Finite Group Databasebeta

← C2×#11722712 #13166383

Subgroups — C2×#11722712 #13166383

Order 1728.

776 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 82 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×#89018C211yesmaximal, characteristic
2C2×C4×C3⋊#157C211yesmaximal, characteristic, complemented
3C2×C3×#3788C211yesmaximal, characteristic, complemented
4#11722712C214yesmaximal, direct product factor
5C2×#63785—31nomaximal
6C22×C3⋊#157—21no—
7C2×C4⋊(C3⋊(C3⋊D3))C411yescharacteristic, complemented
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C32×Dic6C411yescharacteristic, complemented
10C2×#6292—21no—
11C2×#6289—21no—
12#89018—21no—
13#89018—21no—
14#89018C412yescomplemented
15#89018C2212yes—
16C4×C3⋊#157—21no—
17C4×C3⋊#157C2212yescomplemented
18C2×C3×#622—21no—
19C3×#3788—21no—
20C3×#3788C2212yescomplemented
21C2×#3832—31no—
22C2×#3799—62no—
23C2×C4×#157—31no—
24C2×#3788D311yescharacteristic, complemented
25#63785—34no—
26C22×C3⋊(C3⋊D3)—21no—
27C22×C3×C3⋊D3D411yescharacteristic
28C2×C3⋊#157—24no—
29C2×C3⋊#157—22no—
30C2×C33×C4C2×C411yescharacteristic
31C4⋊(C3⋊(C3⋊D3))—21no—
32C4⋊(C3⋊(C3⋊D3))C2×C412yescomplemented
33C2×C3×C3⋊Dic3D411yescharacteristic
34C3×C4×C3⋊D3C2×C411yescharacteristic
35C3×C4×C3⋊D3D411yescharacteristic
36C3×C4×C3⋊D3D411yescharacteristic
37C3×C4×C3⋊D3C2311yescharacteristic
38C2×C32×Dic3—21no—
39C32×Dic6—21no—
40C32×Dic6C2×C412yescomplemented
41#6292—22no—
42#6292—22no—
43#6289—22no—
44#6289—22no—
45C3×#622—22no—
46C2×#825—91nomaximal
47C22×#157—61no—
48C2×C32×Q8—31no—
49#3832—32no—
50#3832—32no—
51#3832—61no—
52#3832—61no—
53C2×C4⋊(C3⋊D3)—31no—
54C2×C4⋊(C3⋊D3)—62no—
55C2×C4⋊(C3⋊D3)—61no—
56C2×C4⋊(C3⋊D3)—122no—
57C2×C4×C3⋊D3—61no—
58C2×C4×C3⋊D3C2×D311yescharacteristic
59C2×C3×C4×D3—62no—
60C2×C3×Dic6—22no—
61C2×#626—64no—
62C2×#624—64no—
63#3799—64no—
64#3799—64no—
65#3799—64no—
66#3799—64no—
67C2×#622—21no—
68C4×#157—32no—
69C4×#157—61no—
70#3788—21no—
71#3788C2×D312yescomplemented
72C22×C33C22⋊C411yescharacteristic
73C2×C3⋊(C3⋊D3)—22no—
74C2×C3⋊(C3⋊D3)—42no—
75C2×C3×C3⋊D3—22no—
76C2×C3×C3⋊D3C22⋊C411yescharacteristic
77C2×C3×C3⋊D3C2×D411yescharacteristic
78C3⋊#157—24no—
79C33×C4C22⋊C411yescharacteristic
80C33×C4C22×C411yescharacteristic
81C3×C3⋊Dic3C22⋊C411yescharacteristic
82C3×C3⋊Dic3C2×D411yescharacteristic
83C32×Dic3—22no—
84C2×#239—182no—
85C2×#235—91no—
86C2×C3×M16—91no—
87C2×C4×Dic3—91no—
88#825—94no—
89C22×C3⋊D3—61no—
90C22×C3⋊D3—64no—
91C22×C3⋊D3—124no—
92C22×C3⋊D3D1211yescharacteristic
93C22×C3×D3—62no—
94C2×#157—64no—
95C2×#157—62no—
96C32×Q8—32no—
97C32×Q8—61no—
98C2×C32×C4—22no—
99C2×C32×C4—42no—
100C2×C32×C4—61no—
101C2×C32×C4C4×D311yescharacteristic
102C2×C3⋊Dic3#6711yescharacteristic
103C4⋊(C3⋊D3)—32no—
104C4⋊(C3⋊D3)—64no—
105C4⋊(C3⋊D3)—62no—
