Finite Group Databasebeta

← #30599666

Subgroups — #30599666

Order 15616.

464 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 78 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#21462686C211yesmaximal, characteristic
2#21461802C211yesmaximal, characteristic
3#21461771C211yesmaximal, characteristic, complemented
4#21461767C211yesmaximal, characteristic, complemented
5#21460700C211yesmaximal, characteristic, complemented
6#21460660C211yesmaximal, characteristic, complemented
7#21444307C211yesmaximal, characteristic
8C22×#11758213C2211yescharacteristic
9#16840795—21no—
10#16840793—21no—
11#16840791—21no—
12D4×#92420—21no—
13#16840787—21no—
14C2×#14519848—21no—
15C2×#14519844—21no—
16C2×#14519841—21no—
17#16840735—21no—
18#16840733C2211yescharacteristic
19C2×#14519834C2211yescharacteristic
20C4×C22⋊#6923C2211yescharacteristic
21#16840638—21no—
22#16840632—21no—
23#16840628—21no—
24#16840272C2211yescharacteristic
25C22⋊C8×D61C2211yescharacteristic
26#16839609C2211yescharacteristic
27#16839500C411yescharacteristic, complemented
28#16839498C411yescharacteristic, complemented
29#16839496C411yescharacteristic, complemented
30C61×#327C411yescharacteristic, complemented
31C2×#11758213—21no—
32C2×#11758213—21no—
33C2×#11758213—42no—
34C2×#11758213D411yescharacteristic
35C2×#11758213D411yescharacteristic
36C2×#11758217—21no—
37C22×#92420—21no—
38C22×#92420—21no—
39C22×#92420D411yescharacteristic
40C22×#92420D411yescharacteristic
41C2×C22⋊#6923—21no—
42C2×C4×#6923—21no—
43#14519848—42no—
44#14519844—21no—
45#14519844—42no—
46C4⋊#92420—21no—
47#14519841—42no—
48#14519834—21no—
49#14519834—21no—
50#14519834—21no—
51C22⋊#92420—21no—
52C22⋊#92420—21no—
53C22⋊#92420—21no—
54#14519832—21no—
55C4×#92420—21no—
56(C2×#6923)⋊C4—21no—
57C2×#11758224D411yescharacteristic
58C22×C4×D61C2311yescharacteristic
59C2×C8×D61—21no—
60#14519759—21no—
61#14519757Q811yescharacteristic
62#14519756C2×C411yescharacteristic
63#14519748Q811yescharacteristic
64#14519745—21no—
65D4×Dic61—21no—
66#14519737—21no—
67#14519735D411yescharacteristic
68C2×#11758198—21no—
69C2×C4×Dic61—21no—
70C22⋊#92418C2×C411yescharacteristic
71#14519689—21no—
72#14519684C2×C411yescharacteristic
73#121×C61C2×C411yescharacteristic
74C22⋊C8×C61C2×C411yescharacteristic
75#95×C61C2×C411yescharacteristic
76C23×D61#4111yescharacteristic
77C22×#6923—41no—
78C2×D4×C61—21no—
79C2×D4×C61C4⋊C411yescharacteristic
80C22×C4×C61—21no—
81C22×C4×C61C22×C411yescharacteristic
82C2×C22⋊D61—21no—
83C2×C22⋊D61—21no—
84C22×Dic61—21no—
85C22×Dic61—21no—
86C22×Dic61C2×D411yescharacteristic
87#11758224—41no—
88#11758224—41no—
89C2×C4×D61—41no—
90C2×C4×D61—41no—
91C2×C4×D61C2×D411yescharacteristic
92C2×C4×D61C2×D411yescharacteristic
93C2×C4×D61C2×Q811yescharacteristic
94C2×C4×D61#4111yescharacteristic
95C2×Dic122C4⋊C411yescharacteristic
96C22⋊#6923—22no—
97C22⋊#6923—41no—
98#11758217—81no—
99C2×#92420—22no—
100C2×#92420—21no—
101C2×#92420—21no—
102C2×#92420—21no—
103C2×#92420—21no—
104C2×#92420—41no—
105C2×#92420—41no—
106C2×#92420—41no—
107C2×#92420—41no—
108C2×#92420—81no—
109C4×#6923—41no—
110C4×#6923—41no—
111#11758213—22no—
112#11758213—22no—
113#11758213—41no—
114#11758213—41no—
115#11758213—41no—
116#11758213—41no—
117C2×C8×C61—21no—
