Finite Group Databasebeta

← C2×#11722710 #13166379

Subgroups — C2×#11722710 #13166379

Order 1728.

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

normal subgroups only

This group has 744 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×#89014C211yesmaximal, characteristic
2C2×C4×C3⋊#157C211yesmaximal, characteristic, complemented
3C2×C3×#3788C211yesmaximal, characteristic, complemented
4#11722710C214yesmaximal, direct product factor
5C2×#63783—31nomaximal
6C22×C3⋊#157—21no—
7C2×#6308C411yescharacteristic, complemented
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C32×D12C411yescharacteristic, complemented
10C2×#6290—21no—
11C2×D3×C3⋊Dic3—21no—
12#89014—21no—
13#89014—21no—
14#89014C412yescomplemented
15#89014C2212yes—
16C4×C3⋊#157—21no—
17C4×C3⋊#157C2212yescomplemented
18C2×C3×#622—21no—
19C3×#3788—21no—
20C3×#3788C2212yescomplemented
21C2×#3830—31no—
22C2×#3795—62no—
23C2×C4×#157—31no—
24C2×#3788D311yescharacteristic, complemented
25#63783—34no—
26C22×C3×C3⋊D3D411yescharacteristic
27C22×C32×D3—21no—
28C2×C3⋊#157—24no—
29C2×C3⋊#157—22no—
30C2×C33×C4C2×C411yescharacteristic
31C2×C3⋊(C3⋊Dic3)—21no—
32#6308—21no—
33#6308C2×C412yescomplemented
34C2×C3×C3⋊Dic3D411yescharacteristic
35C3×C4×C3⋊D3C2×C411yescharacteristic
36C3×C4×C3⋊D3D411yescharacteristic
37C3×C4×C3⋊D3D411yescharacteristic
38C3×C4×C3⋊D3C2311yescharacteristic
39C32×D12—21no—
40C32×D12C2×C412yescomplemented
41#6290—22no—
42#6290—22no—
43D3×C3⋊Dic3—22no—
44D3×C3⋊Dic3—22no—
45C3×#622—22no—
46C2×#825—91nomaximal
47C22×#157—61no—
48C2×C32×D4—31no—
49C2×#638—61no—
50C22×C3⋊Dic3—61no—
51#3830—32no—
52#3830—32no—
53#3830—61no—
54#3830—61no—
55C2×C4×C3⋊D3C2×D311yescharacteristic
56C2×#634—31no—
57C2×#634—62no—
58C2×#634—122no—
59C2×C3×D12—22no—
60C2×C3×C4×D3—62no—
61C2×#625—64no—
62C2×D3×Dic3—64no—
63#3795—64no—
64#3795—64no—
65#3795—64no—
66#3795—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⋊D3C22⋊C411yescharacteristic
75C2×C3×C3⋊D3C2×D411yescharacteristic
76C2×C32×D3—22no—
77C2×C32×D3—42no—
78C3⋊#157—24no—
79C33×C4C22⋊C411yescharacteristic
80C33×C4C22×C411yescharacteristic
81C3⋊(C3⋊Dic3)—22no—
82C3×C3⋊Dic3C22⋊C411yescharacteristic
83C3×C3⋊Dic3C2×D411yescharacteristic
84C2×#237—182no—
85C2×#235—91no—
86C2×C3×M16—91no—
87C2×C4×Dic3—91no—
88#825—94no—
89C23×C32—61no—
90C22×C3⋊D3D1211yescharacteristic
91C22×C3×D3—24no—
92C22×C3×D3—62no—
93C2×#157—64no—
94C2×#157—62no—
95C32×D4—32no—
96C32×D4—61no—
97C2×C32×C4—22no—
98C2×C32×C4—42no—
99C2×C32×C4C4×D311yescharacteristic
100#638—62no—
101#638—62no—
102C2×C3⋊Dic3—61no—
103C2×C3⋊Dic3—62no—
104C2×C3⋊Dic3—62no—
105C2×C3⋊Dic3—64no—
106C2×C3⋊Dic3—124no—
