Finite Group Databasebeta

← #30600710

Subgroups — #30600710

Order 15616.

424 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 88 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#21462616C211yesmaximal, characteristic
2#21462011C211yesmaximal, characteristic
3#21461337C211yesmaximal, characteristic
4C2×#16840622C211yesmaximal, characteristic
5#21460710C211yesmaximal, characteristic
6#21460663C211yesmaximal, characteristic
7#21456765C211yesmaximal, characteristic
8#16840814—21no—
9#16840814—21no—
10#16840813—21no—
11#16840811—21no—
12#16840768—21no—
13#16840765C2211yescharacteristic
14C2×#14519844C2211yescharacteristic
15C2×#14519844C2211yescharacteristic
16#16840726—21no—
17C4×#11758217C2211yescharacteristic
18#16840642—21no—
19#16840642—21no—
20#16840622—21no—
21#16840622—21no—
22#16840622—21no—
23#16840621—21no—
24#16840314C2211yescharacteristic
25C2×#14519775C2211yescharacteristic
26#16840140—21no—
27#16840111—21no—
28#16839971—21no—
29#16839970—21no—
30#16839920C411yescharacteristic
31#16839918C411yescharacteristic
32#16839917C411yescharacteristic
33#16839812C2211yescharacteristic
34C61×#451C411yescharacteristic
35C2×#11758217—21no—
36C2×#11758217D411yescharacteristic
37C2×#11758217D411yescharacteristic
38C22×#92420—21no—
39C22×#92420—21no—
40#14519844—21no—
41#14519844—21no—
42#14519844—21no—
43#14519844—21no—
44#14519844—21no—
45#14519844—21no—
46#14519844D411yescharacteristic
47#14519844D411yescharacteristic
48C4⋊#92420—21no—
49C4⋊#92420—21no—
50C4⋊#92420—21no—
51#14519832—21no—
52#14519832—21no—
53#14519832—21no—
54C4×#92420—21no—
55C4×#92420—21no—
56C4×#92420—21no—
57C22×Dic122C2×C411yescharacteristic
58C2×Dic244—21no—
59C2×#11758202—21no—
60C2×#11758202—21no—
61#14519776D411yescharacteristic
62#14519775—21no—
63#14519775—21no—
64#14519775—21no—
65#14519775—21no—
66#14519775D411yescharacteristic
67#14519767—21no—
68#14519756C2×C411yescharacteristic
69#14519754—21no—
70Q8×Dic61D411yescharacteristic
71#14519742—21no—
72#14519742D411yescharacteristic
73#14519733—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×C4×Dic61C2311yescharacteristic
77#14519706—21no—
78#14519702—21no—
79#14519701—21no—
80#14519700—21no—
81C22⋊#92418C2×C411yescharacteristic
82C8×Dic61—21no—
83C22×Q8×C61C2×C411yescharacteristic
84C2×Q16×C61—21no—
85C22⋊Q8×C61C2×C411yescharacteristic
86#103×C61—21no—
87C22⋊C8×C61C2×C411yescharacteristic
88C2×Q8×C61—21no—
89C2×Q8×C61—21no—
90C2×Q8×C61—81no—
91C2×Q8×C61C22⋊C411yescharacteristic
92C22×C4×C61—21no—
93C22×C4×C61C22×C411yescharacteristic
94C22×Dic61—21no—
95C22×Dic61C2×D411yescharacteristic
96C22×Dic61#4111yescharacteristic
97C2×Dic122—21no—
98C2×Dic122—21no—
99C2×Dic122—21no—
100C2×Dic122—81no—
101C2×Dic122C22⋊C411yescharacteristic
102#11758217—21no—
103#11758217—21no—
104#11758217—21no—
105#11758217—41no—
106#11758217—41no—
107#11758217Q1612yescomplemented
108C2×#92420—21no—
109C2×#92420—21no—
110C2×#92420—21no—
111C2×#92420—21no—
112C2×#92420—41no—
113C2×#92420—41no—
114C2×#92420—41no—
115C2×#92420—41no—
116C2×#92420—81no—
117Q16×C61—81no—
