Finite Group Databasebeta

← C2×#11722784 #13167421

Subgroups — C2×#11722784 #13167421

Order 1728.

616 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 95 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×#89040C211yesmaximal, characteristic
2C2×C3×C4⋊#157C211yesmaximal, characteristic
3C2×#88808C211yesmaximal, characteristic
4#11722784C214yesmaximal, direct product factor
5C2×#63730—31nomaximal
6C22×C3×#157—21no—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C3×#634—21no—
9C2×#6296C412yescomplemented
10C2×#6294—21no—
11#89040—21no—
12#89040—21no—
13#89040C412yescomplemented
14#89040C2212yes—
15C3×C4⋊#157—21no—
16C3×C4⋊#157C2212yes—
17C2×C3×C3⋊(C3⋊C8)—21no—
18#88808—21no—
19#88808C2212yes—
20C2×#3797—31no—
21C2×D3×Dic6—61no—
22C2×C4⋊#157D311yescharacteristic
23C2×#3710—31no—
24C2×D3×C3⋊C8—61no—
25#63730—34no—
26C22×C3×C3⋊D3D411yescharacteristic
27C2×C3×#157—22no—
28C2×C3×#157—42no—
29C2×C33×C4D411yescharacteristic
30C2×C3×C3⋊Dic3—22no—
31C2×C3×C3⋊Dic3C2×C411yescharacteristic
32C3×C4×C3⋊D3C2×C411yescharacteristic
33C3×C4×C3⋊D3D411yescharacteristic
34C3×C4×C3⋊D3D411yescharacteristic
35C3×C4×C3⋊D3C2311yescharacteristic
36C3×#634—22no—
37C3×#634—22no—
38#6296—22no—
39#6296C2×C414yescomplemented
40#6294—22no—
41#6294—22no—
42C3×C3⋊(C3⋊C8)—22no—
43C2×#810—91nomaximal
44C22×#157—21no—
45C2×C4×C3⋊D3C2×D311yescharacteristic
46C2×#634—21no—
47C2×C3×C4×D3—61no—
48C2×C3×C4×D3—61no—
49C2×C3×Dic6—61no—
50C2×C3×Dic6—61no—
51C2×C3×Dic6—121no—
52C2×#627—32no—
53C2×#627—62no—
54C2×#624—61no—
55C2×D3×Dic3—62no—
56#3797—32no—
57#3797—32no—
58#3797—61no—
59#3797—61no—
60D3×Dic6—62no—
61D3×Dic6—62no—
62D3×Dic6—62no—
63D3×Dic6—62no—
64C4⋊#157—21no—
65C4⋊#157C2×D312yes—
66C2×C3⋊(C3⋊C8)—21no—
67C2×C3×C3⋊C8—61no—
68C2×C3×C3⋊C8—61no—
69C2×C3×C3⋊C8—121no—
70#3710—32no—
71#3710—61no—
72D3×C3⋊C8—62no—
73D3×C3⋊C8—62no—
74C22×C33C22⋊C411yescharacteristic
75C2×C3×C3⋊D3C22⋊C411yescharacteristic
76C2×C3×C3⋊D3SD1612yes—
77C2×C3×C3⋊D3Q1612yes—
78C2×C3×C3⋊D3C2×D411yescharacteristic
79C3×#157—42no—
80C33×C4C22⋊C411yescharacteristic
81C33×C4C2×D411yescharacteristic
82C3×C3⋊Dic3—24no—
83C3×C3⋊Dic3C22⋊C411yescharacteristic
84C3×C3⋊Dic3C22×C411yescharacteristic
85C2×D3×Q8—181no—
86C22×Dic6—91no—
87C2×C3×C4⋊C4—91no—
88C22×C3⋊C8—91no—
89C2×D3×C8—181no—
90#810—94no—
91C22×C3⋊D3#6711yescharacteristic
92C22×C3×D3—61no—
93C22×C3×D3—61no—
94C2×#157—22no—
95C2×#157—42no—
96C2×C32×C4—21no—
97C2×C32×C4—21no—
98C2×C32×C4—41no—
99C2×C32×C4—41no—
100C2×C32×C4D1211yescharacteristic
101C2×C3⋊Dic3—22no—
102C2×C3⋊Dic3C4×D311yescharacteristic
103C4×C3⋊D3C4×D311yescharacteristic
104C4×C3⋊D3D1211yescharacteristic
105C4×C3⋊D3#6711yescharacteristic
106C4×C3⋊D3C22×D311yescharacteristic
107#634—22no—
108#634—22no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—122no—
114C3×C4×D3—61no—
115C3×C4×D3—61no—
116C3×C4×D3—61no—
117C3×C4×D3—61no—
118C3×C4×D3—61no—
119C3×C4×D3—61no—
120C3×C4×D3—61no—