107C2×C3⋊Dic3#6711yescharacteristic
108C4×C3⋊D3C4×D311yescharacteristic
109C4×C3⋊D3D1211yescharacteristic
110C4×C3⋊D3#6711yescharacteristic
111C4×C3⋊D3C22×D311yescharacteristic
112#634—32no—
113#634—64no—
114#634—61no—
115#634—64no—
116#634—124no—
117#634—124no—
118C2×C3×Dic3—62no—
119C3×D12—24no—
120C3×D12—24no—
121C3×C4×D3—62no—
122C3×C4×D3—62no—
123C3×C4×D3—62no—
124C3×C4×D3—62no—
125#625—68no—
126#625—68no—
127D3×Dic3—68no—
128D3×Dic3—68no—
129#622—22no—
130C2×#104—271noSylow
131C2×C33#10412yescomplemented
132C2×C33C2×C22⋊C411yescharacteristic
133C3×C3⋊D3—21no—
134C3×C3⋊D3—21no—
135C32×D3—42no—
136C2×C3×D4—62no—
137C22×C3×C4—91no—
138C2×#67—181no—
139C2×#67—184no—
140C22×Dic3—184no—
141C22×Dic3—181no—
142#237—184no—
143#237—184no—
144#237—184no—
145#237—184no—
146#235—92no—
147#235—92no—
148#235—181no—
149#235—181no—
150C2×D12#15711yescharacteristic, complemented
151C2×C4×D3—62no—
152C2×C4×D3—181no—
153C2×Dic6—91no—
154C2×Dic6—182no—
155C2×Dic6—362no—
156C3×M16—92no—
157C3×M16—181no—
158C2×C3×C8—181no—
159C4×Dic3—92no—
160C4×Dic3—181no—
161C22×C32—22no—
162C22×C32—42no—
163C22×C32—62no—
164C22×C32—122no—
165C22×C32(C2×D3)⋊C411yescharacteristic
166C2×C3⋊D3—22no—
167C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
168C2×C3⋊D3C2×D1211yescharacteristic
169C2×C3×D3—28no—
170C2×C3×D3—48no—
171C2×C3×D3—62no—
172C2×C3×D3—62no—
173C2×C3×D3—124no—
174#157—64no—
175C32×C4—22no—
176C32×C4—22no—
177C32×C4—42no—
178C32×C4—42no—
179C32×C4(C2×D3)⋊C411yescharacteristic
180C32×C4C2×C4×D311yescharacteristic
181C3⋊Dic3—62no—
182C3⋊Dic3—68no—
183C3⋊Dic3—128no—
184C3⋊Dic3(C2×D3)⋊C411yescharacteristic
185C3⋊Dic3C2×#6711yescharacteristic
186C3×Dic3—62no—
187C3×Dic3—62no—
188C2×#41—271no—
189C2×M16—91no—
190C2×C42—271no—
191#104—274no—
192C33C2×#10411yescharacteristic, complemented, Sylow
193C23×C3—64no—
194C23×C3—91no—
195C22×D3—21no—
196C22×D3—62no—
197C3×D4—64no—
198C3×D4—64no—
199C2×C3×C4—22no—
200C2×C3×C4—42no—
201C2×C3×C4—91no—
202C2×C3×C4—91no—
203C2×C3×C4—91no—
204C2×C3×C4—91no—
205C2×C3×C4—91no—
206C2×C3×C4C2×#15711yescharacteristic
207#67—182no—
208#67—182no—
209#67—188no—
210#67—188no—
211C2×Dic3—62no—
212C2×Dic3—188no—
213C2×Dic3—188no—
214C2×Dic3—184no—
215C2×Dic3—182no—
216C2×Dic3—184no—
217C2×Dic3—181no—
218C2×Dic3—364no—
219D12—21no—
220D12C2×#15712yescomplemented
221C4×D3—62no—
222C4×D3—62no—
223C4×D3—62no—
224C4×D3—62no—
225C4×D3—182no—
226C4×D3—182no—
227Dic6—92no—
228Dic6—184no—
229Dic6—184no—
230Dic6—181no—
231Dic6—364no—
232Dic6—364no—
233C3×C8—182no—
234C2×C32—22no—
235C2×C32—24no—
236C2×C32—44no—
237C2×C32—42no—
238C2×C32—122no—
239C2×C32#82512yescomplemented
