Finite Group Databasebeta

← #29577801

Subgroups — #29577801

Order 15616.

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

normal subgroups only

This group has 556 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#21450137C211yesmaximal, characteristic
3#21445910C211yesmaximal, characteristic, complemented
4#21445907C211yesmaximal, characteristic, complemented
5#21445804C211yesmaximal, characteristic
6#21443686C211yesmaximal, characteristic, complemented
7C61×#1289C211yesmaximal, characteristic, complemented
8C2×C4×C22⋊D61C2211yescharacteristic
9C22×C4⋊Dic61C2211yescharacteristic
10#16839833—22no—
11#16839830—22no—
12#16839816—21no—
13C2×#14519691—21no—
14#16839724—21no—
15C2×C22⋊#92418C2211yescharacteristic
16#16839702C412yescomplemented
17#16839699C412yescomplemented
18#16839661—21no—
19#16839639—21no—
20#16839637—21no—
21#16839622C2211yescharacteristic
22#16839619—21no—
23#16839528—21no—
24#16839525—22no—
25#16839524—22no—
26C2×C22⋊C8×C61C2211yescharacteristic
27C61×#380C2211yescharacteristic
28C61×#380C2211yescharacteristic
29C22×C22⋊D61D411yescharacteristic
30C23×Dic61—21no—
31C22×C4×D61D411yescharacteristic
32C2×#11758189—22no—
33C22×#92418D412yes—
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×C4⋊Dic61—21no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61—22no—
45C2×C4⋊Dic61C2×C412yes—
46C2×#11758196C2×C411yescharacteristic
47C2×C22⋊Dic61—21no—
48C2×C22⋊Dic61D411yescharacteristic
49C2×#11758198—22no—
50C2×#11758198C2×C411yescharacteristic
51C2×C4×Dic61D411yescharacteristic
52#14519706—22no—
53#14519691—42no—
54C22⋊#92418—21no—
55C22⋊#92418—21no—
56#14519689—22no—
57#14519687—22no—
58#14519684—22no—
59#14519684—22no—
60#14519684—21no—
61C23×C4×C61C2311yescharacteristic
62C22×C8×C61D412yes—
63C2×C4⋊C4×C61—21no—
64C2×C4⋊C4×C61—21no—
65C2×C22⋊C4×C61—21no—
66C2×C22⋊C4×C61—21no—
67C22⋊C8×C61—21no—
68C22⋊C8×C61—21no—
69#95×C61—21no—
70#95×C61C2×C412yes—
71C24×C61C2×D411yescharacteristic
72C23×D61C22⋊C411yescharacteristic
73C22×C4×C61—22no—
74C22×C4×C61—42no—
75C22×C4×C61—41no—
76C22×C4×C61C22×C411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61#4111yescharacteristic
81C22×C4×C61#4111yescharacteristic
82C2×C22⋊D61—21no—
83C2×C22⋊D61—21no—
84C2×C22⋊D61—21no—
85C2×C22⋊D61—21no—
86C2×C22⋊D61—21no—
87C2×C22⋊D61—21no—
88C22×Dic61—22no—
89C22×Dic61—41no—
90C22×Dic61—41no—
91C22×Dic61—41no—
92C22×Dic61—82no—
93C22×Dic61C22⋊C411yescharacteristic
94C22×Dic61C22⋊C411yescharacteristic
95C22×Dic61C22⋊C411yescharacteristic
96C2×C4×D61—21no—
97C2×C4×D61—21no—
98C2×C4×D61—42no—
99C2×C8×C61—22no—
100C2×C8×C61—42no—
101C4⋊C4×C61—42no—
102C4⋊C4×C61—42no—
103C22⋊C4×C61—41no—
104C22⋊C4×C61—41no—
105C22⋊C4×C61—41no—
106C22⋊C4×C61—41no—
107C22⋊Dic61—21no—
108C22⋊Dic61—21no—
109C22⋊Dic61—41no—
110C22⋊Dic61—41no—
111#11758198—21no—
112#11758198—21no—
113#11758198—82no—
114C4⋊Dic61—22no—
115C4⋊Dic61—22no—
116C4⋊Dic61—41no—
117C4⋊Dic61—41no—
118C4⋊Dic61—42no—
119C4⋊Dic61—82no—
120#11758196—21no—
121#11758196—21no—
122C4×Dic61—21no—
123C4×Dic61—21no—
124C2×#92418—22no—
125C2×#92418—42no—
126#11758189—82no—
127C23×C61—41no—
128C23×C61—41no—
129C23×C61C2×C22⋊C411yescharacteristic
130C23×C61#12111yescharacteristic
131C23×C61#12111yescharacteristic
