Finite Group Databasebeta

← #29577789

Subgroups — #29577789

Order 15616.

716 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 109 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#21454845C211yesmaximal, characteristic
2#21449178C211yesmaximal, characteristic
3#21445804C211yesmaximal, characteristic
4#21445754C211yesmaximal, characteristic, complemented
5#21443687C211yesmaximal, characteristic, complemented
6#21443686C211yesmaximal, characteristic, complemented
7C61×#1787C211yesmaximal, characteristic, complemented
8C2×#14519753—21no—
9C2×C4×C22⋊D61C2211yescharacteristic
10#16839833—22no—
11#16839830—22no—
12#16839816—21no—
13C2×#14519701—22no—
14C2×#14519691—21no—
15#16839724—21no—
16C2×C22⋊#92418C2211yescharacteristic
17#16839681C412yescomplemented
18#16839673C412yescomplemented
19#16839669—21no—
20#16839641—21no—
21#16839635—21no—
22#16839622C2211yescharacteristic
23#16839622C2211yescharacteristic
24#16839528—21no—
25C23×Q8×C61C2211yescharacteristic
26C2×#103×C61—22no—
27C2×C22⋊C8×C61C2211yescharacteristic
28C61×#380C2211yescharacteristic
29C22×C22⋊D61D411yescharacteristic
30C22×C4×D61D411yescharacteristic
31C2×#11758189—22no—
32C22×#92418D412yes—
33#14519753—44no—
34C4×C22⋊D61—21no—
35C4×C22⋊D61—21no—
36C4×C22⋊D61—21no—
37C4×C22⋊D61—21no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61—21no—
40C2×#11758196—22no—
41C2×#11758196C2×C411yescharacteristic
42C2×C22⋊Dic61—21no—
43C2×C22⋊Dic61—21no—
44C2×C22⋊Dic61D411yescharacteristic
45C2×#11758198—22no—
46C2×#11758198C2×C411yescharacteristic
47C2×C4×Dic61D411yescharacteristic
48#14519706—22no—
49#14519701—44no—
50#14519691—42no—
51C22⋊#92418—21no—
52C22⋊#92418—21no—
53#14519689—22no—
54#14519687—22no—
55#14519684—22no—
56#14519684—22no—
57#14519684—21no—
58#14519684C2×C412yes—
59C22×Q8×C61—21no—
60C22×Q8×C61—21no—
61C22×Q8×C61—21no—
62C22×Q8×C61—21no—
63C22×Q8×C61—21no—
64C22×Q8×C61—44no—
65C22×Q8×C61C2×C412yes—
66C23×C4×C61—21no—
67C23×C4×C61C2311yescharacteristic
68C22×C8×C61D412yes—
69C2×C4⋊C4×C61—21no—
70C2×C22⋊C4×C61—21no—
71#103×C61—44no—
72C22⋊C8×C61—21no—
73C22⋊C8×C61—21no—
74#95×C61—21no—
75C24×C61C2×D411yescharacteristic
76C23×D61C22⋊C411yescharacteristic
77C2×Q8×C61—24no—
78C2×Q8×C61—24no—
79C2×Q8×C61—24no—
80C2×Q8×C61—22no—
81C2×Q8×C61—22no—
82C2×Q8×C61—44no—
83C2×Q8×C61—42no—
84C2×Q8×C61—42no—
85C2×Q8×C61—42no—
86C2×Q8×C61—44no—
87C2×Q8×C61—44no—
88C2×Q8×C61—41no—
89C2×Q8×C61—41no—
90C22×C4×C61—22no—
91C22×C4×C61—42no—
92C22×C4×C61—44no—
93C22×C4×C61—41no—
94C22×C4×C61—41no—
95C22×C4×C61—41no—
96C22×C4×C61C22×C411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61#4111yescharacteristic
101C22×C4×C61#4111yescharacteristic
102C2×C22⋊D61—21no—
103C2×C22⋊D61—21no—
104C2×C22⋊D61—21no—
105C2×C22⋊D61—21no—
106C2×C22⋊D61—21no—
107C2×C22⋊D61—21no—
108C22×Dic61—22no—
109C22×Dic61—41no—
110C22×Dic61C22⋊C411yescharacteristic
111C22×Dic61C22⋊C411yescharacteristic
112C22×Dic61C22⋊C411yescharacteristic
113C2×C4×D61—21no—
114C2×C4×D61—21no—
115C2×C4×D61—42no—
116C2×C8×C61—22no—
117C2×C8×C61—42no—
118C4⋊C4×C61—42no—
119C22⋊C4×C61—41no—
120C22⋊C4×C61—41no—
121C22⋊Dic61—21no—
122C22⋊Dic61—21no—
123C22⋊Dic61—41no—
124C22⋊Dic61—41no—
125C22⋊Dic61—41no—
126C22⋊Dic61—41no—
127#11758198—21no—
128#11758198—21no—
129#11758198—44no—
130C4⋊Dic61—42no—
131C4⋊Dic61—42no—
132#11758196—21no—
133#11758196—21no—
134#11758196—44no—
135C4×Dic61—21no—
136C4×Dic61—21no—
137C2×#92418—22no—
138C2×#92418—42no—
139#11758189—82no—
140C23×C61—41no—
141C23×C61—41no—