118C2×C8×C61—21no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61C22⋊C411yescharacteristic
121C22⋊C4×C61—21no—
122C22⋊Dic61—21no—
123#11758202—81no—
124#11758202—81no—
125C4⋊Dic61—21no—
126C4⋊Dic61C22⋊C411yescharacteristic
127#11758196—21no—
128#11758196—21no—
129#11758196—21no—
130#11758196—21no—
131#11758196—21no—
132#11758196—41no—
133#11758196—41no—
134#11758196—41no—
135#11758196—41no—
136C4×Dic61—21no—
137C4×Dic61—21no—
138C4×Dic61C2×D411yescharacteristic
139C4×Dic61C2×D411yescharacteristic
140C4×Dic61C2×D411yescharacteristic
141C4×Dic61#4111yescharacteristic
142C2×#92418—21no—
143Dic244—81no—
144C23×C61C4×D411yescharacteristic
145Q8×C61—41no—
146Q8×C61—41no—
147Q8×C61—41no—
148Q8×C61—81no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—81no—
154C2×C4×C61C2×C22⋊C411yescharacteristic
155C2×C4×C61C4×D411yescharacteristic
156C2×Dic61—21no—
157C2×Dic61—21no—
158C2×Dic61—21no—
159C2×Dic61—21no—
160C2×Dic61—41no—
161C2×Dic61—81no—
162C2×Dic61#12011yescharacteristic
163C2×Dic61#12111yescharacteristic
164C2×Dic61#12111yescharacteristic
165C2×Dic61#12311yescharacteristic
166C2×Dic61C2×Q1611yescharacteristic
167C2×Dic61#13511yescharacteristic
168C2×Dic61#13711yescharacteristic
169C2×Dic61#13711yescharacteristic
170Dic122—41no—
171Dic122—41no—
172Dic122—41no—
173Dic122—41no—
174Dic122—41no—
175Dic122—81no—
176#92420—22no—
177#92420—41no—
178#92420—41no—
179#92420—41no—
180#92420—81no—
181#92420—81no—
182C8×C61—41no—
183#92418—41no—
184#9268—611nomaximal, Sylow
185C22×C61—21no—
186C22×C61—21no—
187C22×C61#38611yescharacteristic
188C22×C61C4×Q1611yescharacteristic
189C22×C61#44111yescharacteristic
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—41no—
193C4×C61—41no—
194C4×C61—81no—
195Dic61—21no—
196Dic61—41no—
197Dic61—41no—
198Dic61—81no—
199Dic61#45111yescharacteristic
200Dic61#45211yescharacteristic
201Dic61#46211yescharacteristic
202Dic61#46411yescharacteristic
203Dic61#47111yescharacteristic
204Dic61#47311yescharacteristic
205#3209—611no—
206#2962—611no—
207#1997—611no—
208#1550—611no—
209C2×#333—611no—
210#1301—611no—
211#1283—611no—
212C2×C61—21no—
213C2×C61—21no—
214C2×C61#187811yescharacteristic
215C2×C61#190811yescharacteristic
216C2×C61#191411yescharacteristic
217#533—611no—
218C2×C4⋊Q8—611no—
219#494—1221no—
220#487—1221no—
221#462—1221no—
222#462—1221no—
223#451—611no—
224#451—611no—
225#451—611no—
226#451#692311yescharacteristic
227#446—1221no—
228#446—1221no—
229#446—1221no—
230C8×Q8—1221no—
231#431—611no—
232#424—1221no—
233#423—611no—
234C2×C4⋊C8—611no—
235C2×C4⋊C8—611no—
236#405—1221no—
237C4×M16—611no—
238#343—1221no—
239#343—1221no—
240#333—1221no—
241#333—1221no—
242#333—1221no—
243#332—1221no—
244C61#926811yescharacteristic, complemented, Sylow
245C22×Q8—611no—
246C22×Q8C2×#692311yescharacteristic
247C2×Q16—21no—
248C2×Q16—1221no—
249C2×Q16—1221no—
250C2×Q16—1221no—
251C2×M16—611no—
252C2×M16—611no—
253C2×M16—1221no—
254C22×C8—1221no—
255C22×C8—1221no—
256C4⋊Q8—611no—
257C4⋊Q8—611no—
258C4⋊Q8—1221no—
259C4⋊Q8—1221no—
260C4⋊Q8—1221no—
261C4⋊Q8—1221no—
262#125—1221no—
263C22⋊Q8—611no—
264C22⋊Q8—1221no—
