescharacteristic
55C23×C4×C61D411yescharacteristic
56C4×D4×C61—41no—
57C4×D4×C61—41no—
58C2×C4⋊C4×C61—41no—
59C2×C4⋊C4×C61C2×C411yescharacteristic
60C2×C22⋊C4×C61—22no—
61C2×C22⋊C4×C61D411yescharacteristic
62C2×C22⋊C4×C61D411yescharacteristic
63C2×C42×C61C2×C411yescharacteristic
64#99×C61—22no—
65#99×C61—22no—
66#95×C61—41no—
67C24×C61C2×D411yescharacteristic
68C24×C61C2×D411yescharacteristic
69C23×D61—21no—
70C2×D4×C61—41no—
71C2×D4×C61—41no—
72C2×D4×C61C2×D411yescharacteristic
73C2×D4×C61C2×Q811yescharacteristic
74C2×D4×C61#4111yescharacteristic
75C2×D4×C61#4111yescharacteristic
76C22×C4×C61—21no—
77C22×C4×C61—21no—
78C22×C4×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61—42no—
81C22×C4×C61—42no—
82C22×C4×C61—42no—
83C22×C4×C61C22⋊C411yescharacteristic
84C22×C4×C61C22⋊C411yescharacteristic
85C22×C4×C61C22⋊C411yescharacteristic
86C22×C4×C61C22⋊C411yescharacteristic
87C22×C4×C61C22×C411yescharacteristic
88C2×C22⋊D61—22no—
89C2×C22⋊D61—22no—
90C2×C22⋊D61—22no—
91C2×C22⋊D61—22no—
92C2×C22⋊D61—42no—
93C2×C22⋊D61—42no—
94C2×C22⋊D61—42no—
95C2×C22⋊D61—42no—
96C22×Dic61—21no—
97C22×Dic61—21no—
98C22×Dic61—21no—
99C22×Dic61—42no—
100C2×D244—21no—
101C2×D244—41no—
102C2×D244—41no—
103C2×D244C4⋊C412yescomplemented
104C2×C4×D61—42no—
105C2×C4×D61—42no—
106C2×C4×D61—41no—
107C2×C4×D61—41no—
108C4⋊C4×C61—41no—
109C4⋊C4×C61—82no—
110C22⋊C4×C61—41no—
111C22⋊C4×C61—41no—
112C22⋊C4×C61—42no—
113C22⋊C4×C61—42no—
114C42×C61—21no—
115C42×C61—21no—
116C22⋊Dic61—42no—
117C22⋊Dic61—42no—
118C22⋊Dic61—42no—
119C22⋊Dic61—42no—
120#11758198—42no—
121#11758198—42no—
122#11758198—42no—
123C4⋊Dic61—41no—
124C4⋊Dic61—82no—
125#11758196—42no—
126#11758196—42no—
127C4×Dic61—24no—
128C4×Dic61—21no—
129C4×Dic61—42no—
130C4×Dic61—41no—
131C4×Dic61C4⋊C412yescomplemented
132C23×C61—21no—
133C23×C61—21no—
134C23×C61—21no—
135C23×C61—21no—
136C23×C61—41no—
137C23×C61—41no—
138C23×C61—41no—
139C23×C61—41no—
140C23×C61C2×C22⋊C411yescharacteristic
141C23×C61#12111yescharacteristic
142C23×C61#12111yescharacteristic
143C23×C61C22⋊Q811yescharacteristic
144C23×C61C22⋊Q811yescharacteristic
145C22×D61—22no—
146C22×D61—41no—
147C22×D61—41no—
148C22×D61—41no—
149C22×D61—41no—
150C22×D61—41no—
151C22×D61—41no—
152D4×C61—41no—
153D4×C61—41no—
154D4×C61—41no—
155D4×C61—41no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—22no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—42no—
163C2×C4×C61—42no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—81no—
168C2×C4×C61—81no—
169C2×C4×C61—82no—
170C2×C4×C61—82no—
171C2×C4×C61—82no—
172C2×C4×C61#9911yescharacteristic
173C2×C4×C61#9911yescharacteristic
174C2×C4×C61C2×C4⋊C411yescharacteristic
175C2×C4×C61#11711yescharacteristic
176C22⋊D61—44no—
177C22⋊D61—44no—
178C22⋊D61—44no—
179C22⋊D61—44no—
180C2×Dic61—22no—
181C2×Dic61—22no—
182C2×Dic61—22no—
183C2×Dic61—22no—
184C2×Dic61—41no—
185C2×Dic61—41no—
186C2×Dic61—41no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—82no—
