Finite Group Databasebeta

← C2×#11722782 #13167396

Subgroups — C2×#11722782 #13167396

Order 1728.

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

normal subgroups only

This group has 576 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×#89040C211yesmaximal, characteristic
2C2×C3×#3788C211yesmaximal, characteristic
3C2×#88809C211yesmaximal, characteristic
4#11722782C214yesmaximal, direct product factor
5C2×#63728—31nomaximal
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C2×C3×#634—21no—
8C2×#6296—21no—
9C2×#6294C412yes—
10#89040—21no—
11#89040—21no—
12#89040C412yes—
13#89040C2212yes—
14C2×C3×#622—21no—
15C3×#3788—21no—
16C3×#3788C2212yes—
17C2×C3×C3⋊(C3⋊C8)—21no—
18#88809—21no—
19#88809C2212yes—
20C2×#3797—31no—
21C2×D3×Dic6—61no—
22C2×#3788D311yescharacteristic
23C2×#3712—31no—
24C2×#3711—61no—
25#63728—34no—
26C22×C3×C3⋊D3C2×C411yescharacteristic
27C2×C33×C4D411yescharacteristic
28C2×C3×C3⋊Dic3—22no—
29C2×C3×C3⋊Dic3D411yescharacteristic
30C3×C4×C3⋊D3C2×C411yescharacteristic
31C3×C4×C3⋊D3D411yescharacteristic
32C3×C4×C3⋊D3D411yescharacteristic
33C3×C4×C3⋊D3C2311yescharacteristic
34C3×#634—22no—
35C3×#634—22no—
36#6296—22no—
37#6296—22no—
38#6294—22no—
39#6294C2×C414yes—
40C3×#622—22no—
41C3×C3⋊(C3⋊C8)—22no—
42C2×#824—91nomaximal
43C2×C4×C3⋊D3C2×D311yescharacteristic
44C2×#634—21no—
45C2×C3×C4×D3—61no—
46C2×C3×C4×D3—61no—
47C2×C3×Dic6—61no—
48C2×C3×Dic6—61no—
49C2×C3×Dic6—121no—
50C2×#627—62no—
51C2×#627—61no—
52C2×#624—32no—
53C2×D3×Dic3—62no—
54#3797—32no—
55#3797—32no—
56#3797—61no—
57#3797—61no—
58D3×Dic6—62no—
59D3×Dic6—62no—
60D3×Dic6—62no—
61D3×Dic6—62no—
62C2×#622—21no—
63#3788—21no—
64#3788C2×D312yes—
65C2×C3⋊(C3⋊C8)—21no—
66C2×C3×C3⋊C8—61no—
67C2×C3×C3⋊C8—61no—
68C2×C3×C3⋊C8—121no—
69#3712—32no—
70#3712—61no—
71#3711—62no—
72#3711—62no—
73C22×C33C22⋊C411yescharacteristic
74C2×C3×C3⋊D3—22no—
75C2×C3×C3⋊D3C22⋊C411yescharacteristic
76C2×C3×C3⋊D3C22×C411yescharacteristic
77C33×C4C22⋊C411yescharacteristic
78C33×C4C2×D411yescharacteristic
79C3×C3⋊Dic3—24no—
80C3×C3⋊Dic3C22⋊C411yescharacteristic
81C3×C3⋊Dic3C2×D411yescharacteristic
82C2×D3×Q8—181no—
83C22×Dic6—91no—
84C2×C3×M16—91no—
85C2×#208—91no—
86C2×ΓL(1,𝔽25)—181no—
87#824—94no—
88C22×C3⋊D3C4×D311yescharacteristic
89C22×C3×D3—61no—
90C22×C3×D3—61no—
91C2×C32×C4—21no—
92C2×C32×C4—21no—
93C2×C32×C4—41no—
94C2×C32×C4—41no—
95C2×C32×C4D1211yescharacteristic
96C2×C3⋊Dic3—22no—
97C2×C3⋊Dic3#6711yescharacteristic
98C4×C3⋊D3C4×D311yescharacteristic
99C4×C3⋊D3D1211yescharacteristic
100C4×C3⋊D3#6711yescharacteristic
101C4×C3⋊D3C22×D311yescharacteristic
102#634—22no—