106C4⋊(C3⋊D3)—62no—
107C4⋊(C3⋊D3)—61no—
108C4⋊(C3⋊D3)—64no—
109C4⋊(C3⋊D3)—124no—
110C4⋊(C3⋊D3)—124no—
111C4×C3⋊D3—62no—
112C4×C3⋊D3—62no—
113C4×C3⋊D3C4×D311yescharacteristic
114C4×C3⋊D3D1211yescharacteristic
115C4×C3⋊D3#6711yescharacteristic
116C4×C3⋊D3C22×D311yescharacteristic
117C2×C3×Dic3—24no—
118C2×C3×Dic3—62no—
119C3×C4×D3—62no—
120C3×C4×D3—62no—
121C3×C4×D3—62no—
122C3×C4×D3—62no—
123C3×Dic6—24no—
124C3×Dic6—24no—
125#626—68no—
126#626—68no—
127#624—68no—
128#624—68no—
129#622—22no—
130C2×#104—271noSylow
131C2×C33#10412yescomplemented
132C2×C33C2×C22⋊C411yescharacteristic
133C3⋊(C3⋊D3)—42no—
134C3×C3⋊D3—21no—
135C3×C3⋊D3—21no—
136C2×C3×Q8—62no—
137C22×C3×C4—91no—
138C2×#67—181no—
139C22×Dic3—181no—
140#239—184no—
141#239—184no—
142#239—184no—
143#239—184no—
144#235—92no—
145#235—92no—
146#235—181no—
147#235—181no—
148C2×D12—91no—
149C2×D12—182no—
150C2×D12—184no—
151C2×D12—362no—
152C2×C4×D3—62no—
153C2×C4×D3—181no—
154C2×C4×D3—184no—
155C2×Dic6#15711yescharacteristic, complemented
156C3×M16—92no—
157C3×M16—181no—
158C2×C3×C8—181no—
159C4×Dic3—92no—
160C4×Dic3—181no—
161C22×C32—22no—
162C22×C32—42no—
163C22×C32(C2×D3)⋊C411yescharacteristic
164C2×C3⋊D3—22no—
165C2×C3⋊D3—62no—
166C2×C3⋊D3—68no—
167C2×C3⋊D3—128no—
168C2×C3⋊D3—122no—
169C2×C3⋊D3—128no—
170C2×C3⋊D3—248no—
171C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
172C2×C3⋊D3C2×D1211yescharacteristic
173C2×C3×D3—62no—
174C2×C3×D3—62no—
175C2×C3×D3—124no—
176#157—64no—
177C32×C4—22no—
178C32×C4—22no—
179C32×C4—42no—
180C32×C4—42no—
181C32×C4—62no—
182C32×C4(C2×D3)⋊C411yescharacteristic
183C32×C4C2×C4×D311yescharacteristic
184C3⋊Dic3(C2×D3)⋊C411yescharacteristic
185C3⋊Dic3C2×#6711yescharacteristic
186C3×Dic3—28no—
187C3×Dic3—62no—
188C3×Dic3—62no—
189C2×#41—271no—
190C2×M16—91no—
191C2×C42—271no—
192#104—274no—
193C33C2×#10411yescharacteristic, complemented, Sylow
194C23×C3—91no—
195C22×D3—62no—
196C22×D3—184no—
197C22×D3—181no—
198C22×D3—364no—
199C3×Q8—64no—
200C3×Q8—64no—
201C2×C3×C4—22no—
202C2×C3×C4—42no—
203C2×C3×C4—64no—
204C2×C3×C4—91no—
205C2×C3×C4—91no—
206C2×C3×C4—91no—
207C2×C3×C4—91no—
208C2×C3×C4—91no—
209C2×C3×C4C2×#15711yescharacteristic
210#67—182no—
211#67—182no—
212C2×Dic3—21no—
213C2×Dic3—62no—
214C2×Dic3—182no—
215C2×Dic3—184no—
216D12—92no—
217D12—184no—
218D12—184no—
219D12—188no—
220D12—188no—
221D12—181no—
222D12—364no—
223D12—364no—
224C4×D3—62no—
225C4×D3—62no—
226C4×D3—62no—
227C4×D3—62no—
228C4×D3—182no—
229C4×D3—182no—
230C4×D3—188no—
231C4×D3—188no—
232Dic6—21no—
233Dic6C2×#15712yescomplemented
234C3×C8—182no—
235C2×C32—22no—
236C2×C32—24no—
237C2×C32—44no—
238C2×C32—42no—
239C2×C32#82512yescomplemented
240C2×C32C2×(C2×D3)⋊C411yescharacteristic
241C3⋊D3—21no—
242C3⋊D3—21no—
243C3⋊D3—122no—
244C3⋊D3—128no—
