Finite Group Databasebeta

← #29580257

Subgroups — #29580257

Order 15616.

672 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C2×#16840078C211yesmaximal, characteristic
2#21451563C211yesmaximal, characteristic
3C2×#16839702C211yesmaximal, characteristic, complemented
4#21445910C211yesmaximal, characteristic, complemented
5C2×#16839487C211yesmaximal, characteristic, complemented
6#21443698C211yesmaximal, characteristic, complemented
7C2×C61×#327C211yesmaximal, characteristic
8C2×#14519758—21no—
9C22×C4⋊Dic61C2211yescharacteristic
10C22×#11758196C2211yescharacteristic
11#16840078—22no—
12#16840078—22no—
13C2×#14519708—21no—
14C2×#14519698—21no—
15C2×C22⋊#92418C2211yescharacteristic
16#16839702—22no—
17#16839702—22no—
18#16839693—21no—
19#16839685—22no—
20#16839676—21no—
21#16839642—21no—
22#16839622C2211yescharacteristic
23C2×#14519684—21no—
24C2×#14519684—21no—
25#16839525—22no—
26#16839487—22no—
27C2×#121×C61C2211yescharacteristic
28C2×C22⋊C8×C61C2211yescharacteristic
29C2×#95×C61C2211yescharacteristic
30C61×#327—22no—
31C23×Dic61—21no—
32C23×Dic61D411yescharacteristic
33C23×Dic61D411yescharacteristic
34C22×#92418—21no—
35#14519758—42no—
36#14519758—42no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61D412yes—
43C2×C4⋊Dic61D412yes—
44C2×#11758196—22no—
45C2×#11758196—21no—
46C2×#11758196—21no—
47C2×#11758196—21no—
48C2×#11758196—21no—
49C2×#11758196—21no—
50C2×#11758196—21no—
51C2×C22⋊Dic61—22no—
52C2×C22⋊Dic61—21no—
53C2×C22⋊Dic61—22no—
54#14519708—24no—
55#14519698—24no—
56C22⋊#92418—21no—
57C22⋊#92418D412yes—
58#14519684—22no—
59#14519684—24no—
60#14519684—24no—
61#14519684—21no—
62C22×D4×C61—21no—
63C22×D4×C61D411yescharacteristic
64C23×C4×C61—21no—
65C23×C4×C61C2311yescharacteristic
66C22×C8×C61—21no—
67#121×C61—21no—
68#121×C61—21no—
69#121×C61—21no—
70#121×C61—21no—
71C2×C4⋊C4×C61D411yescharacteristic
72C2×C22⋊C4×C61—21no—
73C22⋊C8×C61—21no—
74C22⋊C8×C61—21no—
75#95×C61—21no—
76#95×C61D412yes—
77C24×C61—21no—
78C24×C61C2×D411yescharacteristic
79C2×D4×C61—21no—
80C2×D4×C61—21no—
81C2×D4×C61—42no—
82C2×D4×C61—41no—
83C2×D4×C61—41no—
84C2×D4×C61—41no—
85C2×D4×C61—41no—
86C2×D4×C61—41no—
87C2×D4×C61—41no—
88C22×C4×C61—22no—
89C22×C4×C61—21no—
90C22×C4×C61—42no—
91C22×C4×C61—42no—
92C22×C4×C61—42no—
93C22×C4×C61—41no—
94C22×C4×C61—41no—
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61#4111yescharacteristic
101C22×Dic61—21no—
102C22×Dic61—21no—
103C22×Dic61—21no—
104C22×Dic61—21no—
105C22×Dic61—21no—
106C22×Dic61—21no—
107C22×Dic61—22no—
108C22×Dic61—21no—
109C22×Dic61—21no—
110C22×Dic61—22no—
111C22×Dic61—21no—
112C22×Dic61—42no—
113C22×Dic61—44no—
114C22×Dic61—41no—
115C22×Dic61—41no—
116C22×Dic61SD1612yes—
117C2×C8×C61—42no—
118C2×C8×C61—41no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61—21no—
121C22⋊C4×C61—41no—
122C22⋊C4×C61—41no—
123C22⋊Dic61—44no—
124C22⋊Dic61—41no—
125C22⋊Dic61—41no—
126C22⋊Dic61—82no—
127C4⋊Dic61—22no—
128C4⋊Dic61—22no—
129C4⋊Dic61—41no—
130C4⋊Dic61—41no—
131C4⋊Dic61—42no—
132#11758196—44no—
133#11758196—41no—
134#11758196—41no—
135#11758196—41no—
136#11758196—41no—
137C2×#92418—22no—
138C2×#92418—42no—
139C23×C61—22no—
140C23×C61—22no—
141C23×C61—41no—
142C23×C61—41no—
143C23×C61—41no—
144C23×C61—42no—
145C23×C61—42no—
146C23×C61#12011yescharacteristic
147C23×C61#12011yescharacteristic
