Finite Group Databasebeta

← #29578043

Subgroups — #29578043

Order 15616.

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

normal subgroups only

This group has 472 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#21455742C211yesmaximal, characteristic
2#21454864C211yesmaximal, characteristic
3#21445899C211yesmaximal, characteristic, complemented
4#21444360C211yesmaximal, characteristic
5#21443725C211yesmaximal, characteristic, complemented
6#21443696C211yesmaximal, characteristic, complemented
7C61×#1824C211yesmaximal, characteristic, complemented
8#16840273C2211yescharacteristic
9#16840097—21no—
10#16840092C412yescomplemented
11#16840080—21no—
12#16840075C412yescomplemented
13#16839837—21no—
14#16839836—21no—
15#16839833—21no—
16C8×C22⋊D61—21no—
17#16839820—21no—
18#16839802—21no—
19#16839761—21no—
20#16839727—21no—
21#16839726C2211yescharacteristic
22#16839726C2211yescharacteristic
23#16839726C2211yescharacteristic
24#16839531—21no—
25#16839531—21no—
26#16839498—21no—
27#16839495—21no—
28C22×C4⋊C4×C61C2211yescharacteristic
29C2×C22⋊C8×C61C2211yescharacteristic
30C61×#380C2211yescharacteristic
31C61×#340—21no—
32C61×#340—21no—
33C2×#11758222D411yescharacteristic
34C2×#11758194—21no—
35C2×#11758194—21no—
36C2×#11758194D411yescharacteristic
37C2×#11758194D411yescharacteristic
38C2×#11758189—21no—
39C2×C8×D61—21no—
40#14519760—21no—
41#14519756C2×C411yescharacteristic
42#14519756D411yescharacteristic
43#14519751C2×C411yescharacteristic
44#14519751D411yescharacteristic
45#14519746—21no—
46C4×C22⋊D61—21no—
47C4×C22⋊D61—21no—
48#14519736D411yescharacteristic
49#14519710—22no—
50#14519708—22no—
51#14519706—21no—
52#14519706—21no—
53#14519700—22no—
54#14519697—22no—
55C22⋊#92418—21no—
56C22⋊#92418—21no—
57C22⋊#92418—21no—
58C22⋊#92418—21no—
59C22⋊#92418—21no—
60C22⋊#92418C2×C412yes—
61#14519689—21no—
62#14519689—21no—
63#14519687—21no—
64C8×Dic61—21no—
65C23×C4×C61—21no—
66C23×C4×C61C2311yescharacteristic
67C22×C8×C61D411yescharacteristic
68C22×C8×C61D411yescharacteristic
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61—21no—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61C2×C412yes—
76C2×C22⋊C4×C61—21no—
77C22⋊C8×C61—21no—
78C22⋊C8×C61—21no—
79#95×C61—21no—
80C24×C61#4111yescharacteristic
81C22×C4×C61—22no—
82C22×C4×C61—41no—
83C22×C4×C61—41no—
84C22×C4×C61—41no—
85C22×C4×C61—41no—
86C22×C4×C61—41no—
87C22×C4×C61—82no—
88C22×C4×C61C22×C411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61#4111yescharacteristic
94C2×C22⋊D61—21no—
95C2×C22⋊D61—21no—
96C2×C22⋊D61—21no—
97#11758222—41no—
98#11758222—41no—
99C2×D244C22⋊C411yescharacteristic
100C2×C4×D61—21no—
101C2×Dic122C22⋊C411yescharacteristic
102C2×C8×C61—21no—
103C2×C8×C61—21no—
104C2×C8×C61—41no—
105C2×C8×C61—41no—
106C4⋊C4×C61—22no—
107C4⋊C4×C61—22no—
108C4⋊C4×C61—41no—
109C4⋊C4×C61—41no—
110C4⋊C4×C61—41no—
111C4⋊C4×C61—41no—
112C22⋊C4×C61—81no—
113C22⋊Dic61—21no—
114C22⋊Dic61—21no—
115C22⋊Dic61—21no—
116#11758198—21no—
