Finite Group Databasebeta

← C2×#11724334 #13181128

Subgroups — C2×#11724334 #13181128

Order 1728.

1144 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 1144 conjugacy classes of subgroups — too many to draw. Only its 143 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×C4⋊#1113C211yesmaximal, characteristic
2C2×#89040C211yesmaximal, characteristic, complemented
3C2×C3×C4×#157C211yesmaximal, characteristic, complemented
4C2×#88939C214yesmaximal, complemented
5#11724334C218yesmaximal, direct product factor
6C2×#64221—31nomaximal
7C22×#1113C2212yes—
8C22×C3×#157C2212yes—
9C2×C3×C4⋊(C3⋊D3)—21no—
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C3×#634—21no—
12C2×#6296—21no—
13C2×#6295—21no—
14C2×#6294C2212yes—
15C4⋊#1113—22no—
16C4⋊#1113C2214yes—
17#89040—22no—
18#89040C2214yescomplemented
19C3×C4×#157C2214yescomplemented
20#88939C22116yescomplemented
21C2×#3802—31no—
22C2×D3×D12—61no—
23C2×#3797—31no—
24C2×D3×Dic6—61no—
25C2×C4×#157D311yescharacteristic, complemented
26C2×D32⋊C4—34no—
27#64221—38no—
28C22×C3×C3⋊D3—22no—
29C22×C3×C3⋊D3C2311yescharacteristic
30C2×#1113—28no—
31C2×#1113—22no—
32C2×#1113C2314yes—
33C2×C3×#157—24no—
34C2×C3×#157C2314yes—
35C2×C33×C4D411yescharacteristic
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3D411yescharacteristic
38C3×C4⋊(C3⋊D3)—24no—
39C3×C4×C3⋊D3D412yes—
40C3×C4×C3⋊D3C2312yes—
41C3×#634—24no—
42#6296—24no—
43#6295—24no—
44#6294—22no—
45#6294C2314yes—
46C2×#875—91nomaximal
47C22×D32—32no—
48C22×D32—62no—
49C22×#157C2×D312yes—
50C2×C4⋊(C3⋊D3)—21no—
51C2×C4×C3⋊D3C2×D311yescharacteristic
52C2×#634—21no—
53C2×C3×D12—61no—
54C2×C3×D12—61no—
55C2×C3×D12—121no—
56C2×C3×C4×D3—61no—
57C2×C3×C4×D3—61no—
58C2×C3×Dic6—61no—
59C2×C3×Dic6—61no—
60C2×C3×Dic6—121no—
61C2×#627—62no—
62C2×#627—61no—
63C2×#626—62no—
64C2×#625—61no—
65C2×#624—32no—
66C2×D3×Dic3—62no—
67#3802—34no—
68#3802—62no—
69D3×D12—64no—
70D3×D12—64no—
71#3797—34no—
72#3797—62no—
73D3×Dic6—64no—
74D3×Dic6—64no—
75C4×#157C2×D314yescomplemented
76D32⋊C4—316no—
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—24no—
79C2×C3×C3⋊D3—44no—
80C2×C3×C3⋊D3C2×D411yescharacteristic
81C2×C3×C3⋊D3#4114yes—
82C2×C3×C3⋊D3C2411yescharacteristic
83#1113—28no—
84#1113—44no—
85C3×#157—24no—
86C33×C4C2×D412yes—
87C3×C3⋊Dic3—24no—
88C3×C3⋊Dic3C2×D412yes—
89C2×D3×Q8—181no—
90C2×D3×D4—181no—
91C22×D12—91no—
92C22×Dic6—91no—
93C2×C3×C42—91no—
94C2×(C2×D3)⋊C4—94no—
95#875—98no—
96C22×C3⋊D3—22no—
97C22×C3⋊D3C22×D311yescharacteristic
98C22×C3×D3—61no—
99C22×C3×D3—61no—
100C22×C3×D3—62no—
101C22×C3×D3—62no—
102C22×C3×D3—122no—
103C2×D32—34no—
104C2×D32—64no—
105C2×D32—64no—
106C2×D32—62no—
107C2×D32—68no—
108C2×D32—128no—
109C2×#157—24no—
110C2×#157C22×D314yes—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—41no—
114C2×C32×C4—41no—
115C2×C32×C4D1211yescharacteristic
116C2×C3⋊Dic3—22no—
117C2×C3⋊Dic3D1211yescharacteristic
118C4⋊(C3⋊D3)—24no—
119C4×C3⋊D3D1212yes—
120C4×C3⋊D3C22×D312yes—