103#634—22no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—61no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—122no—
109C3×C4×D3—61no—
110C3×C4×D3—61no—
111C3×C4×D3—61no—
112C3×C4×D3—61no—
113C3×C4×D3—61no—
114C3×C4×D3—61no—
115C3×C4×D3—61no—
116C3×C4×D3—61no—
117C3×Dic6—62no—
118C3×Dic6—62no—
119C3×Dic6—62no—
120C3×Dic6—62no—
121C3×Dic6—122no—
122C3×Dic6—122no—
123#627—64no—
124#627—64no—
125#627—62no—
126#627—62no—
127#624—34no—
128#624—62no—
129D3×Dic3—64no—
130D3×Dic3—64no—
131#622—22no—
132C3⋊(C3⋊C8)—22no—
133C3×C3⋊C8—62no—
134C3×C3⋊C8—62no—
135C3×C3⋊C8—122no—
136C2×#101—271noSylow
137C2×C33#10112yescomplemented
138C2×C33C2×C22⋊C411yescharacteristic
139C3×C3⋊D3—21no—
140C3×C3⋊D3—21no—
141C2×C3×Q8—181no—
142C22×C3×C4—91no—
143C22×Dic3—92no—
144D3×Q8—182no—
145D3×Q8—182no—
146D3×Q8—182no—
147D3×Q8—182no—
148C2×C4×D3—61no—
149C2×C4×D3—61no—
150C2×C4×D3—182no—
151C2×Dic6—61no—
152C2×Dic6—61no—
153C2×Dic6—92no—
154C2×Dic6—92no—
155C2×Dic6—121no—
156C2×Dic6—181no—
157C2×Dic6—181no—
158C2×Dic6—181no—
159C2×Dic6—182no—
160C3×M16—92no—
161C3×M16—181no—
162C2×C3×C8—181no—
163C2×C3×C8—181no—
164#208—92no—
165#208—181no—
166C2×C3⋊C8—61no—
167C2×C3⋊C8—61no—
168C2×C3⋊C8—121no—
169ΓL(1,𝔽25)—182no—
170ΓL(1,𝔽25)—182no—
171C22×C32—21no—
172C22×C32—21no—
173C22×C32—41no—
174C22×C32—41no—
175C22×C32(C2×D3)⋊C411yescharacteristic
176C2×C3⋊D3—22no—
177C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
178C2×C3⋊D3C2×C4×D311yescharacteristic
179C2×C3×D3—61no—
180C2×C3×D3—61no—
181C2×C3×D3—61no—
182C2×C3×D3—61no—
183C2×C3×D3—64no—
184C2×C3×D3—122no—
185C32×C4—21no—
186C32×C4—21no—
187C32×C4—21no—
188C32×C4—21no—
189C32×C4—41no—
190C32×C4—41no—
191C32×C4—41no—
192C32×C4—41no—
193C32×C4(C2×D3)⋊C411yescharacteristic
194C32×C4C2×D1211yescharacteristic
195C3⋊Dic3—24no—
196C3⋊Dic3(C2×D3)⋊C411yescharacteristic
197C3⋊Dic3C2×#6711yescharacteristic
198C3×Dic3—61no—
199C3×Dic3—61no—
200C3×Dic3—61no—
201C3×Dic3—61no—
202C3×Dic3—64no—
203C3×Dic3—64no—
204C3×Dic3—124no—
205C22×Q8—271no—
206C2×M16—91no—
207C2×M16—271no—
208#101—274no—
209C33C2×#10111yescharacteristic, complemented, Sylow
210C23×C3—91no—
211C22×D3—61no—
212C22×D3—61no—
213C3×Q8—182no—
214C3×Q8—182no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—91no—
220C2×C3×C4—91no—
221C2×C3×C4—91no—
222C2×C3×C4—91no—
223C2×C3×C4—91no—
224C2×C3×C4—182no—
225C2×C3×C4GO+(4,𝔽2)11yescharacteristic
226C2×Dic3—61no—
227C2×Dic3—61no—
228C2×Dic3—62no—
229C2×Dic3—62no—
230C2×Dic3—94no—
231C2×Dic3—122no—
232C2×Dic3—182no—
233C4×D3—61no—
234C4×D3—61no—
235C4×D3—61no—
236C4×D3—61no—
237C4×D3—61no—
238C4×D3—61no—
239C4×D3—61no—
240C4×D3—61no—
241C4×D3—184no—
242C4×D3—184no—
243Dic6—62no—
244Dic6—62no—
245Dic6—62no—
246Dic6—62no—
247Dic6—122no—
