Finite Group Databasebeta

← C2×#11722761 #13167049

Subgroups — C2×#11722761 #13167049

Order 1728.

648 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 79 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×#89042C211yesmaximal, characteristic
2C2×C4×C3⋊#157C211yesmaximal, characteristic, complemented
3C2×C3×#3712C211yesmaximal, characteristic, complemented
4#11722761C214yesmaximal, direct product factor
5C2×#63731—31nomaximal
6C22×C3⋊#157—21no—
7C2×C3×C4×C3⋊D3—21no—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×#6296C411yescharacteristic, complemented
10C2×#6295—21no—
11C2×#6295C411yescharacteristic, complemented
12#89042—21no—
13#89042—21no—
14#89042C412yescomplemented
15#89042C2212yes—
16C4×C3⋊#157—21no—
17C4×C3⋊#157C2212yescomplemented
18C2×C32×C3⋊C8—21no—
19C3×#3712—21no—
20C3×#3712C2212yescomplemented
21C2×#3798—31no—
22C2×#3798—61no—
23C2×C4×#157—31no—
24C2×C3×ΓL(1,𝔽25)—61no—
25C2×#3712D311yescharacteristic, complemented
26#63731—34no—
27C22×C3×C3⋊D3—21no—
28C22×C3×C3⋊D3D411yescharacteristic
29C2×C3⋊#157—24no—
30C2×C3⋊#157—22no—
31C2×C33×C4D411yescharacteristic
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C3×C4×C3⋊D3—22no—
35C3×C4×C3⋊D3—22no—
36C3×C4×C3⋊D3C2×C411yescharacteristic
37C3×C4×C3⋊D3D411yescharacteristic
38C3×C4×C3⋊D3D411yescharacteristic
39C3×C4×C3⋊D3C2311yescharacteristic
40#6296—21no—
41#6296C2×C412yescomplemented
42#6295—22no—
43#6295—22no—
44#6295—21no—
45#6295C2×C412yescomplemented
46C32×C3⋊C8—22no—
47C2×#825—91nomaximal
48C22×#157—61no—
49C2×C4×C3⋊D3—21no—
50C2×C4×C3⋊D3C2×D311yescharacteristic
51C2×C3×C4×D3—61no—
52C2×C3×C4×D3—61no—
53C2×C3×C4×D3—61no—
54C2×C3×C4×D3—61no—
55C2×C3×C4×D3—121no—
56C2×#627—31no—
57C2×#627—61no—
58C2×#626—61no—
59C2×#626—61no—
60C2×#626—61no—
61C2×#625—31no—
62C2×#625—61no—
63#3798—32no—
64#3798—32no—
65#3798—62no—
66#3798—62no—
67#3798—62no—
68#3798—62no—
69#3798—61no—
70#3798—61no—
71C4×#157—32no—
72C4×#157—61no—
73C2×C32×C8—61no—
74C2×C3×C3⋊C8—21no—
75C2×C3×C3⋊C8—21no—
76C2×C3×C3⋊C8—41no—
77C3×ΓL(1,𝔽25)—62no—
78C3×ΓL(1,𝔽25)—62no—
79#3712—21no—
80#3712C2×D312yescomplemented
81C22×C33C22⋊C411yescharacteristic
82C2×C3×C3⋊D3—22no—
83C2×C3×C3⋊D3—22no—
84C2×C3×C3⋊D3—42no—
85C2×C3×C3⋊D3C22⋊C411yescharacteristic
86C2×C3×C3⋊D3C2×D411yescharacteristic
87C3⋊#157—24no—
88C33×C4C22⋊C411yescharacteristic
89C33×C4C2×D411yescharacteristic
90C3×C3⋊Dic3—22no—
91C3×C3⋊Dic3C22⋊C411yescharacteristic
92C3×C3⋊Dic3C22×C411yescharacteristic
93C2×#235—91no—
94C2×#235—181no—
95C2×C3×M16—91no—
96C2×C4×Dic3—91no—
97C2×ΓL(1,𝔽25)—61no—
98#825—94no—
99C22×C3⋊D3—21no—
100C22×C3⋊D3D1211yescharacteristic
101C22×C3×D3—61no—
102C22×C3×D3—61no—
103C22×C3×D3—61no—
104C22×C3×D3—61no—
105C22×C3×D3—121no—
106C2×#157—64no—
107C2×#157—62no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4—41no—
111C2×C32×C4—41no—
112C2×C32×C4#6711yescharacteristic
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3C4×D311yescharacteristic
115C4×C3⋊D3—22no—
116C4×C3⋊D3—22no—
117C4×C3⋊D3C4×D311yescharacteristic
118C4×C3⋊D3D1211yescharacteristic
119C4×C3⋊D3#6711yescharacteristic
120C4×C3⋊D3C22×D311yescharacteristic
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—121no—
126C3×C4×D3—61no—
127C3×C4×D3—61no—
128C3×C4×D3—61no—
