Finite Group Databasebeta

← C2×#72220228 #72252806

Subgroups — C2×#72220228 #72252806

Order 5184.

1078 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 1078 conjugacy classes of subgroups — too many to draw. Only its 73 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×#13049399C211yesmaximal, characteristic
2#72220228C214yesmaximal, direct product factor
3C2×#13049411C212yesmaximal
4C22×#66472—22no—
5C2×#10620150C2211yescharacteristic
6C2×C3⋊(C3⋊#622)C411yescharacteristic, complemented
7C2×C32×#622C411yescharacteristic, complemented
8#13049411—22no—
9#13049411C2214yes—
10#13049399C412yescomplemented
11#13049399C2212yes—
12C2×#88924—62no—
13C22×C3⋊(C3⋊(C3⋊D3))D411yescharacteristic
14C2×#66472—24no—
15C2×#66472—44no—
16C2×C32×C3⋊Dic3C2×C411yescharacteristic
17C2×C32×C3⋊Dic3Q811yescharacteristic
18#10620150C2×C411yescharacteristic
19#10620150D411yescharacteristic
20#10620150Q811yescharacteristic
21#10620150C2311yescharacteristic
22C3⋊(C3⋊#622)C2×C412yescomplemented
23C32×#622C2×C412yescomplemented
24C2×#63768—91nomaximal
25C2×#63745—91nomaximal
26C2×#6289—62no—
27C2×#6289—121no—
28C2×C3⋊#622—62no—
29C2×C3×#622—22no—
30#88924—64no—
31#88924—64no—
32C22×C34C4⋊C411yescharacteristic
33C2×C3⋊(C3⋊(C3⋊D3))C4⋊C411yescharacteristic
34C2×C3⋊(C3⋊(C3⋊D3))SD1614yes—
35C2×C3⋊(C3⋊(C3⋊D3))C2×D411yescharacteristic
36#66472—44no—
37C32×C3⋊Dic3C4⋊C411yescharacteristic
38C32×C3⋊Dic3C4⋊C411yescharacteristic
39C32×C3⋊Dic3C22×C411yescharacteristic
40C32×C3⋊Dic3C2×Q811yescharacteristic
41C2×C4⋊#157—92no—
42C2×C4⋊#157—92no—
43C2×#3787—91no—
44C2×C8×C3⋊D3—91no—
45#63768—94no—
46#63745—94no—
47C22×C3⋊(C3⋊D3)—62no—
48C22×C3⋊(C3⋊D3)—121no—
49C22×C3⋊(C3⋊D3)—244no—
50C2×C3×C3⋊Dic3—22no—
51C2×C3×C3⋊Dic3—41no—
52C2×C32×Dic3—62no—
53C2×C32×Dic3—121no—
54#6289—62no—
55#6289—62no—
56#6289—62no—
57#6289—62no—
58#6289—121no—
59#6289—121no—
60#6289—121no—
61#6289—121no—
62C3⋊#622—64no—
63C3×#622—24no—
64C2×C34#10612yescomplemented
65C2×C34C2×C4⋊C411yescharacteristic
66C3⋊(C3⋊(C3⋊D3))#10612yescomplemented
67C3⋊(C3⋊(C3⋊D3))C2×SD1612yes—
68C22×#157—182no—
69C22×#157—182no—
70C22×#157—368no—
71C2×C4×C3⋊D3—91no—
72C2×C4×C3⋊D3—91no—
73C2×#624—364no—
74C2×#622—91no—
75C2×#622#15711yescharacteristic, complemented
76C4⋊#157—94no—
77C4⋊#157—94no—
78C4⋊#157—182no—
79C4⋊#157—182no—
80#3787—92no—
81#3787—92no—
82C2×C32×C8—91no—
83C2×C3⋊(C3⋊C8)—91no—
84C8×C3⋊D3—92no—
85C8×C3⋊D3—92no—
86C22×C33—22no—
87C22×C33—41no—
88C22×C33—84no—
89C2×C3⋊(C3⋊D3)—62no—
90C2×C3⋊(C3⋊D3)—62no—
91C2×C3⋊(C3⋊D3)—68no—
92C2×C3⋊(C3⋊D3)—121no—
93C2×C3⋊(C3⋊D3)—121no—
94C2×C3⋊(C3⋊D3)—124no—
95C2×C3⋊(C3⋊D3)—244no—