117#11758198—21no—
118C4⋊Dic61C22⋊C411yescharacteristic
119C4⋊Dic61C22⋊C411yescharacteristic
120#11758196—21no—
121#11758196—21no—
122C4×Dic61—21no—
123#11758194—41no—
124#11758194—41no—
125#11758194—41no—
126#11758194—41no—
127#11758194—41no—
128#11758194—41no—
129C2×#92418—22no—
130C2×#92418—21no—
131C2×#92418—21no—
132C2×#92418—41no—
133#11758189—81no—
134C8×D61—81no—
135C23×C61—21no—
136C23×C61—21no—
137C23×C61—41no—
138C23×C61—41no—
139C23×C61C4×D411yescharacteristic
140C23×C61#12111yescharacteristic
141C23×C61#12311yescharacteristic
142C22×D61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—42no—
153C2×C4×C61—82no—
154C2×C4×C61—82no—
155C2×C4×C61—82no—
156C2×C4×C61—81no—
157C2×C4×C61—81no—
158C2×C4×C61—81no—
159C2×C4×C61C2×C22⋊C411yescharacteristic
160C2×C4×C61C4×D411yescharacteristic
161C2×C4×C61#12011yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C22⋊D61—41no—
164C22⋊D61—41no—
165C22⋊D61—81no—
166C22⋊D61—81no—
167C2×Dic61—21no—
168C2×Dic61—21no—
169C2×Dic61—21no—
170D244—41no—
171C4×D61—41no—
172C4×D61—41no—
173Dic122—41no—
174C8×C61—41no—
175C8×C61—41no—
176#92418—42no—
177#92418—41no—
178#92418—41no—
179#92418—81no—
180#9521—611nomaximal, Sylow
181C22×C61—21no—
182C22×C61—21no—
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#38611yescharacteristic
192C22×C61#47111yescharacteristic
193C22×C61#47311yescharacteristic
194C2×D61—41no—
195C2×D61—41no—
196C2×D61—41no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—82no—
201C4×C61—161no—
202C4×C61#45811yescharacteristic
203C4×C61#45811yescharacteristic
204Dic61—41no—
205Dic61—41no—
206Dic61—41no—
207#3092—611no—
208#2976—611no—
209#1824D6111yescharacteristic, complemented
210#1821—611no—
211#1513—611no—
212#1305—611no—
213#1293—611no—
214C2×C61—21no—
215C2×C61—21no—
216C2×C61—41no—
217C2×C61—41no—
218C2×C61#189011yescharacteristic
219C2×C61#190311yescharacteristic
220C2×C61#190811yescharacteristic
221D61—81no—
222#525—611no—
223C22×C4⋊C4C2×D6111yescharacteristic
224#484—1221no—
225#483—612no—
226#481—1221no—
227#480—612no—
228#436—1221no—
229#435—1221no—
230#435—1221no—
231C8×D4—1221no—
232#433—1221no—
233#428—1221no—
234#417—1221no—
235#408—1221no—
236#407—611no—
237#407—611no—
238#407—611no—
239C2×C22⋊C8C2×D6111yescharacteristic
240#380C2×D6111yescharacteristic
241#343—1221no—
242#343—1221no—
243#340—21no—
244#340—21no—
245#328—1221no—
246#327—1221no—
247C61#952111yescharacteristic, complemented, Sylow
248C2×#41—611no—
249C23×C4—21no—
250C23×C4C22×D6111yescharacteristic
251C2×M16—611no—
252C2×M16—611no—
253C2×M16—1221no—
254C2×M16—1221no—
255C2×M16—1221no—
256C22×C8—1221no—
257C22×C8C22⋊D6111yescharacteristic
258C22×C8C22⋊D6111yescharacteristic
259#123—1221no—
260C22⋊Q8—611no—
261C22⋊Q8—611no—
262#121—611no—
263#121—611no—
264#120—1221no—
265C4×D4—1221no—
266C4×D4—1221no—
267#117—611no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4C4×D6112yes—
275C2×C22⋊C4—21no—
276#107—1222no—
277#106—1222no—
278C4⋊C8—1221no—
