Finite Group Databasebeta

← #72220557

Subgroups — #72220557

Order 5184.

1740 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 1740 conjugacy classes of subgroups — too many to draw. Only its 98 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#72220556C212yesmaximal, complemented
2C3⋊D3×C22⋊#157C212yesmaximal
3C22×#10620198C211yesmaximal, characteristic
4C2×#1572C212yesmaximal, complemented
5C22×C3⋊D32C2211yescharacteristic
6C22×#66472C412yescomplemented
7C2×#10620198—24no—
8C2×#10620198C412yescomplemented
9C2×#10620198C2211yescharacteristic
10C2×#10620198C2211yescharacteristic
11C2×C3⋊D3×#157C2214yes—
12#1572—24no—
13#13049596C412yescomplemented
14C32×C22⋊#157C412yescomplemented
15#13049396C414yescomplemented
16D3×C22⋊#157—64no—
17C22×C3⋊(C3⋊(C3⋊D3))C2×C411yescharacteristic
18C22×C32×C3⋊D3C2×C412yes—
19C2×C3⋊D32—22no—
20C2×C3⋊D32—22no—
21C2×C3⋊D32C2×C411yescharacteristic
22C2×C3⋊D32C2×C412yes—
23C2×C3⋊D32C2311yescharacteristic
24C2×#66472—24no—
25C2×#66472C2×C412yes—
26C2×#66472C2×C412yes—
27C2×C3⋊(C3⋊#157)C2×C414yescomplemented
28C2×C32×#157C2×C414yescomplemented
29#10620198—22no—
30#10620198—24no—
31#10620198D414yes—
32C3⋊D3×#157—24no—
33C3⋊D3×#157—24no—
34C22⋊C4×#157—92nomaximal
35C22×D3×C3⋊D3—64no—
36C2×D3×#157—68no—
37#88994—64no—
38C3×C22⋊#157—24no—
39(C2×D3)⋊#157—68no—
40C22×C34C4211yescharacteristic, complemented
41C2×C3⋊(C3⋊(C3⋊D3))—21no—
42C2×C3⋊(C3⋊(C3⋊D3))—21no—
43C2×C3⋊(C3⋊(C3⋊D3))C4211yescharacteristic, complemented
44C2×C3⋊(C3⋊(C3⋊D3))C22×C411yescharacteristic
45C2×C32×C3⋊D3—24no—
46C2×C32×C3⋊D3C4212yescomplemented
47C2×C32×C3⋊D3C22×C412yes—
48C3⋊D32—21no—
49C3⋊D32—21no—
50C3⋊D32—24no—
51C3⋊D32C2×D412yes—
52C3⋊D32#4112yes—
53#66472—22no—
54#66472C22⋊C414yescomplemented
55C3⋊(C3⋊#157)—24no—
56C32×#157—24no—
57C23×#157—92no—
58C2×C22⋊#157—92no—
59C2×C4×#157—94no—
60C22⋊C4×C3⋊D3—92no—
61#63781—94no—
62C22×C3⋊(C3⋊D3)—64no—
63C22×C3⋊(C3⋊D3)—244no—
64C22×C3×C3⋊D3—24no—
65C22×C32×D3—64no—
66C2×D3×C3⋊D3—64no—
67C2×D3×C3⋊D3—64no—
68C2×D3×C3⋊D3—64no—
69C2×D3×C3⋊D3—64no—
70C2×D3×C3⋊D3—128no—
71C2×D3×C3⋊D3—128no—
72C2×C3⋊#157—68no—
73C2×C3×#157—28no—
74D3×#157—128no—
75D3×#157—128no—
76C2×C34—21no—
77C2×C34C2×C4211yescharacteristic
78C3⋊(C3⋊(C3⋊D3))—21no—
79C3⋊(C3⋊(C3⋊D3))C2×C22⋊C411yescharacteristic
80C3⋊(C3⋊(C3⋊D3))#11711yescharacteristic
81C32×C3⋊D3—22no—
82C32×C3⋊D3C4×D414yes—
83C23×C3⋊D3—92no—
84C22×D32—364no—
85C22×#157—92no—
86C22×#157—94no—
87C22×#157—92no—
88C22×#157—94no—
89C22×#157—94no—
90C22×#157—188no—
91C22×#157—364no—
92C2×C4×C3⋊D3—94no—
93C22⋊#157—92no—
94C22⋊#157—94no—
95C22⋊#157#15712yescomplemented
96C4×#157—188no—
97C32×C22⋊C4—92no—
98#3757—92no—
99#3752—94no—
100C22×C33—24no—
101C22×C33—84no—
102C2×C3⋊(C3⋊D3)—64no—
103C2×C3⋊(C3⋊D3)—64no—
104C2×C3⋊(C3⋊D3)—124no—
105C2×C3⋊(C3⋊D3)—124no—
106C2×C3⋊(C3⋊D3)—244no—
107C2×C3⋊(C3⋊D3)—244no—
108C2×C3⋊(C3⋊D3)—248no—
109C2×C3⋊(C3⋊D3)—248no—