96C2×C3⋊(C3⋊D3)—2416no—
97C2×C3⋊(C3⋊D3)—244no—
98C3×C3⋊Dic3—22no—
99C3×C3⋊Dic3—22no—
100C3×C3⋊Dic3—41no—
101C3×C3⋊Dic3—41no—
102C32×Dic3—62no—
103C32×Dic3—62no—
104C32×Dic3—121no—
105C32×Dic3—121no—
106C2×D3×C8—542no—
107C34C2×#10611yescharacteristic, complemented, Sylow
108C22×C3⋊D3—91no—
109C22×C3⋊D3—91no—
110C22×C3⋊D3—364no—
111C22×C3⋊D3—368no—
112C22×C3⋊D3—724no—
113C22×C3⋊D3—724no—
114C22×C3⋊D3—722no—
115C2×#157—184no—
116C2×#157—184no—
117C2×#157—364no—
118C2×#157—364no—
119C2×#157—3632no—
120C2×#157—3616no—
121C2×C32×C4—91no—
122C2×C32×C4—91no—
123C2×C3⋊Dic3PSU(3,𝔽2)11yescharacteristic
124C2×C3⋊Dic3C2×#15711yescharacteristic
125C4×C3⋊D3—91no—
126C4×C3⋊D3—91no—
127C4×C3⋊D3—91no—
128C4×C3⋊D3—91no—
129C4×C3⋊D3—91no—
130C4×C3⋊D3—91no—
131C4×C3⋊D3—91no—
132C4×C3⋊D3—91no—
133C2×C3×Dic3—124no—
134C2×C3×Dic3—124no—
135#624—364no—
136#624—364no—
137#624—364no—
138#624—364no—
139#622—92no—
140#622C2×#15712yescomplemented
141C32×C8—92no—
142C3⋊(C3⋊C8)—92no—
143C2×#106—811noSylow
144C2×C33—24no—
145C2×C33—22no—
146C2×C33—42no—
147C2×C33—41no—
148C2×C33—88no—
149C2×C33—84no—
150C3⋊(C3⋊D3)—64no—
151C3⋊(C3⋊D3)—64no—
152C3⋊(C3⋊D3)—122no—
153C3⋊(C3⋊D3)—122no—
154C3⋊(C3⋊D3)—248no—
155C3⋊(C3⋊D3)—248no—
156C2×C4×D3—542no—
157C2×C4×D3—1081no—
158C2×C3×C8—182no—
159C2×C3⋊C8—542no—
160D3×C8—544no—
161D3×C8—544no—
162C22×C32—44no—
163C22×C32—48no—
164C22×C32—84no—
165C22×C32—84no—
166C22×C32—82no—
167C22×C32#377711yescharacteristic
168C22×C32C4⋊#15711yescharacteristic
169C2×C3⋊D3—94no—
170C2×C3⋊D3—91no—
171C2×C3⋊D3—91no—
172C2×C3⋊D3—94no—
173C2×C3⋊D3—91no—
174C2×C3⋊D3—91no—
175C2×C3⋊D3—364no—
176C2×C3⋊D3—364no—
177C2×C3⋊D3—3616no—
178C2×C3⋊D3—3616no—
179C2×C3⋊D3—368no—
180C2×C3⋊D3—368no—
181C2×C3⋊D3—3616no—
182C2×C3⋊D3—724no—
183C2×C3⋊D3—724no—
184C2×C3⋊D3—7216no—
185C2×C3⋊D3—724no—
186C2×C3⋊D3—724no—
187C2×C3⋊D3—7216no—
188C2×C3⋊D3—722no—
189C2×C3⋊D3—728no—
190C2×C3⋊D3—722no—
191#157—364no—
192#157—364no—
193#157—3632no—
194C32×C4—91no—
195C32×C4—91no—
196C32×C4—91no—
197C32×C4—91no—
198C3⋊Dic3#377711yescharacteristic
199C3⋊Dic3C4⋊#15711yescharacteristic
200C3⋊Dic3C2×PSU(3,𝔽2)11yescharacteristic
201C3⋊Dic3C22×#15711yescharacteristic
202C3×Dic3—124no—
203C3×Dic3—124no—
204C3×Dic3—124no—
205C3×Dic3—124no—
206C22×C8—811no—
207C2×C4⋊C4—812no—
208#106—814no—
209C33—22no—
210C33—41no—
211C33—84no—
212C22×D3—542no—
213C22×D3—1081no—
214C22×D3—2164no—
215C2×C3×C4—182no—
216C2×C3×C4—361no—
217C2×Dic3—62no—
218C2×Dic3—121no—
219C4×D3—542no—
220C4×D3—542no—
221C4×D3—542no—
222C4×D3—542no—
223C4×D3—1081no—
224C4×D3—1081no—
225C4×D3—1081no—
226C4×D3—1081no—
227C3×C8—184no—
228C3⋊C8—544no—
