Finite Group Databasebeta

← #29580459

Subgroups — #29580459

Order 15616.

636 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 101 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, complemented
2#21450312C211yesmaximal, characteristic, complemented
3#21446656C211yesmaximal, characteristic, complemented
4#21445910C211yesmaximal, characteristic, complemented
5#21444387C211yesmaximal, characteristic, complemented
6#21443666C211yesmaximal, characteristic, complemented
7C61×#3186C211yesmaximal, characteristic
8C22×C4⋊Dic61C2211yescharacteristic
9#16840078—22no—
10#16840078—22no—
11C2×#14519708—21no—
12C2×#14519698—21no—
13C2×#14519692—22no—
14C2×C22⋊#92418C2211yescharacteristic
15#16839687—22no—
16#16839665—21no—
17#16839642—22no—
18#16839627—21no—
19#16839622C2211yescharacteristic
20C4×C22⋊Dic61C2211yescharacteristic
21#16839525—22no—
22#16839496—22no—
23C2×#121×C61C2211yescharacteristic
24C2×#121×C61C2211yescharacteristic
25C61×#480—21no—
26C61×#480—21no—
27C61×#466—22no—
28C61×#447—21no—
29C61×#447—21no—
30C2×C22⋊C8×C61C2211yescharacteristic
31C23×Dic61—21no—
32C22×#92418—21no—
33C2×C4⋊Dic61—21no—
34C2×C4⋊Dic61—21no—
35C2×C4⋊Dic61—21no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61D412yes—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—24no—
42C2×C22⋊Dic61D412yes—
43C2×C4×Dic61—21no—
44#14519708—24no—
45#14519698—24no—
46#14519692—44no—
47C22⋊#92418—21no—
48C22⋊#92418D412yes—
49#14519684—24no—
50#14519684—21no—
51#14519684—21no—
52C22×D4×C61—21no—
53C22×D4×C61—21no—
54C22×D4×C61D411yescharacteristic
55C22×D4×C61D411yescharacteristic
56C23×C4×C61C2311yescharacteristic
57C2×D8×C61—22no—
58C22×C8×C61D412yes—
59#121×C61—21no—
60#121×C61—21no—
61#121×C61—21no—
62#121×C61—21no—
63#121×C61—21no—
64#121×C61—21no—
65#121×C61—21no—
66#121×C61—21no—
67C2×C4⋊C4×C61D411yescharacteristic
68C2×C4⋊C4×C61D411yescharacteristic
69C2×C22⋊C4×C61—21no—
70C2×C22⋊C4×C61—21no—
71#107×C61—22no—
72#102×C61—22no—
73#102×C61—22no—
74C22⋊C8×C61—21no—
75C22⋊C8×C61—21no—
76C24×C61—21no—
77C24×C61—21no—
78C24×C61C2×D411yescharacteristic
79C2×D4×C61—21no—
80C2×D4×C61—21no—
81C2×D4×C61—21no—
82C2×D4×C61—21no—
83C2×D4×C61—42no—
84C2×D4×C61—42no—
85C2×D4×C61—41no—
86C2×D4×C61—41no—
87C2×D4×C61—41no—
88C2×D4×C61—41no—
89C2×D4×C61—42no—
90C2×D4×C61—42no—
91C2×D4×C61—42no—
92C2×D4×C61—42no—
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61—42no—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61#4111yescharacteristic
102C22×Dic61—22no—
103C22×Dic61—22no—
104C22×Dic61—41no—
105C22×Dic61—41no—
106C22×Dic61—41no—
107C22×Dic61—82no—
108D8×C61—82no—
109C2×C8×C61—22no—
110C2×C8×C61—42no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61—21no—
115C22⋊C4×C61—41no—
116C22⋊C4×C61—41no—
117C22⋊C4×C61—41no—
118C22⋊C4×C61—41no—
119C22⋊Dic61—22no—
120C22⋊Dic61—22no—
121C22⋊Dic61—42no—
122C22⋊Dic61—84no—
123C4⋊Dic61—22no—
124C4⋊Dic61—22no—
125C4⋊Dic61—41no—
126C4⋊Dic61—41no—
127C4⋊Dic61—42no—
128C4×Dic61—42no—
129C2×#92418—22no—
130C2×#92418—42no—