148C23×C61#12111yescharacteristic
149C23×C61C2×SD1611yescharacteristic
150C23×C61#13511yescharacteristic
151C23×C61AGL(1,ℤ/8)11yescharacteristic
152C23×C61#13711yescharacteristic
153D4×C61—42no—
154D4×C61—81no—
155D4×C61—81no—
156D4×C61—81no—
157D4×C61—81no—
158D4×C61—81no—
159C2×C4×C61—22no—
160C2×C4×C61—22no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—42no—
166C2×C4×C61—42no—
167C2×C4×C61—41no—
168C2×C4×C61—42no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—82no—
172C2×C4×C61—82no—
173C2×C4×C61#12011yescharacteristic
174C2×C4×C61#12111yescharacteristic
175C2×C4×C61#12111yescharacteristic
176C2×C4×C61#12711yescharacteristic
177C2×Dic61—22no—
178C2×Dic61—22no—
179C2×Dic61—22no—
180C2×Dic61—41no—
181C2×Dic61—41no—
182C2×Dic61—41no—
183C2×Dic61—41no—
184C2×Dic61—44no—
185C2×Dic61—42no—
186C2×Dic61—41no—
187C2×Dic61—42no—
188C2×Dic61—42no—
189C2×Dic61—42no—
190C2×Dic61—42no—
191C2×Dic61—42no—
192C2×Dic61—82no—
193C2×Dic61—82no—
194C2×Dic61—81no—
195C8×C61—81no—
196#92418—42no—
197#9921—611nomaximal, Sylow
198C22×C61—22no—
199C22×C61—22no—
200C22×C61—22no—
201C22×C61—22no—
202C22×C61—21no—
203C22×C61—21no—
204C22×C61—21no—
205C22×C61—21no—
206C22×C61—82no—
207C22×C61—82no—
208C22×C61—81no—
209C22×C61—81no—
210C22×C61—81no—
211C22×C61#39211yescharacteristic
212C22×C61#44811yescharacteristic
213C22×C61#44911yescharacteristic
214C22×C61#45011yescharacteristic
215C22×C61#45211yescharacteristic
216C22×C61#45412yes—
217C22×C61#45812yes—
218C22×C61#46511yescharacteristic
219C22×C61#46811yescharacteristic
220C4×C61—21no—
221C4×C61—21no—
222C4×C61—21no—
223C4×C61—21no—
224C4×C61—82no—
225C4×C61—81no—
226Dic61—42no—
227Dic61—82no—
228Dic61—82no—
229Dic61—81no—
230C2×#481—611no—
231#2618—611no—
232C2×#400—611no—
233#1824—611no—
234C2×#327D6111yescharacteristic
235C2×#324—611no—
236#1294—611no—
237C2×C61—21no—
238C2×C61—21no—
239C2×C61—21no—
240C2×C61—21no—
241C2×C61—161no—
242C2×C61#161512yes—
243C2×C61#161612yes—
244C2×C61#200011yescharacteristic
245C2×C61#200311yescharacteristic
246C2×C61#202211yescharacteristic
247C2×#123—1221no—
248C2×#121C2×D6111yescharacteristic
249C22×C4⋊C4—611no—
250C22×C4⋊C4—611no—
251#481—1222no—
252#481—1222no—
253C2×#106—1221no—
254C2×#102—1221no—
255C2×C22⋊C8—611no—
256C2×C22⋊C8C2×D6111yescharacteristic
257#400—1222no—
258#400—1222no—
259#398—1221no—
260#396—1222no—
261#394—1221no—
262#386—1221no—
263#380—611no—
264C2×#95—1221no—
265C2×#95—1221no—
266C2×#95C2×D6111yescharacteristic
267#340—1222no—
268#327—22no—
269#324—1222no—
270C61#992111yescharacteristic, complemented, Sylow
271C22×D4—21no—
272C22×D4D24411yescharacteristic
273C23×C4—21no—
274C23×C4—611no—
275C23×C4—611no—
276C23×C4—1221no—
277C23×C4C22×D6111yescharacteristic
278C22×C8—21no—
279C22×C8—1221no—
280#123—2442no—
281#123—2442no—
282#121—21no—
283#121—21no—
284#121—21no—
285#121—21no—
286C2×C4⋊C4—612no—
287C2×C4⋊C4—612no—
288C2×C4⋊C4—1221no—
289C2×C4⋊C4—1221no—
290C2×C4⋊C4—1221no—
291C2×C4⋊C4—1221no—
292C2×C4⋊C4—1222no—
293C2×C4⋊C4—1221no—
294C2×C4⋊C4—1221no—
295C2×C4⋊C4—1221no—
296C2×C4⋊C4—1221no—
297C2×C4⋊C4—1221no—
298C2×C4⋊C4—1221no—
299C2×C4⋊C4—1221no—
300C2×C4⋊C4D24411yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—1222no—
303C2×C22⋊C4—1222no—
304C2×C22⋊C4—1221no—
305#106—1224no—
306#102—1224no—
307C22⋊C8—21no—
308C22⋊C8—21no—
309C22⋊C8—612no—
310C22⋊C8—1221no—
