Finite Group Databasebeta

← #29578029

Subgroups — #29578029

Order 15616.

644 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 93 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#21455053C211yesmaximal, characteristic
2#21454864C211yesmaximal, characteristic
3#21445743C211yesmaximal, characteristic, complemented
4#21444360C211yesmaximal, characteristic
5#21443725C211yesmaximal, characteristic, complemented
6#21443696C211yesmaximal, characteristic, complemented
7C61×#1787C211yesmaximal, characteristic, complemented
8C2×#14519808—21no—
9#16840273C2211yescharacteristic
10#16839918—21no—
11#16839918C412yescomplemented
12#16839901—21no—
13#16839901C412yescomplemented
14#16839837—21no—
15#16839836—21no—
16#16839833—21no—
17C8×C22⋊D61—21no—
18#16839820—21no—
19#16839771—21no—
20C2×#14519696—21no—
21C2×#14519696—21no—
22#16839727—21no—
23#16839726C2211yescharacteristic
24#16839726C2211yescharacteristic
25#16839726C2211yescharacteristic
26#16839498—21no—
27#16839495—21no—
28C23×Q8×C61C2211yescharacteristic
29C2×#103×C61—21no—
30C2×#103×C61—21no—
31C2×C22⋊C8×C61C2211yescharacteristic
32C61×#380C2211yescharacteristic
33C2×#11758222D411yescharacteristic
34#14519808—42no—
35#14519808—42no—
36C2×#11758202—22no—
37C2×#11758201—22no—
38C2×#11758194—21no—
39C2×#11758194—21no—
40C2×#11758194D411yescharacteristic
41C2×#11758194D411yescharacteristic
42C2×#11758189—21no—
43C2×C8×D61—21no—
44#14519760—21no—
45#14519756C2×C411yescharacteristic
46#14519756D411yescharacteristic
47#14519751C2×C411yescharacteristic
48#14519751D411yescharacteristic
49#14519746—21no—
50C4×C22⋊D61—21no—
51C4×C22⋊D61—21no—
52#14519736D411yescharacteristic
53#14519706—21no—
54#14519706—21no—
55#14519701—22no—
56#14519701—22no—
57#14519696—42no—
58#14519696—42no—
59C22⋊#92418—21no—
60C22⋊#92418—21no—
61C22⋊#92418—21no—
62C22⋊#92418—21no—
63C22⋊#92418—21no—
64C22⋊#92418C2×C412yes—
65#14519689—21no—
66#14519689—21no—
67#14519687—21no—
68C8×Dic61—21no—
69C22×Q8×C61—21no—
70C22×Q8×C61—21no—
71C22×Q8×C61—21no—
72C22×Q8×C61—21no—
73C22×Q8×C61—21no—
74C22×Q8×C61—42no—
75C22×Q8×C61—42no—
76C22×Q8×C61C2×C412yes—
77C23×C4×C61—21no—
78C23×C4×C61C2311yescharacteristic
79C22×C8×C61D411yescharacteristic
80C22×C8×C61D411yescharacteristic
81C2×C4⋊C4×C61—21no—
82C2×C22⋊C4×C61—21no—
83#103×C61—42no—
84#103×C61—42no—
85C22⋊C8×C61—21no—
86C22⋊C8×C61—21no—
87#95×C61—21no—
88C24×C61#4111yescharacteristic
89C2×Q8×C61—22no—
90C2×Q8×C61—22no—
91C2×Q8×C61—22no—
92C2×Q8×C61—22no—
93C2×Q8×C61—22no—
94C2×Q8×C61—22no—
95C2×Q8×C61—22no—
96C2×Q8×C61—22no—
97C2×Q8×C61—44no—
98C2×Q8×C61—41no—
99C2×Q8×C61—41no—
100C2×Q8×C61—41no—
101C2×Q8×C61—41no—
102C2×Q8×C61—41no—
103C2×Q8×C61—41no—
104C2×Q8×C61—44no—
105C2×Q8×C61—44no—
106C2×Q8×C61—41no—
107C2×Q8×C61—41no—
108C22×C4×C61—22no—
109C22×C4×C61—41no—
110C22×C4×C61—41no—
111C22×C4×C61—44no—
112C22×C4×C61—41no—
113C22×C4×C61—41no—
114C22×C4×C61—41no—
115C22×C4×C61C22×C411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61#4111yescharacteristic
121C2×C22⋊D61—21no—
122C2×C22⋊D61—21no—
123C2×C22⋊D61—21no—
124#11758222—41no—
125#11758222—41no—
126C2×D244C22⋊C411yescharacteristic
127C2×C4×D61—21no—
128C2×Dic122C22⋊C411yescharacteristic
129C2×C8×C61—21no—
130C2×C8×C61—21no—
131C2×C8×C61—41no—
132C2×C8×C61—41no—
133C4⋊C4×C61—41no—
134C4⋊C4×C61—41no—
135C22⋊C4×C61—81no—
136C22⋊Dic61—21no—
137C22⋊Dic61—21no—
138C22⋊Dic61—21no—