142C23×C61C2×C22⋊C411yescharacteristic
143C23×C61#12111yescharacteristic
144C23×C61#12111yescharacteristic
145C23×C61AGL(1,ℤ/8)11yescharacteristic
146C23×C61#13711yescharacteristic
147C23×C61#13711yescharacteristic
148C23×C61#13711yescharacteristic
149C22×D61—21no—
150C22×D61—21no—
151C22×D61—21no—
152C22×D61—41no—
153C22×D61—41no—
154Q8×C61—24no—
155Q8×C61—24no—
156Q8×C61—44no—
157Q8×C61—42no—
158Q8×C61—42no—
159Q8×C61—44no—
160C2×C4×C61—22no—
161C2×C4×C61—22no—
162C2×C4×C61—22no—
163C2×C4×C61—24no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—42no—
168C2×C4×C61—44no—
169C2×C4×C61—44no—
170C2×C4×C61—42no—
171C2×C4×C61—81no—
172C2×C4×C61—81no—
173C2×C4×C61—81no—
174C2×C4×C61C4×D411yescharacteristic
175C2×C4×C61C4×D411yescharacteristic
176C2×C4×C61#12011yescharacteristic
177C2×C4×C61#12311yescharacteristic
178C22⋊D61—41no—
179C22⋊D61—41no—
180C22⋊D61—41no—
181C22⋊D61—41no—
182C2×Dic61—21no—
183C2×Dic61—21no—
184C2×Dic61—21no—
185C2×Dic61—21no—
186C2×Dic61—21no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—42no—
190C2×Dic61—42no—
191C2×Dic61—42no—
192C2×Dic61—81no—
193C2×Dic61—81no—
194C2×Dic61—81no—
195C4×D61—42no—
196C4×D61—81no—
197C8×C61—42no—
198#92418—42no—
199#9471—611nomaximal, Sylow
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—21no—
204C22×C61—21no—
205C22×C61—21no—
206C22×C61—41no—
207C22×C61—41no—
208C22×C61—41no—
209C22×C61—41no—
210C22×C61#38611yescharacteristic
211C22×C61#41711yescharacteristic
212C22×C61#41911yescharacteristic
213C22×C61#46811yescharacteristic
214C22×C61#46811yescharacteristic
215C22×C61#47011yescharacteristic
216C22×C61#47011yescharacteristic
217C2×D61—42no—
218C2×D61—41no—
219C2×D61—41no—
220C2×D61—41no—
221C2×D61—81no—
222C4×C61—21no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—44no—
226C4×C61—161no—
227C4×C61B(4,𝔽2)12yes—
228Dic61—41no—
229Dic61—41no—
230Dic61—41no—
231Dic61—82no—
232Dic61—161no—
233#2973—611no—
234#2409—611no—
235#1799—611no—
236#1787—611no—
237#1787D6111yescharacteristic, complemented
238#1289—611no—
239#1289—611no—
240C2×C61—21no—
241C2×C61—21no—
242C2×C61—21no—
243C2×C61—21no—
244C2×C61—21no—
245C2×C61—21no—
246C2×C61#188911yescharacteristic
247C2×C61#189411yescharacteristic
248C2×C61#190911yescharacteristic
249D61—81no—
250C23×Q8C2×D6111yescharacteristic
251C2×C22⋊Q8—1221no—
252C2×C4×D4—611no—
253#435—1222no—
254#435—1222no—
255#432—1221no—
256C2×#103—22no—
257C2×#103—1222no—
258C2×#99—1221no—
259#407—1221no—
260C2×C22⋊C8—611no—
261C2×C22⋊C8C2×D6111yescharacteristic
262#395—612no—
263#393—612no—
264#391—1221no—
265#386—1221no—
266#385—1221no—
267#380—611no—
268#380—611no—
269#380C2×D6111yescharacteristic
270#342—1221no—
271C61#947111yescharacteristic, complemented, Sylow
272C22×Q8—21no—
273C22×Q8—21no—
274C22×Q8—21no—
275C22×Q8—21no—
276C22×Q8—21no—
277C22×Q8—44no—
278C22×Q8C4×D6112yes—
279C22×D4—611no—
280C23×C4—21no—
281C23×C4—611no—
282C23×C4C22×D6111yescharacteristic
283C2×M16—1222no—
284C22×C8—612no—
285C22×C8C22⋊D6112yes—
286C22⋊Q8—2444no—
287C4×D4—1221no—
288C4×D4—1221no—
289C4×D4—1221no—
290C4×D4—1221no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—611no—
293C2×C4⋊C4—1222no—
294C2×C4⋊C4—1221no—
295C2×C4⋊C4—1221no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—611no—
298C2×C22⋊C4—611no—
299C2×C22⋊C4—1222no—
300C2×C22⋊C4—1221no—
301C2×C22⋊C4—1221no—
302C2×C42—611no—
303C4⋊C8—1222no—
304#103—44no—
305#103—2444no—
306#99—2442no—
307C22⋊C8—21no—
308C22⋊C8—21no—
309C22⋊C8—1222no—
310C22⋊C8—1221no—