118C4⋊C4×C61C4⋊C411yescharacteristic
119C22⋊C4×C61—21no—
120C22⋊Dic61—21no—
121C22⋊Dic61—21no—
122#11758198—21no—
123#11758198—21no—
124#11758198—42no—
125C4⋊Dic61—21no—
126C4⋊Dic61C4⋊C411yescharacteristic
127#11758196—21no—
128C4×Dic61—21no—
129C4×Dic61—41no—
130C4×Dic61—41no—
131C2×#92418—21no—
132C8×D61—41no—
133C8×D61—41no—
134C23×C61—21no—
135C23×C61C4×D411yescharacteristic
136C22×D61—21no—
137C22×D61—21no—
138C22×D61—41no—
139C22×D61—41no—
140C22×D61C22⋊Q811yescharacteristic
141C22×D61#12311yescharacteristic
142C2×#6923—42no—
143C2×#6923—81no—
144C2×#6923—81no—
145D4×C61—41no—
146D4×C61—41no—
147D4×C61—41no—
148C2×C4×C61—21no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61C2×C4⋊C411yescharacteristic
154C2×C4×C61C4×D411yescharacteristic
155C22⋊D61—81no—
156C22⋊D61—81no—
157C2×Dic61—21no—
158C2×Dic61—21no—
159C2×Dic61—22no—
160C2×Dic61—21no—
161C2×Dic61—21no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61#12011yescharacteristic
167C2×Dic61C22⋊Q811yescharacteristic
168C4×D61—21no—
169C4×D61—21no—
170C4×D61—21no—
171C4×D61—21no—
172C4×D61—41no—
173C4×D61—41no—
174Dic122—41no—
175#92420—42no—
176#92420—41no—
177#92420—41no—
178#92420—41no—
179#92420—41no—
180C8×C61—41no—
181#92418—41no—
182#8875—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—41no—
186C22×C61—41no—
187C22×C61—41no—
188C22×C61#38511yescharacteristic
189C22×C61#44311yescharacteristic
190C22×C61#44411yescharacteristic
191C2×D61—21no—
192C2×D61—21no—
193C2×D61—21no—
194C2×D61—21no—
195C2×D61—21no—
196C2×D61—21no—
197C2×D61—81no—
198C2×D61—81no—
199#6923—82no—
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—41no—
203C4×C61—81no—
204Dic61—21no—
205Dic61—42no—
206Dic61—41no—
207Dic61—41no—
208Dic61B(4,𝔽2)11yescharacteristic
209Dic61#45811yescharacteristic
210#2976—611no—
211#1871—611no—
212#1855—611no—
213#1851—611no—
214#1501—611no—
215#1297—611no—
216#1282—611no—
217C2×C61—21no—
218C2×C61—21no—
219C2×C61—81no—
220C2×C61#186111yescharacteristic
221C2×C61#186611yescharacteristic
222C2×C61#186711yescharacteristic
223D61—41no—
224D61—41no—
225C22×M16—611no—
226#525—611no—
227#436—1221no—
228#435—1221no—
229#435—1221no—
230C8×D4—1221no—
231#433—1221no—
232C2×#108—1221no—
233C2×C4⋊C8—1221no—
234C2×#104—1221no—
235#408—1221no—
236#407—611no—
237C2×C22⋊C8—611no—
238C2×C22⋊C8—611no—
239C4×C22⋊C4—611no—
240C4×C22⋊C4—611no—
241#341—1221no—
242#339—1221no—
243#336—1221no—
244#328—611no—
245#328—611no—
246#327—611no—
247#327#692311yescharacteristic, complemented
248C61#887511yescharacteristic, complemented, Sylow
249C2×#41—611no—
250C23×C4—611no—
251C2×M16—611no—
252C2×M16—611no—
253C2×M16—1221no—
254C2×M16—1221no—
255C2×M16—1221no—
256C2×M16—2442no—
257C22×C8—611no—
258C22×C8—611no—
259C22×C8—1221no—
260C22×C8—1221no—
261C22×C8—1221no—
262#123—1221no—
263C22⋊Q8—611no—
264C22⋊Q8—611no—
265#121—611no—
266#121C2×#692311yescharacteristic
267#120—1221no—
268C4×D4—1221no—
269C4×D4—1221no—
270#117—611no—
271C2×C22⋊C4—1221no—
272C2×C22⋊C4—1221no—
273C2×C42—1221no—
274C2×C42—1221no—
275#108—2442no—
276C4⋊C8—1221no—
277C4⋊C8—1221no—
278C4⋊C8—2442no—
279#104—2442no—
280C22⋊C8—611no—
281C22⋊C8—1221no—
282C22⋊C8—1221no—
283C22⋊C8—1221no—
284C22⋊C8—1221no—
285C22⋊C8—1221no—
286C22⋊C8—1221no—
287C22⋊C8—1221no—