129C3×C4×D3—61no—
130C3×C4×D3—61no—
131C3×C4×D3—61no—
132C3×C4×D3—61no—
133C3×C4×D3—61no—
134C3×C4×D3—62no—
135C3×C4×D3—62no—
136C3×C4×D3—62no—
137C3×C4×D3—62no—
138C3×C4×D3—122no—
139C3×C4×D3—122no—
140#627—32no—
141#627—62no—
142#627—62no—
143#627—61no—
144#626—62no—
145#626—62no—
146#626—62no—
147#626—62no—
148#626—62no—
149#626—62no—
150#625—32no—
151#625—62no—
152#625—62no—
153#625—61no—
154C32×C8—62no—
155C3×C3⋊C8—22no—
156C3×C3⋊C8—22no—
157C3×C3⋊C8—42no—
158C2×#104—271noSylow
159C2×C33#10412yescomplemented
160C2×C33C2×C22⋊C411yescharacteristic
161C3×C3⋊D3—21no—
162C3×C3⋊D3—21no—
163C3×C3⋊D3—42no—
164C22×C3×C4—91no—
165C22×C3×C4—181no—
166C2×#67—181no—
167C2×#67—181no—
168C2×#67—181no—
169C22×Dic3—181no—
170#235—92no—
171#235—92no—
172#235—182no—
173#235—182no—
174#235—181no—
175#235—181no—
176#235—182no—
177#235—182no—
178C2×D12—91no—
179C2×D12—181no—
180C2×C4×D3—61no—
181C2×C4×D3—61no—
182C2×C4×D3—61no—
183C2×C4×D3—61no—
184C2×C4×D3—121no—
185C2×Dic6—91no—
186C2×Dic6—181no—
187C3×M16—92no—
188C3×M16—181no—
189C2×C3×C8—61no—
190C2×C3×C8—61no—
191C2×C3×C8—121no—
192C4×Dic3—92no—
193C4×Dic3—181no—
194C2×C3⋊C8—21no—
195ΓL(1,𝔽25)—62no—
196ΓL(1,𝔽25)—62no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32—41no—
200C22×C32—41no—
201C22×C32(C2×D3)⋊C411yescharacteristic
202C2×C3⋊D3—22no—
203C2×C3⋊D3—22no—
204C2×C3⋊D3—42no—
205C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
206C2×C3⋊D3C2×D1211yescharacteristic
207C2×C3×D3—61no—
208C2×C3×D3—61no—
209C2×C3×D3—61no—
210C2×C3×D3—61no—
211C2×C3×D3—62no—
212C2×C3×D3—62no—
213C2×C3×D3—122no—
214C2×C3×D3—122no—
215C2×C3×D3—122no—
216C2×C3×D3—122no—
217C2×C3×D3—124no—
218C2×C3×D3—122no—
219#157—64no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—41no—
225C32×C4—41no—
226C32×C4—41no—
227C32×C4—41no—
228C32×C4(C2×D3)⋊C411yescharacteristic
229C32×C4C2×#6711yescharacteristic
230C3⋊Dic3—22no—
231C3⋊Dic3(C2×D3)⋊C411yescharacteristic
232C3⋊Dic3C2×C4×D311yescharacteristic
233C3×Dic3—61no—
234C3×Dic3—61no—
235C3×Dic3—61no—
236C3×Dic3—61no—
237C3×Dic3—62no—
238C3×Dic3—62no—
239C3×Dic3—122no—
240C2×#41—271no—
241C2×M16—91no—
242C2×C42—271no—
243#104—274no—
244C33C2×#10411yescharacteristic, complemented, Sylow
245C23×C3—91no—
246C23×C3—181no—
247C22×D3—61no—
248C22×D3—61no—
249C22×D3—61no—
250C22×D3—61no—
251C22×D3—121no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—91no—
257C2×C3×C4—91no—
258C2×C3×C4—91no—
259C2×C3×C4—91no—
260C2×C3×C4—91no—
261C2×C3×C4—182no—
262C2×C3×C4—182no—
263C2×C3×C4—181no—
264C2×C3×C4GO+(4,𝔽2)11yescharacteristic
265#67—182no—
266#67—182no—
267#67—182no—
268#67—182no—
269#67—182no—
270#67—182no—
271C2×Dic3—61no—
272C2×Dic3—61no—
273C2×Dic3—61no—
274C2×Dic3—61no—
275C2×Dic3—121no—
276C2×Dic3—182no—
277C2×Dic3—184no—
278D12—92no—
279D12—181no—
280D12—182no—
281D12—182no—
282C4×D3—61no—
283C4×D3—61no—
284C4×D3—61no—
285C4×D3—61no—
286C4×D3—61no—
287C4×D3—61no—
288C4×D3—61no—
289C4×D3—61no—
290C4×D3—62no—
291C4×D3—62no—
292C4×D3—62no—
293C4×D3—62no—
294C4×D3—122no—
295C4×D3—122no—
296Dic6—92no—
297Dic6—182no—
298Dic6—181no—
299Dic6—182no—
300C3×C8—62no—
301C3×C8—62no—
302C3×C8—122no—
303C3⋊C8—22no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32—22no—
307C2×C32—22no—