279C4⋊C8—1221no—
280#103—1222no—
281#102—1222no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—612no—
285C22⋊C8—1221no—
286C22⋊C8—1221no—
287C22⋊C8—1221no—
288C22⋊C8—1221no—
289C22⋊C8—1221no—
290C22⋊C8—1221no—
291C22⋊C8—1221no—
292#97—1221no—
293C4×C8—1221no—
294#95—21no—
295C24#1175822411yescharacteristic
296#41—2441no—
297#41—2441no—
298C2×Q8—611no—
299C2×D4—611no—
300C2×D4—1221no—
301C2×D4—1221no—
302C2×D4—1221no—
303C22×C4—22no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—82no—
310C22×C4—1221no—
311C22×C4C2×C4×D6111yescharacteristic
312C22×C4#1175822211yescharacteristic
313C22×C4D4×D6111yescharacteristic
314C22×C4D4×D6111yescharacteristic
315C22×C4D4×D6111yescharacteristic
316C22×C4C2×C22⋊D6111yescharacteristic
317M16—4881no—
318M16—2441no—
319M16—2441no—
320M16—2441no—
321M16—2441no—
322M16—2441no—
323M16—2441no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—1222no—
329C2×C8—1221no—
330C2×C8—1221no—
331C2×C8—4881no—
332C2×C8—2441no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—611no—
340C4⋊C4—611no—
341C4⋊C4—1221no—
342C4⋊C4—1221no—
343C22⋊C4—81no—
344C22⋊C4—1221no—
345C22⋊C4—1221no—
346C22⋊C4—1221no—
347C22⋊C4—1221no—
348C22⋊C4—1221no—
349C42—1221no—
350C23—21no—
351C23—21no—
352C23—41no—
353C23—41no—
354C23—1221no—
355C23#1451973711yescharacteristic
356C23#1451974811yescharacteristic
357C23#1451975911yescharacteristic
358Q8—2441no—
359D4—4881no—
360D4—4881no—
361D4—2441no—
362D4—2441no—
363D4—2441no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—1221no—
381C2×C4—1221no—
382C2×C4—1221no—
383C2×C4—2441no—
384C2×C4—2441no—
385C2×C4C22⋊C4×D6111yescharacteristic
386C2×C4C4×C22⋊D6111yescharacteristic
387C2×C4#1451974511yescharacteristic
388C2×C4#1451975011yescharacteristic
389C8—41no—
390C8—41no—
391C8—4881no—
392C8—2442no—
393C8—2441no—
394C8—2441no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—42no—
404C22—42no—
405C22—2441no—
406C22—2441no—
407C22—2441no—
408C22#1683963811yescharacteristic
409C22#1684002211yescharacteristic
410C22#1684003411yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—82no—
415C4—161no—
416C4—2441no—
417C4—2441no—
418C4—2441no—
419C4#1683995111yescharacteristic
420C4#1683995111yescharacteristic
421C2—21no—
422C2—21no—
423C2—41no—
424C2—41no—
425C2—4881no—
426C2#2144622911yescharacteristic
427C2#2144629611yescharacteristic
428C2#2144633711yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1824C2
socleC22×C61#386
prime radical#16840273C22
cosocleC23×C4×C61C23
hypercenter#1824D61
hypocenterC61#9521
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
3. #21445899C2semidirect
5. #21443725C2semidirect
6. #21443696C2semidirect
7. C61×#1824C2semidirect
10. #16840092C4semidirect
12. #16840075C4semidirect
209. #1824D61semidirect
247. C61#9521semidirect

Maximal subgroups

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

subgroupindex
#214557422
#214548642
#214458992
#214443602
#214437252
#214436962
C61×#18242
#952161