229C2×C32—44no—
230C2×C32—48no—
231C2×C32—416no—
232C2×C32—48no—
233C2×C32—88no—
234C2×C32—84no—
235C2×C32—88no—
236C2×C32—84no—
237C2×C32—84no—
238C2×C32—82no—
239C2×C32#6374512yescomplemented
240C2×C32#6376812yescomplemented
241C2×C32C2×#377711yescharacteristic
242C2×C32C2×C4⋊#15711yescharacteristic
243C3⋊D3—92no—
244C3⋊D3—92no—
245C3⋊D3—92no—
246C3⋊D3—92no—
247C3⋊D3—368no—
248C3⋊D3—368no—
249C3⋊D3—368no—
250C3⋊D3—3616no—
251C3⋊D3—368no—
252C3⋊D3—728no—
253C3⋊D3—728no—
254C3⋊D3—728no—
255C3⋊D3—728no—
256C3⋊D3—724no—
257C3⋊D3—724no—
258C22×C4—811no—
259C22×C4—1622no—
260C2×C8—91no—
261C2×C8—812no—
262C2×C8—812no—
263C2×C8—811no—
264C4⋊C4—814no—
265C4⋊C4—1622no—
266C22×C3—22no—
267C22×C3—41no—
268C22×C3—84no—
269C2×D3—542no—
270C2×D3—542no—
271C2×D3—548no—
272C2×D3—1081no—
273C2×D3—1081no—
274C2×D3—1084no—
275C2×D3—21616no—
276C2×D3—2164no—
277C2×D3—2164no—
278C3×C4—182no—
279C3×C4—182no—
280C3×C4—361no—
281C3×C4—361no—
282Dic3—62no—
283Dic3—62no—
284Dic3—121no—
285Dic3—121no—
286C32—44no—
287C32—48no—
288C32—84no—
289C32—84no—
290C32—82no—
291C32C2×#6374511yescharacteristic, complemented
292C32C2×#6376811yescharacteristic, complemented
293C23—811no—
294C2×C4—91no—
295C2×C4—91no—
296C2×C4—3244no—
297C2×C4—811no—
298C2×C4—811no—
299C2×C4—811no—
300C2×C4—811no—
301C2×C4—1624no—
302C8—92no—
303C8—812no—
304C2×C3—24no—
305C2×C3—22no—
306C2×C3—42no—
307C2×C3—41no—
308C2×C3—88no—
309C2×C3—84no—
310D3—544no—
311D3—544no—
312D3—1082no—
313D3—1082no—
314D3—2168no—
315D3—2168no—
316C22—814no—
317C22—811no—
318C22—811no—
319C22#1304978311yescharacteristic
320C4—91no—
321C4—91no—
322C4—91no—
323C4—91no—
324C4—3244no—
325C3—22no—
326C3—41no—
327C3—84no—
328C2—812no—
329C2—812no—
330C2#7222022812yesdirect product factor
331C2C2×#1304978311yescharacteristic
subgroupG/H
center, hypercenterC22#13049783
commutatorC32×C3⋊Dic3C22×C4
FrattiniC2C2×#13049783
Fitting, socleC22×C34C4⋊C4
prime radicalC22×C3⋊(C3⋊(C3⋊D3))D4
cosocle#10620150C23
hypocenterC34C2×#106
LCSG ▷ C32×C3⋊Dic3 ▷ C2×C34 ▷ C34
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. #72220228C2direct (N × H)
6. C2×C3⋊(C3⋊#622)C4semidirect
7. C2×C32×#622C4semidirect
10. #13049399C4semidirect
22. C3⋊(C3⋊#622)C2×C4semidirect
23. C32×#622C2×C4semidirect
64. C2×C34#106semidirect
66. C3⋊(C3⋊(C3⋊D3))#106semidirect
75. C2×#622#157semidirect
107. C34C2×#106semidirect
140. #622C2×#157semidirect
239. C2×C32#63745semidirect
240. C2×C32#63768semidirect
291. C32C2×#63745semidirect
292. C32C2×#63768semidirect
330. C2#72220228direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#130493992
#722202282
C2×#130494112
C2×#637689
C2×#637459