131C23×C61—41no—
132C23×C61—41no—
133C23×C61—42no—
134C23×C61—42no—
135C23×C61—41no—
136C23×C61—41no—
137C23×C61—42no—
138C23×C61—42no—
139C23×C61—81no—
140C23×C61—81no—
141C23×C61#12011yescharacteristic
142C23×C61#12111yescharacteristic
143C23×C61#12711yescharacteristic
144C23×C61#13512yes—
145C23×C61AGL(1,ℤ/8)12yes—
146D4×C61—42no—
147D4×C61—42no—
148D4×C61—42no—
149D4×C61—81no—
150D4×C61—82no—
151D4×C61—82no—
152D4×C61—81no—
153D4×C61—81no—
154D4×C61—81no—
155D4×C61—81no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—22no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—42no—
165C2×C4×C61—42no—
166C2×C4×C61#12011yescharacteristic
167C2×C4×C61#12011yescharacteristic
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×Dic61—42no—
171C2×Dic61—42no—
172C2×Dic61—44no—
173C2×Dic61—42no—
174C2×Dic61—42no—
175C2×Dic61—82no—
176C2×Dic61—82no—
177C2×Dic61—81no—
178C2×Dic61—81no—
179C2×Dic61—81no—
180C8×C61—42no—
181#92418—42no—
182#9950—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—42no—
190C22×C61—42no—
191C22×C61—84no—
192C22×C61—81no—
193C22×C61—81no—
194C22×C61—81no—
195C22×C61—81no—
196C22×C61—81no—
197C22×C61—81no—
198C22×C61—162no—
199C22×C61#39211yescharacteristic
200C22×C61#44912yes—
201C22×C61#46312yes—
202C22×C61#49511yescharacteristic
203C22×C61#49611yescharacteristic
204C4×C61—21no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—81no—
208C4×C61—81no—
209C4×C61#45812yes—
210Dic61—82no—
211Dic61—82no—
212Dic61—161no—
213#3186D6111yescharacteristic
214C2×#481—611no—
215#2504—611no—
216#1964—611no—
217#1824—611no—
218#1519—611no—
219#1282—611no—
220C2×C61—22no—
221C2×C61—21no—
222C2×C61—22no—
223C2×C61—21no—
224C2×C61—161no—
225C2×C61—161no—
226C2×C61#200811yescharacteristic
227C2×C61#201211yescharacteristic
228C2×C61#202211yescharacteristic
229C2×#121C2×D6111yescharacteristic
230C2×#121C2×D6111yescharacteristic
231C22×C4⋊C4—611no—
232#481—1222no—
233#481—1222no—
234#480—21no—
235#480—21no—
236#466—22no—
237#447—21no—
238#447—21no—
239C2×#106—1221no—
240C2×#102—1221no—
241C2×#99—1222no—
242C2×C22⋊C8—611no—
243C2×C22⋊C8C2×D6111yescharacteristic
244#397—1222no—
245#390—1221no—
246#386—1222no—
247#382—1221no—
248#380—611no—
249C4×C22⋊C4—611no—
250#340—1222no—
251#327—1222no—
252C61#995011yescharacteristic, complemented, Sylow
253C22×D4—21no—
254C22×D4—21no—
255C22×D4C22⋊D6111yescharacteristic
256C22×D4C22⋊D6111yescharacteristic
257C23×C4—1221no—
258C23×C4C22×D6111yescharacteristic
259C2×D8—22no—
260C22×C8—1221no—
261C22×C8C22⋊D6112yes—
262#121—21no—
263#121—21no—
264#121—21no—
265#121—21no—
266#121—21no—
267#121—21no—
268#121—21no—
269#121—21no—
270C2×C4⋊C4—612no—
271C2×C4⋊C4—1221no—
272C2×C4⋊C4—1221no—
273C2×C4⋊C4—1221no—
274C2×C4⋊C4—1221no—
275C2×C4⋊C4—1221no—
276C2×C4⋊C4—1221no—
277C2×C4⋊C4C22⋊D6111yescharacteristic
278C2×C4⋊C4C22⋊D6111yescharacteristic
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—612no—
282C2×C22⋊C4—1224no—
283C2×C22⋊C4—1221no—
284C2×C42—1221no—
285#107—22no—
286#106—1224no—