139#11758202—44no—
140#11758201—44no—
141#11758198—21no—
142#11758198—21no—
143C4⋊Dic61C22⋊C411yescharacteristic
144C4⋊Dic61C22⋊C411yescharacteristic
145#11758196—21no—
146#11758196—21no—
147C4×Dic61—21no—
148#11758194—41no—
149#11758194—41no—
150#11758194—41no—
151#11758194—41no—
152#11758194—41no—
153#11758194—41no—
154C2×#92418—22no—
155C2×#92418—21no—
156C2×#92418—21no—
157C2×#92418—41no—
158#11758189—81no—
159C8×D61—81no—
160C23×C61—21no—
161C23×C61—21no—
162C23×C61—41no—
163C23×C61—41no—
164C23×C61C4×D411yescharacteristic
165C23×C61#12111yescharacteristic
166C23×C61#12311yescharacteristic
167C22×D61—21no—
168Q8×C61—24no—
169Q8×C61—24no—
170Q8×C61—42no—
171Q8×C61—42no—
172Q8×C61—42no—
173Q8×C61—42no—
174Q8×C61—42no—
175Q8×C61—42no—
176C2×C4×C61—21no—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—24no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—81no—
193C2×C4×C61—81no—
194C2×C4×C61—81no—
195C2×C4×C61C2×C22⋊C411yescharacteristic
196C2×C4×C61C4×D411yescharacteristic
197C2×C4×C61#12011yescharacteristic
198C2×C4×C61#12111yescharacteristic
199C22⋊D61—41no—
200C22⋊D61—41no—
201C22⋊D61—81no—
202C22⋊D61—81no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—21no—
206D244—41no—
207C4×D61—41no—
208C4×D61—41no—
209Dic122—41no—
210C8×C61—41no—
211C8×C61—41no—
212#92418—42no—
213#92418—41no—
214#92418—41no—
215#92418—81no—
216#9519—611nomaximal, Sylow
217C22×C61—21no—
218C22×C61—21no—
219C22×C61—21no—
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—21no—
224C22×C61—21no—
225C22×C61—42no—
226C22×C61—42no—
227C22×C61#38611yescharacteristic
228C22×C61#47111yescharacteristic
229C22×C61#47311yescharacteristic
230C2×D61—41no—
231C2×D61—41no—
232C2×D61—41no—
233C4×C61—21no—
234C4×C61—21no—
235C4×C61—21no—
236C4×C61—44no—
237C4×C61—161no—
238C4×C61B(4,𝔽2)11yescharacteristic
239C4×C61B(4,𝔽2)11yescharacteristic
240Dic61—41no—
241Dic61—41no—
242Dic61—41no—
243#3004—611no—
244#2976—611no—
245#1787D6111yescharacteristic, complemented
246#1784—611no—
247#1513—611no—
248#1305—611no—
249#1293—611no—
250C2×C61—21no—
251C2×C61—21no—
252C2×C61—41no—
253C2×C61—41no—
254C2×C61#188911yescharacteristic
255C2×C61#190211yescharacteristic
256C2×C61#190811yescharacteristic
257D61—81no—
258C23×Q8C2×D6111yescharacteristic
259C2×#137—1221no—
260#525—611no—
261#451—612no—
262#451—1221no—
263#448—612no—
264#448—1221no—
265#436—1221no—
266#435—1221no—
267#435—1221no—
268C8×D4—1221no—
269#433—1221no—
270#419—1221no—
271C2×#103—21no—
272C2×#103—21no—
273C2×#101—1221no—
274C2×#101—1221no—
275#408—1221no—
276#407—611no—
277#407—611no—
278#407—611no—
279C2×C22⋊C8C2×D6111yescharacteristic
280#380C2×D6111yescharacteristic
281#328—1221no—
282#327—1221no—
283C61#951911yescharacteristic, complemented, Sylow
284C2×#41—611no—
285C22×Q8—21no—
286C22×Q8—21no—
287C22×Q8—21no—
288C22×Q8—21no—
289C22×Q8—21no—
290C22×Q8—42no—
291C22×Q8—42no—
292C22×Q8C4×D6112yes—
293C23×C4—21no—
294C23×C4C22×D6111yescharacteristic
295#137—2442no—
296#137—2442no—
297C2×Q16—1222no—
298C2×SD16—1222no—
299C2×M16—611no—
300C2×M16—611no—
301C2×M16—1221no—
302C2×M16—1221no—
303C2×M16—1221no—
304C22×C8—1221no—
305C22×C8C22⋊D6111yescharacteristic
306C22×C8C22⋊D6111yescharacteristic
307#123—1221no—
308C22⋊Q8—611no—
309C22⋊Q8—611no—
310#121—611no—
311#121—611no—
312#120—1221no—
313C4×D4—1221no—
314C4×D4—1221no—
315#117—611no—
316C2×C4⋊C4—21no—