132C23×C61AGL(1,ℤ/8)11yescharacteristic
133C23×C61#13711yescharacteristic
134C23×C61#13711yescharacteristic
135C23×C61#13711yescharacteristic
136C22×D61—21no—
137C22×D61—21no—
138C22×D61—21no—
139C22×D61—41no—
140C22×D61—41no—
141C2×C4×C61—22no—
142C2×C4×C61—22no—
143C2×C4×C61—22no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—42no—
148C2×C4×C61—42no—
149C2×C4×C61—42no—
150C2×C4×C61—81no—
151C2×C4×C61—81no—
152C2×C4×C61—81no—
153C2×C4×C61C4×D411yescharacteristic
154C2×C4×C61C4×D411yescharacteristic
155C2×C4×C61#12011yescharacteristic
156C2×C4×C61#12311yescharacteristic
157C22⋊D61—41no—
158C22⋊D61—41no—
159C22⋊D61—41no—
160C22⋊D61—41no—
161C2×Dic61—21no—
162C2×Dic61—21no—
163C2×Dic61—21no—
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—42no—
169C2×Dic61—42no—
170C2×Dic61—42no—
171C2×Dic61—82no—
172C2×Dic61—82no—
173C2×Dic61—81no—
174C2×Dic61—81no—
175C2×Dic61—81no—
176C4×D61—42no—
177C4×D61—81no—
178C8×C61—42no—
179#92418—42no—
180#9473—611nomaximal, Sylow
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—41no—
188C22×C61—41no—
189C22×C61—41no—
190C22×C61—41no—
191C22×C61#38611yescharacteristic
192C22×C61#41711yescharacteristic
193C22×C61#41911yescharacteristic
194C22×C61#46811yescharacteristic
195C22×C61#46811yescharacteristic
196C22×C61#47011yescharacteristic
197C22×C61#47011yescharacteristic
198C2×D61—42no—
199C2×D61—41no—
200C2×D61—41no—
201C2×D61—41no—
202C2×D61—81no—
203C4×C61—21no—
204C4×C61—21no—
205C4×C61—21no—
206C4×C61—82no—
207C4×C61—161no—
208C4×C61#45812yes—
209Dic61—41no—
210Dic61—41no—
211Dic61—41no—
212Dic61—82no—
213Dic61—161no—
214#2973—611no—
215#2487—611no—
216#1824—611no—
217#1824—611no—
218#1799—611no—
219#1289—611no—
220#1289D6111yescharacteristic, complemented
221C2×C61—21no—
222C2×C61—21no—
223C2×C61—21no—
224C2×C61—21no—
225C2×C61—21no—
226C2×C61—21no—
227C2×C61#189011yescharacteristic
228C2×C61#189411yescharacteristic
229C2×C61#191011yescharacteristic
230D61—81no—
231C2×C4×D4—611no—
232C22×C4⋊C4—611no—
233#435—1222no—
234#435—1222no—
235#432—1221no—
236C2×#99—1221no—
237#407—1221no—
238C2×C22⋊C8—611no—
239C2×C22⋊C8C2×D6111yescharacteristic
240#400—612no—
241#399—612no—
242#389—1221no—
243#386—1221no—
244#385—1221no—
245#380—611no—
246#380—1221no—
247#380C2×D6111yescharacteristic
248#380C2×D6111yescharacteristic
249#342—1221no—
250#340—1222no—
251#340—1222no—
252C61#947311yescharacteristic, complemented, Sylow
253C22×D4—611no—
254C23×C4—611no—
255C23×C4—1221no—
256C23×C4C22×D6111yescharacteristic
257C2×M16—1222no—
258C22×C8—612no—
259C22×C8D24412yes—
260C4×D4—1221no—
261C4×D4—1221no—
262C4×D4—1221no—
263C4×D4—1221no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—611no—
267C2×C4⋊C4—612no—
268C2×C4⋊C4—1221no—
269C2×C4⋊C4—1221no—
270C2×C4⋊C4—1221no—
271C2×C4⋊C4—1221no—
272C2×C4⋊C4—1222no—
273C2×C4⋊C4—1221no—
274C2×C4⋊C4—1221no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—611no—
278C2×C22⋊C4—611no—
279C2×C22⋊C4—1222no—
280C2×C22⋊C4—1221no—
281C2×C42—611no—
282C4⋊C8—1222no—
283#99—2442no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8—1222no—
287C22⋊C8—1221no—
288C22⋊C8—1221no—
289#97—1222no—
290#95—21no—
291#95—1222no—
292#95—1222no—
293#95—1221no—
294#95C4×D6112yes—
295C24—611no—