288C22⋊C8C2×#692311yescharacteristic
289#97—1221no—
290C4×C8—1221no—
291#95—611no—
292#95—1221no—
293#95C2×#692311yescharacteristic
294C24—611no—
295#41—2441no—
296#41—2441no—
297C2×Q8—611no—
298C2×D4—21no—
299C2×D4—1221no—
300C2×D4—1221no—
301C2×D4C4⋊#692311yescharacteristic
302C22×C4—21no—
303C22×C4—611no—
304C22×C4—611no—
305C22×C4—611no—
306C22×C4—611no—
307C22×C4—611no—
308C22×C4—1221no—
309C22×C4—1221no—
310C22×C4—2441no—
311C22×C4—2441no—
312C22×C4—2441no—
313C22×C4C22×#692311yescharacteristic
314M16—1222no—
315M16—1222no—
316M16—4881no—
317M16—2441no—
318M16—2441no—
319M16—2441no—
320M16—2441no—
321C2×C8—21no—
322C2×C8—1221no—
323C2×C8—1221no—
324C2×C8—1221no—
325C2×C8—1221no—
326C2×C8—1222no—
327C2×C8—1221no—
328C2×C8—4881no—
329C2×C8—2441no—
330C2×C8—2441no—
331C2×C8—2441no—
332C2×C8—2441no—
333C2×C8—2441no—
334C2×C8—2441no—
335C4⋊C4—611no—
336C4⋊C4—1221no—
337C4⋊C4—1221no—
338C4⋊C4C4⋊#692311yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—1221no—
341C22⋊C4—1221no—
342C22⋊C4—1221no—
343C22⋊C4—1221no—
344C22⋊C4—1222no—
345C22⋊C4—2442no—
346C22⋊C4—2441no—
347C42—1221no—
348C42—2441no—
349C42—2441no—
350C42—2441no—
351C42—2441no—
352C23—21no—
353C23—611no—
354C23—611no—
355C23—1221no—
356C23—1221no—
357C23—2441no—
358C23—2441no—
359C23D4×#692311yescharacteristic
360Q8—2441no—
361D4—41no—
362D4—41no—
363D4—41no—
364D4—4881no—
365D4—4881no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—611no—
372C2×C4—611no—
373C2×C4—1221no—
374C2×C4—1221no—
375C2×C4—1221no—
376C2×C4—1221no—
377C2×C4—1221no—
378C2×C4—1221no—
379C2×C4—1222no—
380C2×C4—1221no—
381C2×C4—1221no—
382C2×C4—4881no—
383C2×C4—4881no—
384C2×C4—2441no—
385C2×C4—2441no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4—2442no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4C2×C4⋊#692311yescharacteristic
392C2×C4D4×#692311yescharacteristic
393C8—41no—
394C8—2441no—
395C8—2441no—
396C8—2441no—
397C8—2442no—
398C8—2441no—
399C8—2441no—
400C22—21no—
401C22—21no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—1221no—
406C22—1221no—
407C22—1221no—
408C22—1221no—
409C22—1221no—
410C22—1221no—
411C22—4881no—
412C22—4881no—
413C22#1684070911yescharacteristic
414C22#1684080411yescharacteristic
415C22#1684080711yescharacteristic
416C4—21no—
417C4—21no—
418C4—41no—
419C4—81no—
420C4—611no—
421C4—611no—
422C4—1221no—
423C4—4882no—
424C4—2442no—
425C4—2441no—
426C4—2441no—
427C2—21no—
428C2—21no—
429C2—81no—
430C2—2441no—
431C2—2441no—
432C2#2146178111yescharacteristic
433C2#2146178611yescharacteristic
434C2#2146179011yescharacteristic
subgroupG/H
centerC22#16840709
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#327C4
socleC22×C61#385
prime radical#16840272C22
cosocleC22×C4×D61C23
hypercenter#327#6923
hypocenterC61#8875
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
3. #21461771C2semidirect
4. #21461767C2semidirect
5. #21460700C2semidirect
6. #21460660C2semidirect
27. #16839500C4semidirect
28. #16839498C4semidirect
29. #16839496C4semidirect
30. C61×#327C4semidirect
247. #327#6923semidirect
248. C61#8875semidirect

Maximal subgroups

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

subgroupindex
#214626862
#214618022
#214617712
#214617672
#214607002
#214606602
#214443072
#887561