311C22⋊C8—1221no—
312#97—1222no—
313#95—21no—
314#95—612no—
315#95—1222no—
316#95—1222no—
317#95—1221no—
318C24—611no—
319C24D4×D6111yescharacteristic
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—24no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—44no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×D4—1221no—
334C2×D4—1221no—
335C2×D4—1221no—
336C2×D4—1221no—
337C2×D4—1221no—
338C2×D4—1221no—
339C22×C4—22no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—611no—
346C22×C4—611no—
347C22×C4—611no—
348C22×C4—1221no—
349C22×C4—1221no—
350C22×C4—1222no—
351C22×C4—2442no—
352C22×C4—2441no—
353C22×C4C2×C4×D6111yescharacteristic
354C22×C4#1175822211yescharacteristic
355C22×C4D4×D6111yescharacteristic
356C22×C4D4×D6111yescharacteristic
357C22×C4#1175822411yescharacteristic
358C22×C4C2×C22⋊D6111yescharacteristic
359M16—4882no—
360C2×C8—22no—
361C2×C8—42no—
362C2×C8—1222no—
363C2×C8—2442no—
364C4⋊C4—42no—
365C4⋊C4—1221no—
366C4⋊C4—1221no—
367C4⋊C4—2442no—
368C4⋊C4—2444no—
369C4⋊C4—2442no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—1221no—
373C22⋊C4—1221no—
374C22⋊C4—1221no—
375C22⋊C4—1221no—
376C22⋊C4—2444no—
377C22⋊C4—2441no—
378C22⋊C4—2441no—
379C22⋊C4—2441no—
380C22⋊C4—2441no—
381C42—1221no—
382C42—1221no—
383C23—41no—
384C23—41no—
385C23—1221no—
386C23—1221no—
387C23—1221no—
388C23—2441no—
389C23—2441no—
390C23C22⋊C4×D6111yescharacteristic
391C23#1451974811yescharacteristic
392C23#1451975011yescharacteristic
393C23#1451980411yescharacteristic
394C23#1451980611yescharacteristic
395C23#1451980911yescharacteristic
396C23#1451980911yescharacteristic
397Q8—24no—
398Q8—24no—
399Q8—44no—
400Q8—42no—
401Q8—42no—
402Q8—44no—
403D4—2441no—
404D4—2441no—
405D4—2441no—
406D4—2441no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—1221no—
422C2×C4—1221no—
423C2×C4—1221no—
424C2×C4—1221no—
425C2×C4—1221no—
426C2×C4—4881no—
427C2×C4—4881no—
428C2×C4—4881no—
429C2×C4—4881no—
430C2×C4—2442no—
431C2×C4—2442no—
432C2×C4—2442no—
433C2×C4—2442no—
434C2×C4—2441no—
435C2×C4—2441no—
436C2×C4#1451973711yescharacteristic
437C2×C4C4×C22⋊D6111yescharacteristic
438C2×C4#1451974511yescharacteristic
439C2×C4#1451975911yescharacteristic
440C8—42no—
441C8—2442no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—4881no—
453C22—2442no—
454C22—2441no—
455C22—2441no—
456C22—2441no—
457C22#1683963811yescharacteristic
458C22#1683976211yescharacteristic
459C22#1683976911yescharacteristic
460C22#1684000511yescharacteristic
461C22#1684001111yescharacteristic
462C22#1684001711yescharacteristic
463C22#1684001911yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—44no—
468C4—161no—
469C4—9761no—
470C4—4882no—
471C4—2441no—
472C4—2441no—
473C4—2441no—
474C4#1683994812yes—
475C2—21no—
476C2—21no—
477C2—21no—
478C2—21no—
479C2—21no—
480C2—21no—
481C2—4881no—
482C2#2144622411yescharacteristic
483C2#2144624911yescharacteristic
484C2#2144633811yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1787C2
socleC22×C61#386
prime radicalC22×C22⋊D61D4
cosocleC23×C4×C61C23
hypercenter#1787D61
hypocenterC61#9471
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
4. #21445754C2semidirect
5. #21443687C2semidirect
6. #21443686C2semidirect
7. C61×#1787C2semidirect
17. #16839681C4semidirect
18. #16839673C4semidirect
237. #1787D61semidirect
271. C61#9471semidirect

Maximal subgroups

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

subgroupindex
#214548452
#214491782
#214458042
#214457542
#214436872
#214436862
C61×#17872
#947161