121#634—24no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—62no—
126C2×C3×Dic3—122no—
127C3×D12—64no—
128C3×D12—64no—
129C3×D12—124no—
130C3×C4×D3—62no—
131C3×C4×D3—62no—
132C3×C4×D3—62no—
133C3×C4×D3—62no—
134C3×Dic6—64no—
135C3×Dic6—64no—
136C3×Dic6—124no—
137#627—68no—
138#627—64no—
139#626—68no—
140#625—64no—
141#624—34no—
142#624—62no—
143D3×Dic3—64no—
144D3×Dic3—64no—
145C2×#124—271noSylow
146C2×C33#12412yescomplemented
147C2×C33C22×D411yescharacteristic
148C3×C3⋊D3—44no—
149C3×C3⋊D3#12412yescomplemented
150C3×C3⋊D3C2×#4112yes—
151C23×D3—92no—
152C23×D3—182no—
153C2×C3×Q8—181no—
154C2×C3×D4—181no—
155C22×C3×C4—91no—
156C22×C3×C4—92no—
157C2×#67—182no—
158C22×Dic3—92no—
159D3×Q8—184no—
160D3×Q8—184no—
161D3×D4—184no—
162D3×D4—184no—
163C2×D12—61no—
164C2×D12—61no—
165C2×D12—94no—
166C2×D12—121no—
167C2×D12—182no—
168C2×D12—181no—
169C2×C4×D3—61no—
170C2×C4×D3—61no—
171C2×C4×D3—182no—
172C2×Dic6—61no—
173C2×Dic6—61no—
174C2×Dic6—94no—
175C2×Dic6—121no—
176C2×Dic6—182no—
177C2×Dic6—181no—
178C2×Dic6—182no—
179C3×C42—94no—
180(C2×D3)⋊C4—916no—
181C22×C32—21no—
182C22×C32—21no—
183C22×C32—41no—
184C22×C32—41no—
185C22×C32C2×D1211yescharacteristic
186C2×C3⋊D3—24no—
187C2×C3⋊D3—44no—
188C2×C3⋊D3C2×D1211yescharacteristic
189C2×C3⋊D3#23514yes—
190C2×C3⋊D3C23×D311yescharacteristic
191C2×C3×D3—61no—
192C2×C3×D3—61no—
193C2×C3×D3—64no—
194C2×C3×D3—64no—
195C2×C3×D3—61no—
196C2×C3×D3—61no—
197C2×C3×D3—64no—
198C2×C3×D3—64no—
199C2×C3×D3—124no—
200C2×C3×D3—124no—
201C2×C3×D3—124no—
202C2×C3×D3—244no—
203D32—68no—
204D32—128no—
205D32—128no—
206D32—124no—
207#157—24no—
208C32×C4—22no—
209C32×C4—22no—
210C32×C4—42no—
211C32×C4—42no—
212C32×C4C2×D1212yes—
213C3⋊Dic3—24no—
214C3⋊Dic3C2×D1212yes—
215C3×Dic3—62no—
216C3×Dic3—62no—
217C3×Dic3—64no—
218C3×Dic3—64no—
219C3×Dic3—124no—
220C22×Q8—271no—
221C22×D4—271no—
222#124—278no—
223C2×C22⋊C4—274no—
224C2×C42—91no—
225C33C2×#12411yescharacteristic, complemented, Sylow
226C23×C3—91no—
227C23×C3—182no—
228C22×D3—61no—
229C22×D3—61no—
230C22×D3—62no—
231C22×D3—62no—
232C22×D3—94no—
233C22×D3—122no—
234C22×D3—184no—
235C22×D3—182no—
236C22×D3—184no—
237C22×D3—188no—
238C22×D3—368no—
239C3×Q8—184no—
240C3×D4—184no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—91no—
246C2×C3×C4—92no—
247C2×C3×C4—92no—
248C2×C3×C4—94no—
249C2×C3×C4—182no—
250C2×C3×C4—184no—
251C2×C3×C4GO+(4,𝔽2)11yescharacteristic
252#67—188no—
253C2×Dic3—61no—
254C2×Dic3—61no—
255C2×Dic3—62no—
256C2×Dic3—62no—
257C2×Dic3—94no—
258C2×Dic3—122no—
259C2×Dic3—182no—
260D12—64no—
261D12—64no—
262D12—124no—
263D12—184no—
264C4×D3—62no—
265C4×D3—62no—
266C4×D3—62no—
267C4×D3—62no—
268C4×D3—184no—
269C4×D3—184no—
270Dic6—64no—
271Dic6—64no—
272Dic6—124no—
273Dic6—184no—
274Dic6—188no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32—22no—
278C2×C32—22no—
279C2×C32—42no—
280C2×C32—42no—
281C2×C32—41no—
282C2×C32—41no—
283C2×C32#87512yescomplemented
284C2×C32C22×D1211yescharacteristic
285C3⋊D3—44no—
286C3⋊D3#87512yescomplemented