287#102—22no—
288#102—22no—
289#102—1224no—
290#99—2444no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—612no—
294C22⋊C8—1221no—
295#95—1224no—
296#95—1221no—
297#95—1221no—
298C24—21no—
299C24—21no—
300C24D4×D6111yescharacteristic
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—42no—
318C22×C4—1222no—
319C22×C4—1222no—
320C22×C4—4882no—
321C22×C4—2441no—
322C22×C4—2441no—
323C22×C4—2441no—
324C22×C4D4×D6111yescharacteristic
325C22×C4D4×D6111yescharacteristic
326C22×C4#1175822411yescharacteristic
327C22×C4C2×C22⋊D6111yescharacteristic
328C22×C4C2×C22⋊D6111yescharacteristic
329C22×C4C2×C22⋊D6111yescharacteristic
330D8—82no—
331C2×C8—22no—
332C2×C8—42no—
333C2×C8—1222no—
334C2×C8—2442no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—1222no—
340C4⋊C4—1222no—
341C4⋊C4—2442no—
342C4⋊C4—2441no—
343C4⋊C4—2441no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—1222no—
349C22⋊C4—1222no—
350C22⋊C4—4884no—
351C22⋊C4—2442no—
352C42—2442no—
353C23—41no—
354C23—41no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—41no—
359C23—41no—
360C23—42no—
361C23—81no—
362C23—81no—
363C23#1451974511yescharacteristic
364C23#1451974911yescharacteristic
365C23#1451977311yescharacteristic
366C23#1451979612yes—
367C23#1451980112yes—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—81no—
372D4—82no—
373D4—81no—
374D4—81no—
375D4—81no—
376D4—81no—
377D4—82no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—4881no—
389C2×C4—4881no—
390C2×C4—4881no—
391C2×C4—4882no—
392C2×C4—4882no—
393C2×C4—2442no—
394C2×C4—2442no—
395C2×C4—2444no—
396C2×C4—2442no—
397C2×C4—2442no—
398C2×C4#1451974511yescharacteristic
399C2×C4#1451974611yescharacteristic
400C2×C4#1451974811yescharacteristic
401C2×C4#1451974911yescharacteristic
402C8—42no—
403C8—2442no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—42no—
411C22—42no—
412C22—84no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—162no—
420C22#1683967011yescharacteristic
421C22#1683990412yes—
422C22#1683997512yes—
423C22#1684014511yescharacteristic
424C22#1684015011yescharacteristic
425C4—21no—
426C4—21no—
427C4—21no—
428C4—81no—
429C4—81no—
430C4—9761no—
431C4—4882no—
432C4—4882no—
433C4#1683995212yes—
434C2—21no—
435C2—22no—
436C2—22no—
437C2—21no—
438C2—161no—
439C2—161no—
440C2#2144686911yescharacteristic
441C2#2144689411yescharacteristic
442C2#2144694811yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
Fitting, prime radicalC61×#3186C2
socleC22×C61#392
hypercenter#3186D61
hypocenterC61#9950
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3186

Complements

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

NH ≅ G/Ntype
1. C2×#16840078C2semidirect
2. #21450312C2semidirect
3. #21446656C2semidirect
4. #21445910C2semidirect
5. #21444387C2semidirect
6. #21443666C2semidirect
252. C61#9950semidirect

Maximal subgroups

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

subgroupindex
C2×#168400782
#214503122
#214466562
#214459102
#214443872
#214436662
C61×#31862
#995061