308C2×C32—42no—
309C2×C32—42no—
310C2×C32—41no—
311C2×C32—41no—
312C2×C32#82512yescomplemented
313C2×C32C2×(C2×D3)⋊C411yescharacteristic
314C3⋊D3—21no—
315C3⋊D3—21no—
316C3⋊D3—42no—
317C3×D3—121no—
318C3×D3—121no—
319C3×D3—121no—
320C3×D3—121no—
321C3×D3—122no—
322C3×D3—124no—
323C3×D3—122no—
324#41—272no—
325#41—272no—
326#41—541no—
327#41—541no—
328C2×Q8—271no—
329C2×D4—271no—
330C2×D4—541no—
331C22×C4—91no—
332C22×C4—181no—
333C22×C4—541no—
334M16—92no—
335M16—181no—
336C2×C8—61no—
337C42—272no—
338C42—541no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—91no—
344C22×C3—91no—
345C22×C3—182no—
346C22×C3—182no—
347C22×C3—362no—
348C22×C3D32⋊C411yescharacteristic
349C2×D3—61no—
350C2×D3—61no—
351C2×D3—61no—
352C2×D3—61no—
353C2×D3—62no—
354C2×D3—62no—
355C2×D3—122no—
356C2×D3—122no—
357C2×D3—122no—
358C2×D3—122no—
359C2×D3—124no—
360C2×D3—122no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—41no—
366C3×C4—41no—
367C3×C4—41no—
368C3×C4—41no—
369C3×C4—91no—
370C3×C4—91no—
371C3×C4—182no—
372C3×C4D32⋊C411yescharacteristic
373C3×C4C2×GO+(4,𝔽2)11yescharacteristic
374Dic3—61no—
375Dic3—61no—
376Dic3—61no—
377Dic3—61no—
378Dic3—62no—
379Dic3—62no—
380Dic3—122no—
381Dic3—184no—
382C32—21no—
383C32—21no—
384C32—41no—
385C32—41no—
386C32C2×#82511yescharacteristic, complemented
387C23—91no—
388C23—181no—
389Q8—272no—
390Q8—541no—
391D4—272no—
392D4—541no—
393D4—542no—
394D4—542no—
395C2×C4—91no—
396C2×C4—91no—
397C2×C4—91no—
398C2×C4—91no—
399C2×C4—91no—
400C2×C4—181no—
401C2×C4—182no—
402C2×C4—182no—
403C2×C4—542no—
404C2×C4—544no—
405C2×C4D32⋊D311yescharacteristic
406C8—62no—
407C2×C3—22no—
408C2×C3—22no—
409C2×C3—21no—
410C2×C3—21no—
411C2×C3—42no—
412C2×C3—42no—
413C2×C3—41no—
414C2×C3—41no—
415C2×C3—181no—
416C2×C3—181no—
417C2×C3—362no—
418C2×C3#6373112yescomplemented
419C2×C3C2×D32⋊C411yescharacteristic
420D3—121no—
421D3—121no—
422D3—121no—
423D3—121no—
424D3—122no—
425D3—124no—
426D3—122no—
427C22—91no—
428C22—91no—
429C22—182no—
430C22—182no—
431C22—362no—
432C22#8893211yescharacteristic
433C4—91no—
434C4—91no—
435C4—182no—
436C4—544no—
437C4#8893211yescharacteristic
438C4C2×D32⋊D311yescharacteristic
439C3—21no—
440C3—21no—
441C3—41no—
442C3—41no—
443C3C2×#6373111yescharacteristic, complemented
444C2—181no—
445C2—181no—
446C2—362no—
447C2#1172276112yesdirect product factor
448C2C2×#8893211yescharacteristic
subgroupG/H
center, hypercenterC2×C4D32⋊D3
commutatorC3×C3⋊Dic3C22×C4
FrattiniC4C2×D32⋊D3
FittingC2×C33×C4D4
socleC22×C33C22⋊C4
prime radicalC2×#89042C2
cosocleC3×C4×C3⋊D3C23
hypocenterC33C2×#104
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C3⋊#157C2semidirect
3. C2×C3×#3712C2semidirect
4. #11722761C2direct (N × H)
9. C2×#6296C4semidirect
11. C2×#6295C4semidirect
14. #89042C4semidirect
17. C4×C3⋊#157C22semidirect
20. C3×#3712C22semidirect
25. C2×#3712D3semidirect
41. #6296C2×C4semidirect
45. #6295C2×C4semidirect
80. #3712C2×D3semidirect
159. C2×C33#104semidirect
244. C33C2×#104semidirect
312. C2×C32#825semidirect
386. C32C2×#825semidirect
418. C2×C3#63731semidirect
443. C3C2×#63731semidirect
447. C2#11722761direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890422
C2×C4×C3⋊#1572
C2×C3×#37122
#117227612
C2×#637313
C2×#8259