287C3⋊D3C2×#23512yes—
288C3×D3—62no—
289C3×D3—62no—
290C3×D3—62no—
291C3×D3—62no—
292C3×D3—124no—
293C3×D3—124no—
294C3×D3—244no—
295C24—272no—
296C2×Q8—181no—
297C2×Q8—274no—
298C2×Q8—541no—
299C2×Q8—542no—
300C2×D4—181no—
301C2×D4—274no—
302C2×D4—542no—
303C2×D4—541no—
304C22×C4—91no—
305C22×C4—92no—
306C22×C4—272no—
307C22⋊C4—2716no—
308C42—94no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—91no—
314C22×C3—91no—
315C22×C3—94no—
316C22×C3—184no—
317C22×C3—364no—
318C22×C3C2×GO+(4,𝔽2)11yescharacteristic
319C2×D3—61no—
320C2×D3—61no—
321C2×D3—61no—
322C2×D3—61no—
323C2×D3—64no—
324C2×D3—64no—
325C2×D3—64no—
326C2×D3—64no—
327C2×D3—124no—
328C2×D3—124no—
329C2×D3—124no—
330C2×D3—188no—
331C2×D3—244no—
332C2×D3—368no—
333C2×D3—368no—
334C2×D3—364no—
335C3×C4—22no—
336C3×C4—22no—
337C3×C4—42no—
338C3×C4—42no—
339C3×C4—92no—
340C3×C4—184no—
341C3×C4—184no—
342C3×C4C2×GO+(4,𝔽2)12yes—
343Dic3—62no—
344Dic3—62no—
345Dic3—64no—
346Dic3—64no—
347Dic3—124no—
348C32—21no—
349C32—21no—
350C32—41no—
351C32—41no—
352C32C2×#87511yescharacteristic, complemented
353C23—91no—
354C23—182no—
355C23—274no—
356C23—542no—
357C23—548no—
358Q8—184no—
359Q8—544no—
360D4—184no—
361D4—544no—
362C2×C4—91no—
363C2×C4—92no—
364C2×C4—92no—
365C2×C4—94no—
366C2×C4—184no—
367C2×C4—182no—
368C2×C4—274no—
369C2×C4—542no—
370C2×C4#157⋊D311yescharacteristic
371C2×C3—22no—
372C2×C3—22no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—42no—
376C2×C3—42no—
377C2×C3—41no—
378C2×C3—41no—
379C2×C3—92no—
380C2×C3—92no—
381C2×C3—364no—
382C2×C3#6422112yescomplemented
383C2×C3C22×GO+(4,𝔽2)11yescharacteristic
384D3—62no—
385D3—62no—
386D3—62no—
387D3—62no—
388D3—124no—
389D3—124no—
390D3—244no—
391C22—94no—
392C22—91no—
393C22—91no—
394C22—184no—
395C22—364no—
396C22—548no—
397C22—1084no—
398C22C2×#157⋊D311yescharacteristic
399C4—92no—
400C4—184no—
401C4—184no—
402C4C2×#157⋊D312yes—
403C3—21no—
404C3—21no—
405C3—41no—
406C3—41no—
407C3C2×#6422111yescharacteristic, complemented
408C2—92no—
409C2—92no—
410C2—364no—
411C2#1172433412yesdirect product factor
412C2C22×#157⋊D311yescharacteristic
subgroupG/H
centerC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊D3C24
FrattiniC2C22×#157⋊D3
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C4⋊#1113C2
hypercenterC2×C4#157⋊D3
hypocenterC33C2×#124
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#89040C2semidirect
3. C2×C3×C4×#157C2semidirect
4. C2×#88939C2semidirect
5. #11724334C2direct (N × H)
18. #89040C22semidirect
19. C3×C4×#157C22semidirect
20. #88939C22semidirect
25. C2×C4×#157D3semidirect
75. C4×#157C2×D3semidirect
146. C2×C33#124semidirect
149. C3×C3⋊D3#124semidirect
225. C33C2×#124semidirect
283. C2×C32#875semidirect
286. C3⋊D3#875semidirect
352. C32C2×#875semidirect
382. C2×C3#64221semidirect
407. C3C2×#64221semidirect
411. C2#11724334direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊#11132
C2×#890402
C2×C3×C4×#1572
C2×#889392
#117243342
C2×#642213
C2×#8759