265C22⋊Q8C2×#692311yescharacteristic
266C4×Q8—611no—
267C4×Q8—611no—
268C4×Q8—1221no—
269#117—1221no—
270C2×C4⋊C4—1221no—
271C2×C4⋊C4—1221no—
272C2×C42—611no—
273C4⋊C8—611no—
274C4⋊C8—611no—
275C4⋊C8—1221no—
276C4⋊C8—1221no—
277C4⋊C8—1221no—
278C4⋊C8—1221no—
279C4⋊C8—1221no—
280C4⋊C8—1221no—
281C4⋊C8—1221no—
282C4⋊C8—1221no—
283C4⋊C8—1221no—
284C4⋊C8—1221no—
285#103—21no—
286#103—1221no—
287#103—1221no—
288#103—1221no—
289C22⋊C8—611no—
290C22⋊C8C2×#692311yescharacteristic
291#97—1221no—
292#97—1221no—
293#97—1221no—
294C4×C8—1221no—
295C4×C8—1221no—
296C4×C8—1221no—
297C4×C8—1221no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—81no—
301C2×Q8—611no—
302C2×Q8—1221no—
303C2×Q8—1221no—
304C2×Q8—1221no—
305C2×Q8—4881no—
306C2×Q8C22⋊#692311yescharacteristic
307C22×C4—21no—
308C22×C4—611no—
309C22×C4—611no—
310C22×C4—1221no—
311C22×C4C22×#692311yescharacteristic
312Q16—81no—
313Q16—4881no—
314Q16—4881no—
315Q16—4881no—
316M16—612no—
317M16—1221no—
318M16—1221no—
319M16—1221no—
320M16—2441no—
321M16—2441no—
322C2×C8—21no—
323C2×C8—1221no—
324C2×C8—1221no—
325C2×C8—1221no—
326C2×C8—1221no—
327C2×C8—1221no—
328C2×C8—4881no—
329C2×C8—2441no—
330C2×C8—2441no—
331C2×C8—2441no—
332C2×C8—2441no—
333C4⋊C4—21no—
334C4⋊C4—611no—
335C4⋊C4—1221no—
336C4⋊C4—1221no—
337C4⋊C4—1221no—
338C4⋊C4—1221no—
339C4⋊C4—1221no—
340C4⋊C4—1221no—
341C4⋊C4—2441no—
342C4⋊C4—2441no—
343C4⋊C4—2441no—
344C4⋊C4—2441no—
345C4⋊C4C22⋊#692311yescharacteristic
346C22⋊C4—21no—
347C22⋊C4—1221no—
348C42—611no—
349C42—611no—
350C42—611no—
351C42—611no—
352C42—1221no—
353C42—1221no—
354C23D4×#692311yescharacteristic
355Q8—41no—
356Q8—41no—
357Q8—41no—
358Q8—81no—
359Q8—4881no—
360Q8—2441no—
361Q8—2441no—
362Q8—2441no—
363Q8—2441no—
364Q8—2441no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—81no—
370C2×C4—611no—
371C2×C4—611no—
372C2×C4—611no—
373C2×C4—611no—
374C2×C4—611no—
375C2×C4—611no—
376C2×C4—611no—
377C2×C4—611no—
378C2×C4—1221no—
379C2×C4—1221no—
380C2×C4—1221no—
381C2×C4—1221no—
382C2×C4—4881no—
383C2×C4—2441no—
384C2×C4C2×C22⋊#692311yescharacteristic
385C2×C4D4×#692311yescharacteristic
386C8—41no—
387C8—1222no—
388C8—4881no—
389C8—4881no—
390C8—2441no—
391C8—2441no—
392C8—2441no—
393C8—2441no—
394C22—21no—
395C22—21no—
396C22#1684071011yescharacteristic
397C22Q16×#692311yescharacteristic
398C22#1684080211yescharacteristic
399C4—21no—
400C4—21no—
401C4—41no—
402C4—41no—
403C4—81no—
404C4—611no—
405C4—611no—
406C4—611no—
407C4—611no—
408C4—611no—
409C4—611no—
410C4—1221no—
411C4—4881no—
412C4—2441no—
413C4—2441no—
414C2—21no—
415C2—21no—
416C2#2146181311yescharacteristic
417C2#2146185911yescharacteristic
418C2#2146187011yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#451C4
socleC22×C61#386
prime radicalC23×C61C4×D4
cosocleC2×C4×Dic61C23
hypercenter#451#6923
hypocenterC61#9268
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
107. #11758217Q16semidirect
244. C61#9268semidirect

Maximal subgroups

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

subgroupindex
#214626162
#214620112
#214613372
C2×#168406222
#214607102
#214606632
#214567652
#926861