296C24D4×D6111yescharacteristic
297C2×D4—1221no—
298C2×D4—1221no—
299C2×D4—1221no—
300C2×D4—1221no—
301C2×D4—1221no—
302C2×D4—1221no—
303C22×C4—22no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—611no—
307C22×C4—611no—
308C22×C4—611no—
309C22×C4—1221no—
310C22×C4—1221no—
311C22×C4—1222no—
312C22×C4—4882no—
313C22×C4—2442no—
314C22×C4—2441no—
315C22×C4—2441no—
316C22×C4—2441no—
317C22×C4C2×C4×D6111yescharacteristic
318C22×C4C2×D24411yescharacteristic
319C22×C4#1175822211yescharacteristic
320C22×C4D4×D6111yescharacteristic
321C22×C4D4×D6111yescharacteristic
322C22×C4#1175822611yescharacteristic
323M16—4882no—
324C2×C8—22no—
325C2×C8—42no—
326C2×C8—1222no—
327C2×C8—2442no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—1221no—
331C4⋊C4—1221no—
332C4⋊C4—1222no—
333C4⋊C4—1222no—
334C4⋊C4—4882no—
335C4⋊C4—2442no—
336C4⋊C4—2441no—
337C4⋊C4—2441no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—1221no—
343C22⋊C4—1221no—
344C22⋊C4—1221no—
345C22⋊C4—1221no—
346C22⋊C4—4882no—
347C22⋊C4—2441no—
348C22⋊C4—2441no—
349C42—1221no—
350C42—1221no—
351C23—41no—
352C23—41no—
353C23—1221no—
354C23—1221no—
355C23—1221no—
356C23—2441no—
357C23—2441no—
358C23C22⋊C4×D6111yescharacteristic
359C23#1451975011yescharacteristic
360C23#1451975211yescharacteristic
361C23#1451980111yescharacteristic
362C23#1451980611yescharacteristic
363C23#1451980611yescharacteristic
364C23#1451980911yescharacteristic
365D4—2441no—
366D4—2441no—
367D4—2441no—
368D4—2441no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—1221no—
382C2×C4—1221no—
383C2×C4—1221no—
384C2×C4—1221no—
385C2×C4—1221no—
386C2×C4—4881no—
387C2×C4—4881no—
388C2×C4—4881no—
389C2×C4—4881no—
390C2×C4—4882no—
391C2×C4—4882no—
392C2×C4—2442no—
393C2×C4—2442no—
394C2×C4—2442no—
395C2×C4—2442no—
396C2×C4—2441no—
397C2×C4—2441no—
398C2×C4#1451973911yescharacteristic
399C2×C4C4×D24411yescharacteristic
400C2×C4#1451974711yescharacteristic
401C2×C4#1451976211yescharacteristic
402C8—42no—
403C8—2442no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—4881no—
415C22—2442no—
416C22—2441no—
417C22—2441no—
418C22—2441no—
419C22#1683964011yescharacteristic
420C22#1683976011yescharacteristic
421C22#1683976911yescharacteristic
422C22#1684000711yescharacteristic
423C22#1684000911yescharacteristic
424C22#1684001911yescharacteristic
425C22#1684002111yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—82no—
430C4—161no—
431C4—9761no—
432C4—4882no—
433C4—2441no—
434C4—2441no—
435C4—2441no—
436C4#1683995312yes—
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—4881no—
444C2#2144623111yescharacteristic
445C2#2144625111yescharacteristic
446C2#2144634511yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1289C2
socleC22×C61#386
prime radicalC22×C22⋊D61D4
cosocleC23×C4×C61C23
hypercenter#1289D61
hypocenterC61#9473
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
3. #21445910C2semidirect
4. #21445907C2semidirect
6. #21443686C2semidirect
7. C61×#1289C2semidirect
16. #16839702C4semidirect
17. #16839699C4semidirect
220. #1289D61semidirect
252. C61#9473semidirect

Maximal subgroups

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

subgroupindex
#214548452
#214501372
#214459102
#214459072
#214458042
#214436862
C61×#12892
#947361