Finite Group Databasebeta

← C2×#11720705 #13146674

Subgroups — C2×#11720705 #13146674

Order 1728.

631 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 631 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C22×#6198C211yesmaximal, characteristic
2C22×#6188C211yesmaximal, characteristic, complemented
3C2×#88449C211yesmaximal, characteristic, complemented
4#11720705C214yesmaximal, direct product factor
5C23×(C3⋊D3)⋊C3—21no—
6C22×#1084C2211yescharacteristic
7C22×(C3⋊Dic3)⋊C3—21no—
8C2×#6198—21no—
9C2×#6198—22no—
10C2×#6198C414yescomplemented
11C2×#6198C2212yes—
12C2×#6188—22no—
13C2×#6188C2212yescomplemented
14#88449—21no—
15#88449C2212yescomplemented
16C22×#626—61no—
17C2×Dic3⋊Dic3—61no—
18C23×H(3,𝔽3)D411yescharacteristic
19C22×(C3⋊D3)⋊C3—24no—
20C22×(C3⋊D3)⋊C3—22no—
21C22×(C3⋊D3)⋊C3—42no—
22C22×(C3⋊D3)⋊C3—42no—
23C2×#1084C2×C412yes—
24C2×#1084D411yescharacteristic
25C2×#1084D412yes—
26C2×#1084C2311yescharacteristic
27C2×(C3⋊Dic3)⋊C3—22no—
28C2×(C3⋊Dic3)⋊C3—42no—
29#6198—24no—
30#6198—22no—
31#6198C2×C414yescomplemented
32#6188—22no—
33C2×#821—91nomaximal
34C23×C3⋊D3—21no—
35C23×C3×D3—61no—
36C22×C3⋊Dic3—21no—
37C22×C3×Dic3—61no—
38C22×C3×Dic3—61no—
39C22×C3×Dic3—61no—
40C2×#626—64no—
41C2×#626—62no—
42C2×#626—62no—
43C2×#626—64no—
44Dic3⋊Dic3—62no—
45Dic3⋊Dic3—62no—
46C22×H(3,𝔽3)C22⋊C412yes—
47C22×H(3,𝔽3)D812yes—
48C22×H(3,𝔽3)SD1612yes—
49C22×H(3,𝔽3)C2×D411yescharacteristic
50C2×(C3⋊D3)⋊C3—24no—
51C2×(C3⋊D3)⋊C3—48no—
52C2×(C3⋊D3)⋊C3—42no—
53C2×(C3⋊D3)⋊C3—42no—
54#1084C22⋊C412yes—
55#1084C22×C411yescharacteristic
56#1084C2×D411yescharacteristic
57(C3⋊Dic3)⋊C3—42no—
58C22×#67—181no—
59C22×D12—91no—
60C22×C3×C8—91no—
61C2×C4⋊Dic3—91no—
62C2×Dic3⋊C4—181no—
63#821—94no—
64C23×C32—21no—
65C23×C32—21no—
66C22×C3⋊D3—22no—
67C22×C3⋊D3—24no—
68C22×C3⋊D3—42no—
69C22×C3⋊D3—42no—
70C22×C3×D3—62no—
71C22×C3×D3—64no—
72C22×C3×D3—122no—
73C22×C3×D3—122no—
74C2×C3⋊Dic3—22no—
75C2×C3⋊Dic3—42no—
76C2×C3×Dic3—62no—
77C2×C3×Dic3—62no—
78C2×C3×Dic3—61no—
79C2×C3×Dic3—61no—
80C2×C3×Dic3—62no—
81C2×C3×Dic3—62no—
82C2×C3×Dic3—61no—
83C2×C3×Dic3—61no—
84C2×C3×Dic3—62no—
85C2×C3×Dic3—122no—
86#626—64no—
87#626—68no—
88#626—64no—
89C2×#102—271noSylow
90C2×H(3,𝔽3)#10214yescomplemented
91C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
92C2×H(3,𝔽3)C2×D811yescharacteristic
93C2×H(3,𝔽3)C2×SD1611yescharacteristic
94(C3⋊D3)⋊C3—44no—
95C24×C3—181no—
96C23×D3—61no—
97C23×D3—181no—
98C22×C3×C4—91no—
99C22×C3×C4—181no—
100C2×#67—182no—
101C2×#67—184no—
102C2×#67—184no—
103C2×#67—182no—
104C22×Dic3—61no—
105C22×Dic3—61no—
106C22×Dic3—61no—
107C22×Dic3—181no—
108C2×D12—92no—
109C2×D12—94no—
110C2×D12—181no—
111C2×D12—182no—
112C2×C3×C8—92no—
113C2×C3×C8—182no—
114C4⋊Dic3—92no—
115C4⋊Dic3—181no—
116Dic3⋊C4—182no—
117Dic3⋊C4—182no—
118C22×C32—21no—
119C22×C32—21no—
120C22×C32—22no—
121C22×C32—22no—
122C22×C32—22no—
123C22×C32—22no—
124C22×C32—22no—
125C22×C32—22no—
126C2×C3⋊D3—24no—
127C2×C3⋊D3—42no—
128C2×C3⋊D3—42no—
129C2×C3⋊D3—48no—
130C2×C3×D3—64no—
131C2×C3×D3—122no—
132C2×C3×D3—122no—
133C2×C3×D3—128no—
134C3⋊Dic3—42no—
135C3×Dic3—62no—
136C3×Dic3—62no—
137C3×Dic3—61no—
138C3×Dic3—61no—
139C3×Dic3—61no—
140C3×Dic3—61no—
141C3×Dic3—122no—
142C22×D4—271no—
143C22×C8—91no—
144C2×C4⋊C4—271no—
145#102—274no—
146H(3,𝔽3)C2×#10211yescharacteristic, complemented, Sylow