190C2×Dic61—82no—
191C2×Dic61—82no—
192D244—42no—
193D244—42no—
194C4×D61—82no—
195C4×D61—82no—
196#12797—611nomaximal, Sylow
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—41no—
204C22×C61—41no—
205C22×C61—41no—
206C22×C61—41no—
207C22×C61—41no—
208C22×C61—41no—
209C22×C61—41no—
210C22×C61—41no—
211C22×C61—81no—
212C22×C61—81no—
213C22×C61#38011yescharacteristic
214C22×C61C2×#9911yescharacteristic
215C22×C61#41011yescharacteristic
216C2×D61—42no—
217C2×D61—42no—
218C2×D61—42no—
219C2×D61—81no—
220C2×D61—81no—
221C2×D61—81no—
222C2×D61—81no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—21no—
226C4×C61—41no—
227C4×C61—41no—
228C4×C61—41no—
229C4×C61—162no—
230C4×C61#35312yescomplemented
231Dic61—24no—
232Dic61—42no—
233Dic61—81no—
234Dic61—162no—
235C4×#127—611no—
236C2×#353—612no—
237#1799—611no—
238#1799D6111yescharacteristic, complemented
239#1393—612no—
240C2×C61—21no—
241C2×C61—21no—
242C2×C61—41no—
243C2×C61—41no—
244C2×C61—41no—
245C2×C61—41no—
246C2×C61#179911yescharacteristic
247C2×C61C2×#35311yescharacteristic
248C2×C61#212611yescharacteristic
249D61—82no—
250C2×#127—611no—
251C2×C4×D4—611no—
252C2×C4×D4—1221no—
253C2×C4×D4—1221no—
254C2×C4×D4C2×D6111yescharacteristic
255C2×#99—612no—
256C2×#99C2×D6112yes—
257#390—611no—
258#390—611no—
259#390—1221no—
260#390—1221no—
261#384—611no—
262#380—21no—
263#380—1221no—
264C43—611no—
265#353—1224no—
266#342—21no—
267#342—1221no—
268C61#1279711yescharacteristic, complemented, Sylow
269C22×D4—611no—
270C22×D4—1221no—
271C22×D4—1221no—
272C22×D4C22×D6111yescharacteristic
273C23×C4—1221no—
274C23×C4D24411yescharacteristic
275C23×C4C22⋊D6111yescharacteristic
276#127—612no—
277#127—612no—
278#127—1221no—
279#127—1221no—
280C4×D4—41no—
281C4×D4—41no—
282C4×D4—2441no—
283C4×D4—2441no—
284C4×D4—2444no—
285C4×D4—2444no—
286C2×C4⋊C4—41no—
287C2×C4⋊C4—611no—
288C2×C4⋊C4—1221no—
289C2×C4⋊C4—1221no—
290C2×C4⋊C4—2441no—
291C2×C4⋊C4C4×D6111yescharacteristic
292C2×C22⋊C4—22no—
293C2×C22⋊C4—1221no—
294C2×C22⋊C4—1221no—
295C2×C22⋊C4—1221no—
296C2×C22⋊C4—1221no—
297C2×C22⋊C4—1221no—
298C2×C22⋊C4—1222no—
299C2×C22⋊C4D24411yescharacteristic
300C2×C22⋊C4C22⋊D6111yescharacteristic
301C2×C42—611no—
302C2×C42—611no—
303C2×C42—1221no—
304C2×C42—1221no—
305C2×C42C4×D6111yescharacteristic
306#99—22no—
307#99—22no—
308#99—1222no—
309#99—1222no—
310#95—41no—
311#95—2441no—
312C24—1221no—
313C24C2×D24411yescharacteristic
314C24C2×C22⋊D6111yescharacteristic
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—612no—
318C2×D4—1222no—
319C2×D4—1222no—
320C2×D4—1222no—
321C2×D4—1222no—
322C2×D4—1221no—
323C2×D4—2441no—
324C2×D4—2441no—
325C2×D4—2442no—
326C2×D4—2442no—
327C2×D4—2442no—
328C2×D4—2442no—
329C2×D4D4×D6111yescharacteristic
330C2×D4#1175822411yescharacteristic
331C2×D4Q8×D6111yescharacteristic
332C2×D4#1175822611yescharacteristic
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—1221no—
341C22×C4—1221no—