110C2×C3×C3⋊D3—24no—
111C2×C3×C3⋊D3—24no—
112C2×C3×C3⋊D3—48no—
113C2×C32×D3—64no—
114C2×C32×D3—64no—
115C2×C32×D3—128no—
116D3×C3⋊D3—68no—
117D3×C3⋊D3—68no—
118D3×C3⋊D3—128no—
119D3×C3⋊D3—128no—
120D3×C3⋊D3—124no—
121D3×C3⋊D3—124no—
122C3⋊#157—128no—
123C3×#157—48no—
124D3×C22⋊C4—544no—
125C34C4×C22⋊C411yescharacteristic, complemented, Sylow
126C23×C32—92no—
127C22×C3⋊D3—92no—
128C22×C3⋊D3—92no—
129C22×C3⋊D3—92no—
130C22×C3⋊D3—92no—
131C22×C3⋊D3—92no—
132C22×C3⋊D3—184no—
133C22×C3⋊D3—184no—
134C22×C3⋊D3—364no—
135C22×C3⋊D3—364no—
136C22×C3⋊D3—728no—
137C22×C3⋊D3—724no—
138C22×C3⋊D3C2×#15712yes—
139C22×C3×D3—128no—
140C2×D32—364no—
141C2×D32—368no—
142C2×D32—364no—
143C2×D32—3616no—
144C2×D32—3616no—
145C2×#157—94no—
146C2×#157—98no—
147C2×#157—94no—
148C2×#157—94no—
149C2×#157—184no—
150C2×#157—184no—
151C2×#157—188no—
152C2×#157—188no—
153C2×#157—184no—
154C2×#157—364no—
155C2×#157—364no—
156C2×#157—3616no—
157C2×#157C2×#15714yescomplemented
158C2×C32×C4—94no—
159C2×C3⋊Dic3—94no—
160C4×C3⋊D3—184no—
161C4×C3⋊D3—184no—
162C4×C22⋊C4—811noSylow
163C2×C33—24no—
164C2×C33—44no—
165C2×C33—88no—
166C2×C33—84no—
167C3⋊(C3⋊D3)—64no—
168C3⋊(C3⋊D3)—64no—
169C3⋊(C3⋊D3)—124no—
170C3⋊(C3⋊D3)—248no—
171C3⋊(C3⋊D3)—244no—
172C3⋊(C3⋊D3)—244no—
173C3×C3⋊D3—28no—
174C3×C3⋊D3—44no—
175C32×D3—68no—
176C32×D3—124no—
177C23×D3—544no—
178C2×C4×D3—548no—
179C3×C22⋊C4—184no—
180#217—544no—
181(C2×D3)⋊C4—548no—
182C22×C32—44no—
183C22×C32—44no—
184C22×C32—88no—
185C22×C32—84no—
186C22×C32—92no—
187C22×C32—92no—
188C22×C32—184no—
189C22×C32C4×#15712yescomplemented
190C2×C3⋊D3—24no—
191C2×C3⋊D3—92no—
192C2×C3⋊D3—94no—
193C2×C3⋊D3—94no—
194C2×C3⋊D3—92no—
195C2×C3⋊D3—182no—
196C2×C3⋊D3—182no—
197C2×C3⋊D3—184no—
198C2×C3⋊D3—182no—
199C2×C3⋊D3—182no—
200C2×C3⋊D3—184no—
201C2×C3⋊D3—364no—
202C2×C3⋊D3—364no—
203C2×C3⋊D3—368no—
204C2×C3⋊D3—368no—
205C2×C3⋊D3—364no—
206C2×C3⋊D3—368no—
207C2×C3⋊D3—364no—
208C2×C3⋊D3—368no—
209C2×C3⋊D3—728no—
210C2×C3⋊D3—728no—
211C2×C3⋊D3—7216no—
212C2×C3⋊D3—7216no—
213C2×C3⋊D3—724no—
214C2×C3⋊D3—724no—
215C2×C3⋊D3—728no—
216C2×C3⋊D3—728no—
217C2×C3⋊D3C4×#15712yescomplemented
218C2×C3⋊D3C22×#15712yes—
219C2×C3×D3—128no—
220C2×C3×D3—128no—
221C2×C3×D3—1232no—
222D32—368no—
223D32—368no—
224D32—3632no—
225D32—368no—
226D32—368no—
227#157—24no—
228#157—98no—
229#157—184no—
230#157—184no—
231#157—368no—
232#157—368no—
233C32×C4—184no—
234C3⋊Dic3—184no—
235C23×C4—811no—
236C2×C22⋊C4—812no—
237C2×C42—812no—
238#95—812no—
239C33—24no—
240C33—84no—
241C23×C3—184no—
242C22×D3—64no—
243C22×D3—544no—
244C22×D3—544no—
245C22×D3—544no—
246C22×D3—544no—
247C22×D3—544no—
248C22×D3—1088no—
249C22×D3—1088no—
250C22×D3—2164no—
251C2×C3×C4—188no—
252C2×Dic3—548no—
253C4×D3—1088no—
254C4×D3—1088no—
255C2×C32—22no—
256C2×C32—44no—
257C2×C32—48no—
258C2×C32—48no—
259C2×C32—44no—
260C2×C32—88no—
261C2×C32—816no—
262C2×C32—88no—