121C3×C4×D3—61no—
122C3×Dic6—62no—
123C3×Dic6—62no—
124C3×Dic6—62no—
125C3×Dic6—62no—
126C3×Dic6—122no—
127C3×Dic6—122no—
128#627—34no—
129#627—64no—
130#627—64no—
131#627—62no—
132#624—62no—
133#624—62no—
134D3×Dic3—64no—
135D3×Dic3—64no—
136C3⋊(C3⋊C8)—22no—
137C3×C3⋊C8—62no—
138C3×C3⋊C8—62no—
139C3×C3⋊C8—122no—
140C2×#103—271noSylow
141C2×C33#10312yescomplemented
142C2×C33C2×C22⋊C411yescharacteristic
143C3×C3⋊D3#10312yescomplemented
144C3×C3⋊D3C2×SD1611yescharacteristic
145C3×C3⋊D3C2×Q1611yescharacteristic
146C2×C3×Q8—181no—
147C22×C3×C4—91no—
148C22×C3×C4—181no—
149C22×Dic3—181no—
150D3×Q8—182no—
151D3×Q8—182no—
152D3×Q8—182no—
153D3×Q8—182no—
154C2×C4×D3—61no—
155C2×C4×D3—61no—
156C2×C4×D3—182no—
157C2×Dic6—61no—
158C2×Dic6—61no—
159C2×Dic6—92no—
160C2×Dic6—92no—
161C2×Dic6—92no—
162C2×Dic6—121no—
163C2×Dic6—182no—
164C2×Dic6—181no—
165C2×Dic6—181no—
166C2×C3×C8—181no—
167C3×C4⋊C4—92no—
168C3×C4⋊C4—181no—
169C2×C3⋊C8—61no—
170C2×C3⋊C8—61no—
171C2×C3⋊C8—92no—
172C2×C3⋊C8—121no—
173C2×C3⋊C8—181no—
174D3×C8—182no—
175D3×C8—182no—
176C22×C32—21no—
177C22×C32—21no—
178C22×C32—41no—
179C22×C32—41no—
180C22×C32(C2×D3)⋊C411yescharacteristic
181C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
182C2×C3⋊D3#21412yes—
183C2×C3⋊D3#21612yes—
184C2×C3⋊D3C2×#6711yescharacteristic
185C2×C3×D3—61no—
186C2×C3×D3—61no—
187C2×C3×D3—62no—
188C2×C3×D3—62no—
189C2×C3×D3—62no—
190C2×C3×D3—62no—
191C2×C3×D3—61no—
192C2×C3×D3—61no—
193#157—42no—
194C32×C4—21no—
195C32×C4—21no—
196C32×C4—21no—
197C32×C4—21no—
198C32×C4—41no—
199C32×C4—41no—
200C32×C4—41no—
201C32×C4—41no—
202C32×C4(C2×D3)⋊C411yescharacteristic
203C32×C4C2×D1211yescharacteristic
204C3⋊Dic3—24no—
205C3⋊Dic3(C2×D3)⋊C411yescharacteristic
206C3⋊Dic3C2×C4×D311yescharacteristic
207C3×Dic3—61no—
208C3×Dic3—61no—
209C3×Dic3—61no—
210C3×Dic3—61no—
211C3×Dic3—64no—
212C3×Dic3—64no—
213C3×Dic3—124no—
214C22×Q8—271no—
215C22×C8—271no—
216C2×C4⋊C4—91no—
217#103—274no—
218C33C2×#10311yescharacteristic, complemented, Sylow
219C23×C3—91no—
220C22×D3—61no—
221C22×D3—61no—
222C3×Q8—182no—
223C3×Q8—182no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—91no—
229C2×C3×C4—91no—
230C2×C3×C4—91no—
231C2×C3×C4—91no—
232C2×C3×C4—91no—
233C2×C3×C4—182no—
234C2×C3×C4—182no—
235C2×C3×C4—362no—
236C2×C3×C4GO+(4,𝔽2)11yescharacteristic
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—62no—
240C2×Dic3—62no—
241C2×Dic3—122no—
242C2×Dic3—182no—
243C2×Dic3—182no—
244C4×D3—61no—
245C4×D3—61no—
246C4×D3—61no—
247C4×D3—61no—
248C4×D3—61no—
249C4×D3—61no—
250C4×D3—61no—
251C4×D3—61no—
252C4×D3—184no—
253C4×D3—184no—
254Dic6—62no—
255Dic6—62no—
256Dic6—62no—
257Dic6—62no—
258Dic6—94no—
259Dic6—122no—
260Dic6—122no—
261Dic6—184no—
262Dic6—182no—
263Dic6—184no—
264C3×C8—182no—
265C3⋊C8—62no—
266C3⋊C8—62no—
267C3⋊C8—122no—
268C2×C32—21no—
269C2×C32—21no—
270C2×C32—22no—
271C2×C32—22no—
272C2×C32—42no—
273C2×C32—42no—
274C2×C32—41no—
275C2×C32—41no—