311#95—21no—
312#95—1224no—
313#95—1224no—
314#95—1222no—
315#95—1221no—
316#95C22⋊D6112yes—
317C24—21no—
318C24D4×D6111yescharacteristic
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—42no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—612no—
336C22×C4—1221no—
337C22×C4—1221no—
338C22×C4—1221no—
339C22×C4—1221no—
340C22×C4—1221no—
341C22×C4—1221no—
342C22×C4—1221no—
343C22×C4—1222no—
344C22×C4—1221no—
345C22×C4—1222no—
346C22×C4—1221no—
347C22×C4—2444no—
348C22×C4—2441no—
349C22×C4—2441no—
350C22×C4—2442no—
351C22×C4C2×D24411yescharacteristic
352C22×C4#1175822211yescharacteristic
353C22×C4D4×D6111yescharacteristic
354C22×C4D4×D6111yescharacteristic
355C22×C4D4×D6111yescharacteristic
356C22×C4C2×C22⋊D6111yescharacteristic
357C2×C8—42no—
358C2×C8—41no—
359C2×C8—1222no—
360C2×C8—2442no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—1222no—
364C4⋊C4—1222no—
365C4⋊C4—2442no—
366C4⋊C4—2444no—
367C4⋊C4—2441no—
368C4⋊C4—2441no—
369C4⋊C4—2441no—
370C4⋊C4—2441no—
371C4⋊C4—2441no—
372C4⋊C4—2441no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—4882no—
376C22⋊C4—2444no—
377C22⋊C4—2441no—
378C22⋊C4—2441no—
379C23—22no—
380C23—22no—
381C23—42no—
382C23—42no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23#1451974511yescharacteristic
387C23#1451974711yescharacteristic
388C23#1451975011yescharacteristic
389C23SD16×D6111yescharacteristic
390C23#1451979611yescharacteristic
391C23#1451980111yescharacteristic
392C23#1451980611yescharacteristic
393D4—42no—
394D4—81no—
395D4—81no—
396D4—81no—
397D4—81no—
398D4—81no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—1222no—
414C2×C4—1222no—
415C2×C4—1222no—
416C2×C4—4882no—
417C2×C4—4882no—
418C2×C4—4881no—
419C2×C4—2442no—
420C2×C4—2444no—
421C2×C4—2442no—
422C2×C4—2442no—
423C2×C4—2442no—
424C2×C4—2442no—
425C2×C4—2442no—
426C2×C4—2441no—
427C2×C4—2441no—
428C2×C4—2441no—
429C2×C4—2441no—
430C2×C4—2441no—
431C2×C4#1451974711yescharacteristic
432C2×C4#1451975011yescharacteristic
433C2×C4#1451975111yescharacteristic
434C2×C4#1451977311yescharacteristic
435C8—81no—
436C8—2442no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—82no—
449C22—82no—
450C22#1683967111yescharacteristic
451C22#1683989911yescharacteristic
452C22#1683990411yescharacteristic
453C22#1683991411yescharacteristic
454C22#1683992511yescharacteristic
455C22#1683993612yes—
456C22#1683995312yes—
457C22#1683999011yescharacteristic
458C22#1684000711yescharacteristic
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—82no—
464C4—81no—
465C4—4881no—
466C4—4882no—
467C4—4882no—
468C4—2442no—
469C2—21no—
470C2—21no—
471C2—21no—
472C2—21no—
473C2—161no—
474C2#2144485112yes—
475C2#2144485712yes—
476C2#2144682311yescharacteristic
477C2#2144684011yescharacteristic
478C2#2144694911yescharacteristic
subgroupG/H
centerC23#14519747
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC2×C61×#327C2
socleC23×C61#120
prime radicalC2×#121×C61C22
hypercenterC2×#327D61
hypocenterC61#9921
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
3. C2×#16839702C2semidirect
4. #21445910C2semidirect
5. C2×#16839487C2semidirect
6. #21443698C2semidirect
270. C61#9921semidirect

Maximal subgroups

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

subgroupindex
C2×#168400782
#214515632
C2×#168397022
#214459102
C2×#168394872
#214436982
C2×C61×#3272
#992161