147C23×C3—61no—
148C23×C3—61no—
149C23×C3—184no—
150C23×C3—182no—
151C23×C3—362no—
152C23×C3—362no—
153C23×C3GO+(4,𝔽2)11yescharacteristic
154C22×D3—62no—
155C22×D3—64no—
156C22×D3—122no—
157C22×D3—122no—
158C22×D3—184no—
159C22×D3—182no—
160C22×D3—362no—
161C22×D3—362no—
162C2×C3×C4—91no—
163C2×C3×C4—91no—
164C2×C3×C4—92no—
165C2×C3×C4—92no—
166C2×C3×C4—182no—
167C2×C3×C4—362no—
168#67—184no—
169#67—184no—
170#67—188no—
171C2×Dic3—61no—
172C2×Dic3—61no—
173C2×Dic3—62no—
174C2×Dic3—62no—
175C2×Dic3—62no—
176C2×Dic3—62no—
177C2×Dic3—61no—
178C2×Dic3—61no—
179C2×Dic3—62no—
180C2×Dic3—122no—
181C2×Dic3—182no—
182C2×Dic3—362no—
183D12—94no—
184D12—182no—
185D12—184no—
186C3×C8—182no—
187C2×C32—21no—
188C2×C32—21no—
189C2×C32—21no—
190C2×C32—21no—
191C2×C32—21no—
192C2×C32—21no—
193C2×C32—24no—
194C2×C32—24no—
195C3⋊D3—44no—
196C3×D3—124no—
197C24—181no—
198C2×D4—272no—
199C2×D4—274no—
200C2×D4—542no—
201C2×D4—541no—
202C22×C4—91no—
203C22×C4—181no—
204C2×C8—92no—
205C2×C8—182no—
206C4⋊C4—272no—
207C4⋊C4—541no—
208C22×C3—62no—
209C22×C3—62no—
210C22×C3—62no—
211C22×C3—62no—
212C22×C3—62no—
213C22×C3—62no—
214C22×C3—61no—
215C22×C3—61no—
216C22×C3—184no—
217C22×C3—368no—
218C22×C3—362no—
219C22×C3—362no—
220C22×C3D32⋊C412yes—
221C22×C3#377412yes—
222C22×C3#377512yes—
223C22×C3C2×GO+(4,𝔽2)11yescharacteristic
224C2×D3—64no—
225C2×D3—122no—
226C2×D3—122no—
227C2×D3—128no—
228C2×D3—184no—
229C2×D3—368no—
230C2×D3—362no—
231C2×D3—362no—
232C3×C4—91no—
233C3×C4—91no—
234C3×C4—92no—
235C3×C4—362no—
236Dic3—61no—
237Dic3—61no—
238Dic3—62no—
239Dic3—61no—
240Dic3—61no—
241Dic3—62no—
242Dic3—122no—
243Dic3—362no—
244C32—21no—
245C32—21no—
246C23—184no—
247C23—182no—
248C23—362no—
249C23—362no—
250C23#1088⋊C211yescharacteristic
251D4—274no—
252D4—542no—
253D4—544no—
254C2×C4—92no—
255C2×C4—92no—
256C2×C4—91no—
257C2×C4—91no—
258C2×C4—182no—
259C2×C4—362no—
260C8—182no—
261C2×C3—64no—
262C2×C3—64no—
263C2×C3—61no—
264C2×C3—61no—
265C2×C3—61no—
266C2×C3—61no—
267C2×C3—61no—
268C2×C3—61no—
269C2×C3—364no—
270C2×C3#6373914yes—
271C2×C3C2×D32⋊C411yescharacteristic
272C2×C3C2×#377411yescharacteristic
273C2×C3C2×#377511yescharacteristic
274D3—124no—
275D3—364no—
276C22—184no—
277C22—368no—
278C22—362no—
279C22—362no—
280C22#8858712yes—
281C22#8858812yes—
282C22(((C3⋊D3)⋊C3)⋊C2)⋊C412yes—
283C22C2×#1088⋊C211yescharacteristic
284C4—91no—
285C4—91no—
286C4—92no—
287C4—362no—
288C3—61no—
289C3—61no—
290C3C2×#6373911yescharacteristic
291C2—364no—
292C2#1172070514yesdirect product factor
293C2C2×#8858711yescharacteristic
294C2C2×#8858811yescharacteristic
295C2C2×(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
subgroupG/H
center, hypercenterC23#1088⋊C2
commutator#1084C22×C4
FrattiniC22×C3C2×GO+(4,𝔽2)
FittingC23×H(3,𝔽3)D4
socleC23×C3GO+(4,𝔽2)
prime radicalC22×#6198C2
cosocleC2×#1084C23
hypocenterH(3,𝔽3)C2×#102
LCSG ▷ #1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×#6188C2semidirect
3. C2×#88449C2semidirect
4. #11720705C2direct (N × H)
10. C2×#6198C4semidirect
13. C2×#6188C22semidirect
15. #88449C22semidirect
31. #6198C2×C4semidirect
90. C2×H(3,𝔽3)#102semidirect
146. H(3,𝔽3)C2×#102semidirect
292. C2#11720705direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#61982
C22×#61882
C2×#884492
#117207052
C2×#8219