276C2×C32#81012yescomplemented
277C2×C32C2×(C2×D3)⋊C411yescharacteristic
278C3⋊D3#81012yescomplemented
279C3⋊D3C2×#21411yescharacteristic
280C3⋊D3C2×#21611yescharacteristic
281C3×D3—62no—
282C3×D3—62no—
283C3×D3—61no—
284C3×D3—61no—
285C3×D3—61no—
286C3×D3—61no—
287C2×Q8—181no—
288C2×Q8—272no—
289C2×Q8—272no—
290C2×Q8—272no—
291C2×Q8—541no—
292C2×Q8—541no—
293C22×C4—91no—
294C22×C4—181no—
295C22×C4—541no—
296C2×C8—181no—
297C2×C8—272no—
298C2×C8—541no—
299C4⋊C4—92no—
300C4⋊C4—181no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—91no—
306C22×C3—91no—
307C22×C3—92no—
308C22×C3—92no—
309C22×C3D32⋊C411yescharacteristic
310C2×D3—61no—
311C2×D3—61no—
312C2×D3—61no—
313C2×D3—61no—
314C2×D3—62no—
315C2×D3—62no—
316C2×D3—62no—
317C2×D3—62no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—41no—
323C3×C4—41no—
324C3×C4—41no—
325C3×C4—41no—
326C3×C4—91no—
327C3×C4—91no—
328C3×C4—184no—
329C3×C4—362no—
330C3×C4D32⋊C411yescharacteristic
331C3×C4C2×GO+(4,𝔽2)11yescharacteristic
332Dic3—61no—
333Dic3—61no—
334Dic3—61no—
335Dic3—61no—
336Dic3—64no—
337Dic3—64no—
338Dic3—124no—
339C32—21no—
340C32—21no—
341C32—41no—
342C32—41no—
343C32C2×#81011yescharacteristic, complemented
344C23—91no—
345Q8—182no—
346Q8—182no—
347Q8—274no—
348Q8—542no—
349C2×C4—91no—
350C2×C4—91no—
351C2×C4—91no—
352C2×C4—91no—
353C2×C4—91no—
354C2×C4—182no—
355C2×C4—182no—
356C2×C4—362no—
357C2×C4—542no—
358C2×C4—542no—
359C2×C4#157⋊D311yescharacteristic
360C8—182no—
361C2×C3—22no—
362C2×C3—22no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—42no—
366C2×C3—42no—
367C2×C3—41no—
368C2×C3—41no—
369C2×C3—91no—
370C2×C3—91no—
371C2×C3—92no—
372C2×C3#6373012yescomplemented
373C2×C3C2×D32⋊C411yescharacteristic
374D3—62no—
375D3—62no—
376D3—61no—
377D3—61no—
378D3—61no—
379D3—61no—
380C22—92no—
381C22—92no—
382C22—91no—
383C22—91no—
384C22#8893911yescharacteristic
385C4—91no—
386C4—91no—
387C4—184no—
388C4—362no—
389C4#8893911yescharacteristic
390C4C2×#157⋊D311yescharacteristic
391C3—21no—
392C3—21no—
393C3—41no—
394C3—41no—
395C3C2×#6373011yescharacteristic, complemented
396C2—91no—
397C2—92no—
398C2—91no—
399C2#1172278412yesdirect product factor
400C2C2×#8893911yescharacteristic
subgroupG/H
centerC22#88939
commutatorC3×C3⋊Dic3C22×C4
FrattiniC4C2×#157⋊D3
FittingC2×C33×C4D4
socleC22×C33C22⋊C4
prime radicalC22×C3×C3⋊D3D4
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4#157⋊D3
hypocenterC33C2×#103
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722784C2direct (N × H)
9. C2×#6296C4semidirect
13. #89040C4semidirect
39. #6296C2×C4semidirect
141. C2×C33#103semidirect
143. C3×C3⋊D3#103semidirect
218. C33C2×#103semidirect
276. C2×C32#810semidirect
278. C3⋊D3#810semidirect
343. C32C2×#810semidirect
372. C2×C3#63730semidirect
395. C3C2×#63730semidirect
399. C2#11722784direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890402
C2×C3×C4⋊#1572
C2×#888082
#117227842
C2×#637303
C2×#8109