240C2×C32C2×(C2×D3)⋊C411yescharacteristic
241C3⋊D3—21no—
242C3⋊D3—21no—
243C3×D3—48no—
244C3×D3—122no—
245C3×D3—122no—
246#41—272no—
247#41—272no—
248#41—541no—
249#41—541no—
250C2×Q8—271no—
251C2×D4—31no—
252C2×D4—541no—
253C22×C4—91no—
254C22×C4—541no—
255C22×C4—541no—
256M16—92no—
257M16—181no—
258C2×C8—181no—
259C42—272no—
260C42—541no—
261C22×C3—22no—
262C22×C3—42no—
263C22×C3—68no—
264C22×C3—91no—
265C22×C3—91no—
266C22×C3—128no—
267C22×C3—182no—
268C22×C3C22⋊#15711yescharacteristic
269C2×D3—22no—
270C2×D3—42no—
271C2×D3—62no—
272C2×D3—62no—
273C2×D3—124no—
274C3×C4—22no—
275C3×C4—22no—
276C3×C4—42no—
277C3×C4—42no—
278C3×C4—91no—
279C3×C4—91no—
280C3×C4C22⋊#15711yescharacteristic
281C3×C4C22×#15711yescharacteristic
282Dic3—62no—
283Dic3—62no—
284Dic3—188no—
285Dic3—184no—
286Dic3—182no—
287Dic3—368no—
288C32—22no—
289C32—42no—
290C32C2×#82511yescharacteristic, complemented
291C23—61no—
292C23—91no—
293Q8—272no—
294Q8—541no—
295D4—32no—
296D4—61no—
297D4—542no—
298D4—542no—
299C2×C4—91no—
300C2×C4—91no—
301C2×C4—91no—
302C2×C4—91no—
303C2×C4—91no—
304C2×C4—542no—
305C2×C4—542no—
306C2×C4—542no—
307C2×C4—544no—
308C2×C4—541no—
309C2×C4D3×#15711yescharacteristic
310C8—182no—
311C2×C3—24no—
312C2×C3—22no—
313C2×C3—44no—
314C2×C3—42no—
315C2×C3—128no—
316C2×C3—181no—
317C2×C3—181no—
318C2×C3#6378312yescomplemented
319C2×C3C2×C22⋊#15711yescharacteristic
320D3—42no—
321D3—122no—
322D3—122no—
323C22—62no—
324C22—91no—
325C22—91no—
326C22—122no—
327C22—182no—
328C22(C2×D3)⋊#15711yescharacteristic
329C4—91no—
330C4—91no—
331C4—544no—
332C4—542no—
333C4(C2×D3)⋊#15711yescharacteristic
334C4C2×D3×#15711yescharacteristic
335C3—22no—
336C3—42no—
337C3C2×#6378311yescharacteristic, complemented
338C2—122no—
339C2—181no—
340C2—181no—
341C2#1172271012yesdirect product factor
342C2C2×(C2×D3)⋊#15711yescharacteristic
subgroupG/H
centerC22(C2×D3)⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC2×C33×C4C2×C4
socleC22×C33C22⋊C4
prime radicalC2×#89014C2
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. #11722710C2direct (N × H)
7. C2×#6308C4semidirect
9. C2×C32×D12C4semidirect
14. #89014C4semidirect
17. C4×C3⋊#157C22semidirect
20. C3×#3788C22semidirect
24. C2×#3788D3semidirect
33. #6308C2×C4semidirect
40. C32×D12C2×C4semidirect
71. #3788C2×D3semidirect
131. C2×C33#104semidirect
150. C2×D12#157semidirect
192. C33C2×#104semidirect
220. D12C2×#157semidirect
239. C2×C32#825semidirect
290. C32C2×#825semidirect
318. C2×C3#63783semidirect
337. C3C2×#63783semidirect
341. C2#11722710direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890142
C2×C4×C3⋊#1572
C2×C3×#37882
#117227102
C2×#637833
C2×#8259