245C3⋊D3—248no—
246C3×D3—122no—
247C3×D3—122no—
248#41—272no—
249#41—272no—
250#41—541no—
251#41—541no—
252C2×Q8—31no—
253C2×D4—271no—
254C2×D4—541no—
255C22×C4—91no—
256C22×C4—541no—
257C22×C4—541no—
258M16—92no—
259M16—181no—
260C2×C8—181no—
261C42—272no—
262C42—541no—
263C22×C3—22no—
264C22×C3—42no—
265C22×C3—91no—
266C22×C3—91no—
267C22×C3—182no—
268C22×C3C22⋊#15711yescharacteristic
269C2×D3—62no—
270C2×D3—62no—
271C2×D3—124no—
272C2×D3—188no—
273C2×D3—182no—
274C2×D3—368no—
275C2×D3—368no—
276C2×D3—362no—
277C2×D3—728no—
278C3×C4—22no—
279C3×C4—22no—
280C3×C4—42no—
281C3×C4—42no—
282C3×C4—68no—
283C3×C4—91no—
284C3×C4—91no—
285C3×C4C22⋊#15711yescharacteristic
286C3×C4C22×#15711yescharacteristic
287Dic3—22no—
288Dic3—62no—
289Dic3—62no—
290Dic3—184no—
291C32—22no—
292C32—42no—
293C32C2×#82511yescharacteristic, complemented
294C23—91no—
295C23—541no—
296Q8—32no—
297Q8—61no—
298D4—272no—
299D4—542no—
300D4—542no—
301D4—541no—
302C2×C4—61no—
303C2×C4—91no—
304C2×C4—91no—
305C2×C4—91no—
306C2×C4—91no—
307C2×C4—91no—
308C2×C4—542no—
309C2×C4—542no—
310C2×C4—542no—
311C2×C4—544no—
312C2×C4D3×#15711yescharacteristic
313C8—182no—
314C2×C3—24no—
315C2×C3—22no—
316C2×C3—44no—
317C2×C3—42no—
318C2×C3—181no—
319C2×C3—181no—
320C2×C3#6378512yescomplemented
321C2×C3C2×C22⋊#15711yescharacteristic
322D3—122no—
323D3—122no—
324D3—368no—
325D3—362no—
326D3—728no—
327C22—91no—
328C22—91no—
329C22—182no—
330C22—542no—
331C22—1082no—
332C22(C2×D3)⋊#15711yescharacteristic
333C4—62no—
334C4—91no—
335C4—91no—
336C4—544no—
337C4(C2×D3)⋊#15711yescharacteristic
338C4C2×D3×#15711yescharacteristic
339C3—22no—
340C3—42no—
341C3C2×#6378511yescharacteristic, complemented
342C2—181no—
343C2—181no—
344C2—1082no—
345C2#1172271212yesdirect product factor
346C2C2×(C2×D3)⋊#15711yescharacteristic
subgroupG/H
centerC22(C2×D3)⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC2×C33×C4C2×C4
socleC22×C33C22⋊C4
prime radicalC2×#89018C2
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33C2×#104
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C3⋊#157C2semidirect
3. C2×C3×#3788C2semidirect
4. #11722712C2direct (N × H)
7. C2×C4⋊(C3⋊(C3⋊D3))C4semidirect
9. C2×C32×Dic6C4semidirect
14. #89018C4semidirect
17. C4×C3⋊#157C22semidirect
20. C3×#3788C22semidirect
24. C2×#3788D3semidirect
32. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
40. C32×Dic6C2×C4semidirect
71. #3788C2×D3semidirect
131. C2×C33#104semidirect
155. C2×Dic6#157semidirect
193. C33C2×#104semidirect
233. Dic6C2×#157semidirect
239. C2×C32#825semidirect
293. C32C2×#825semidirect
320. C2×C3#63785semidirect
341. C3C2×#63785semidirect
345. C2#11722712direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890182
C2×C4×C3⋊#1572
C2×C3×#37882
#117227122
C2×#637853
C2×#8259