263C2×C32—84no—
264C2×C32—94no—
265C2×C32—182no—
266C2×C32C2×C4×#15712yes—
267C3⋊D3—22no—
268C3⋊D3—92no—
269C3⋊D3—92no—
270C3⋊D3—182no—
271C3⋊D3—364no—
272C3⋊D3—364no—
273C3⋊D3—368no—
274C3⋊D3—364no—
275C3⋊D3—364no—
276C3⋊D3—368no—
277C3⋊D3—728no—
278C3⋊D3—728no—
279C3⋊D3—7216no—
280C3⋊D3—724no—
281C3⋊D3—724no—
282C3⋊D3—728no—
283C3⋊D3D4×#15714yes—
284C3×D3—1216no—
285C3×D3—1216no—
286C24—811no—
287C22×C4—814no—
288C22×C4—811no—
289C22×C4—812no—
290C22×C4—811no—
291C22×C4—812no—
292C22×C4—1624no—
293C22⋊C4—92no—
294C22⋊C4—814no—
295C22⋊C4—812no—
296C42—1624no—
297C22×C3—24no—
298C22×C3—84no—
299C22×C3—184no—
300C22×C3—184no—
301C22×C3—368no—
302C2×D3—64no—
303C2×D3—64no—
304C2×D3—128no—
305C2×D3—548no—
306C2×D3—548no—
307C2×D3—544no—
308C2×D3—544no—
309C2×D3—1088no—
310C2×D3—1084no—
311C2×D3—1084no—
312C2×D3—1088no—
313C2×D3—1084no—
314C2×D3—1084no—
315C2×D3—2164no—
316C2×D3—2164no—
317C2×D3—2168no—
318C2×D3—2168no—
319C3×C4—368no—
320Dic3—1088no—
321C32—44no—
322C32—44no—
323C32—88no—
324C32—84no—
325C32C22⋊C4×#15712yescomplemented
326C23—92no—
327C23—811no—
328C23—812no—
329C23—811no—
330C23—811no—
331C23—1622no—
332C23—1622no—
333C2×C4—94no—
334C2×C4—814no—
335C2×C4—812no—
336C2×C4—814no—
337C2×C4—812no—
338C2×C4—1624no—
339C2×C4—1624no—
340C2×C4—1624no—
341C2×C4—1624no—
342C2×C4—1622no—
343C2×C3—24no—
344C2×C3—44no—
345C2×C3—88no—
346C2×C3—84no—
347C2×C3—188no—
348C2×C3—364no—
349D3—68no—
350D3—124no—
351D3—544no—
352D3—544no—
353D3—1084no—
354D3—2164no—
355D3—2164no—
356D3—2168no—
357C22—92no—
358C22—92no—
359C22—184no—
360C22—812no—
361C22—812no—
362C22—811no—
363C22—811no—
364C22—1621no—
365C22—1621no—
366C22—1624no—
367C22—1621no—
368C22—1621no—
369C22#157211yescharacteristic, complemented
370C4—184no—
371C4—814no—
372C4—1624no—
373C4—1622no—
374C3—24no—
375C3—84no—
376C2—21no—
377C2—94no—
378C2—182no—
379C2—811no—
380C2—811no—
381C2—1621no—
382C2C2×#157211yescharacteristic
subgroupG/H
center, FrattiniC2C2×#1572
commutator, socleC2×C34C2×C42
FittingC22×C34C42
prime radicalC22×C3⋊D32C22
cosocleC2×C3⋊D32C23
hypercenterC22#1572
hypocenterC34C4×C22⋊C4
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. #72220556C2semidirect
4. C2×#1572C2semidirect
6. C22×#66472C4semidirect
8. C2×#10620198C4semidirect
13. #13049596C4semidirect
14. C32×C22⋊#157C4semidirect
15. #13049396C4semidirect
27. C2×C3⋊(C3⋊#157)C2×C4semidirect
28. C2×C32×#157C2×C4semidirect
40. C22×C34C42semidirect
43. C2×C3⋊(C3⋊(C3⋊D3))C42semidirect
46. C2×C32×C3⋊D3C42semidirect
54. #66472C22⋊C4semidirect
95. C22⋊#157#157semidirect
125. C34C4×C22⋊C4semidirect
157. C2×#157C2×#157semidirect
189. C22×C32C4×#157semidirect
217. C2×C3⋊D3C4×#157semidirect
325. C32C22⋊C4×#157semidirect
369. C22#1572semidirect

Maximal subgroups

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

subgroupindex
#722205562
C3⋊D3×C22⋊#1572
C22×#106201982
C2×#15722
C22⋊C4×#1579