317C2×C22⋊C4—21no—
318C4⋊C8—1221no—
319C4⋊C8—1221no—
320#103—42no—
321#103—42no—
322#103—1222no—
323#103—1222no—
324#101—2442no—
325#101—2442no—
326C22⋊C8—21no—
327C22⋊C8—21no—
328C22⋊C8—612no—
329C22⋊C8—1221no—
330C22⋊C8—1221no—
331C22⋊C8—1221no—
332C22⋊C8—1221no—
333C22⋊C8—1221no—
334C22⋊C8—1221no—
335C22⋊C8—1221no—
336#97—1221no—
337C4×C8—1221no—
338#95—21no—
339C24#1175822411yescharacteristic
340#41—2441no—
341#41—2441no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—44no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8—611no—
362C2×D4—611no—
363C2×D4—1221no—
364C2×D4—1221no—
365C2×D4—1221no—
366C22×C4—22no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—44no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—1221no—
374C22×C4C2×C4×D6111yescharacteristic
375C22×C4#1175822211yescharacteristic
376C22×C4D4×D6111yescharacteristic
377C22×C4D4×D6111yescharacteristic
378C22×C4D4×D6111yescharacteristic
379C22×C4C2×C22⋊D6111yescharacteristic
380Q16—2444no—
381SD16—2444no—
382M16—4881no—
383M16—2441no—
384M16—2441no—
385M16—2441no—
386M16—2441no—
387M16—2441no—
388M16—2441no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—1222no—
394C2×C8—1221no—
395C2×C8—1221no—
396C2×C8—4881no—
397C2×C8—2441no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—611no—
401C4⋊C4—611no—
402C4⋊C4—1221no—
403C4⋊C4—1221no—
404C22⋊C4—81no—
405C22⋊C4—1221no—
406C22⋊C4—1221no—
407C22⋊C4—1221no—
408C22⋊C4—1221no—
409C22⋊C4—1221no—
410C42—1221no—
411C23—21no—
412C23—21no—
413C23—41no—
414C23—41no—
415C23—1221no—
416C23#1451973711yescharacteristic
417C23#1451974811yescharacteristic
418C23#1451975911yescharacteristic
419Q8—24no—
420Q8—24no—
421Q8—42no—
422Q8—42no—
423Q8—42no—
424Q8—42no—
425Q8—42no—
426Q8—42no—
427Q8—2441no—
428D4—4881no—
429D4—4881no—
430D4—2441no—
431D4—2441no—
432D4—2441no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—24no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—1221no—
453C2×C4—1221no—
454C2×C4—1221no—
455C2×C4—2441no—
456C2×C4—2441no—
457C2×C4C22⋊C4×D6111yescharacteristic
458C2×C4C4×C22⋊D6111yescharacteristic
459C2×C4#1451974511yescharacteristic
460C2×C4#1451975011yescharacteristic
461C8—41no—
462C8—41no—
463C8—4881no—
464C8—2442no—
465C8—2441no—
466C8—2441no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—42no—
476C22—42no—
477C22—2441no—
478C22—2441no—
479C22—2441no—
480C22#1683963811yescharacteristic
481C22#1684002211yescharacteristic
482C22#1684003411yescharacteristic
483C4—21no—
484C4—21no—
485C4—21no—
486C4—44no—
487C4—161no—
488C4—2441no—
489C4—2441no—
490C4—2441no—
491C4#1683994811yescharacteristic
492C4#1683994811yescharacteristic
493C2—21no—
494C2—21no—
495C2—41no—
496C2—41no—
497C2—4881no—
498C2#2144622411yescharacteristic
499C2#2144628911yescharacteristic
500C2#2144633711yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1787C2
socleC22×C61#386
prime radical#16840273C22
cosocleC23×C4×C61C23
hypercenter#1787D61
hypocenterC61#9519
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
3. #21445743C2semidirect
5. #21443725C2semidirect
6. #21443696C2semidirect
7. C61×#1787C2semidirect
11. #16839918C4semidirect
13. #16839901C4semidirect
245. #1787D61semidirect
283. C61#9519semidirect

Maximal subgroups

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

subgroupindex
#214550532
#214548642
#214457432
#214443602
#214437252
#214436962
C61×#17872
#951961