342C22×C4—1221no—
343C22×C4—2442no—
344C22×C4—2442no—
345C22×C4—2442no—
346C22×C4—2441no—
347C22×C4—2441no—
348C22×C4#1175819811yescharacteristic
349C22×C4#1175819811yescharacteristic
350C22×C4#1175819811yescharacteristic
351C22×C4#1175819811yescharacteristic
352C22×C4C2×C4×D6111yescharacteristic
353C4⋊C4—41no—
354C4⋊C4—82no—
355C4⋊C4—4882no—
356C4⋊C4—2441no—
357C4⋊C4—2442no—
358C4⋊C4—2442no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—2442no—
364C22⋊C4—2442no—
365C22⋊C4—2442no—
366C22⋊C4—2442no—
367C22⋊C4—2442no—
368C22⋊C4—2442no—
369C22⋊C4—2442no—
370C42—21no—
371C42—21no—
372C42—612no—
373C42—1224no—
374C42—1221no—
375C42—2441no—
376C42—2442no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—1222no—
386C23—2441no—
387C23—2441no—
388C23—2441no—
389C23—2441no—
390C23—2441no—
391C23—2441no—
392C23C2×#1175819811yescharacteristic
393C23#1451974811yescharacteristic
394C23#1451975211yescharacteristic
395C23#1451975311yescharacteristic
396C23#1451975711yescharacteristic
397D4—41no—
398D4—41no—
399D4—41no—
400D4—41no—
401D4—2442no—
402D4—2442no—
403D4—2444no—
404D4—2444no—
405D4—2444no—
406D4—2444no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—1222no—
424C2×C4—1222no—
425C2×C4—1222no—
426C2×C4—1222no—
427C2×C4—4882no—
428C2×C4—4882no—
429C2×C4—4882no—
430C2×C4—4882no—
431C2×C4—4882no—
432C2×C4—2441no—
433C2×C4—2441no—
434C2×C4—2441no—
435C2×C4—2441no—
436C2×C4—2441no—
437C2×C4#1451969111yescharacteristic
438C2×C4#1451969111yescharacteristic
439C2×C4C4⋊C4×D6111yescharacteristic
440C2×C4#1451973511yescharacteristic
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—81no—
456C22—81no—
457C22—4881no—
458C22—4881no—
459C22—4881no—
460C22—4881no—
461C22—2442no—
462C22—2442no—
463C22—2442no—
464C22#1683961911yescharacteristic
465C22C2×#1451969111yescharacteristic
466C22#1683973511yescharacteristic
467C4—21no—
468C4—21no—
469C4—21no—
470C4—41no—
471C4—41no—
472C4—41no—
473C4—162no—
474C4—1224no—
475C4—9762no—
476C4—4881no—
477C4—2442no—
478C4#1683955412yescomplemented
479C2—21no—
480C2—21no—
481C2—41no—
482C2—41no—
483C2—41no—
484C2—41no—
485C2—4882no—
486C2#2144580411yescharacteristic
487C2C2×#1683955411yescharacteristic
488C2#2144748211yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC22×D4C22×D61
FittingC61×#1799C2
socleC22×C61C2×#99
prime radicalC2×#14519773C22
cosocleC22×D4×C61C23
hypercenter#1799D61
hypocenterC61#12797
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #21445807C2semidirect
4. #21443943C2semidirect
5. C61×#1799C2semidirect
7. C2×C4×D244C4semidirect
12. #16839663C4semidirect
15. #16839631C4semidirect
17. C42×Dic61C4semidirect
103. C2×D244C4⋊C4semidirect
131. C4×Dic61C4⋊C4semidirect
230. C4×C61#353semidirect
238. #1799D61semidirect
268. C61#12797semidirect
478. C4#16839554semidirect

Maximal subgroups

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

subgroupindex
C4×#145197732
C2×#168395542
#214458072
#214439432
C61×#17992
#1279761