248Dic6—122no—
249Dic6—182no—
250Dic6—182no—
251Dic6—184no—
252Dic6—184no—
253C3×C8—182no—
254C3×C8—182no—
255C3⋊C8—62no—
256C3⋊C8—62no—
257C3⋊C8—122no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—22no—
261C2×C32—22no—
262C2×C32—42no—
263C2×C32—42no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32#82412yescomplemented
267C2×C32C2×(C2×D3)⋊C411yescharacteristic
268C3⋊D3—21no—
269C3⋊D3—21no—
270C3×D3—62no—
271C3×D3—62no—
272C3×D3—121no—
273C3×D3—121no—
274C2×Q8—181no—
275C2×Q8—272no—
276C2×Q8—272no—
277C2×Q8—541no—
278C2×Q8—541no—
279C2×Q8—541no—
280C22×C4—91no—
281C22×C4—272no—
282M16—92no—
283M16—181no—
284M16—272no—
285M16—541no—
286C2×C8—181no—
287C2×C8—181no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—91no—
293C22×C3—91no—
294C22×C3—182no—
295C22×C3D32⋊C411yescharacteristic
296C2×D3—61no—
297C2×D3—61no—
298C2×D3—61no—
299C2×D3—61no—
300C2×D3—64no—
301C2×D3—122no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—41no—
307C3×C4—41no—
308C3×C4—41no—
309C3×C4—41no—
310C3×C4—91no—
311C3×C4—91no—
312C3×C4—184no—
313C3×C4D32⋊C411yescharacteristic
314C3×C4C2×GO+(4,𝔽2)11yescharacteristic
315Dic3—61no—
316Dic3—61no—
317Dic3—61no—
318Dic3—61no—
319Dic3—64no—
320Dic3—64no—
321Dic3—124no—
322C32—21no—
323C32—21no—
324C32—41no—
325C32—41no—
326C32C2×#82411yescharacteristic, complemented
327C23—91no—
328Q8—182no—
329Q8—182no—
330Q8—542no—
331Q8—542no—
332C2×C4—91no—
333C2×C4—91no—
334C2×C4—91no—
335C2×C4—91no—
336C2×C4—91no—
337C2×C4—182no—
338C2×C4—274no—
339C2×C4—542no—
340C2×C4#157⋊D311yescharacteristic
341C8—182no—
342C8—182no—
343C2×C3—22no—
344C2×C3—22no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—42no—
348C2×C3—42no—
349C2×C3—41no—
350C2×C3—41no—
351C2×C3—181no—
352C2×C3—181no—
353C2×C3#6372812yescomplemented
354C2×C3C2×D32⋊C411yescharacteristic
355D3—62no—
356D3—62no—
357D3—121no—
358D3—121no—
359C22—91no—
360C22—91no—
361C22—182no—
362C22#8893911yescharacteristic
363C4—91no—
364C4—91no—
365C4—184no—
366C4#8893911yescharacteristic
367C4C2×#157⋊D311yescharacteristic
368C3—21no—
369C3—21no—
370C3—41no—
371C3—41no—
372C3C2×#6372811yescharacteristic, complemented
373C2—181no—
374C2—181no—
375C2#1172278212yesdirect product factor
376C2C2×#8893911yescharacteristic
subgroupG/H
centerC22#88939
commutatorC2×C3×C3⋊D3C22×C4
FrattiniC4C2×#157⋊D3
FittingC2×C33×C4D4
socleC22×C33C22⋊C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4#157⋊D3
hypocenterC33C2×#101
LCSG ▷ C2×C3×C3⋊D3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722782C2direct (N × H)
137. C2×C33#101semidirect
209. C33C2×#101semidirect
266. C2×C32#824semidirect
326. C32C2×#824semidirect
353. C2×C3#63728semidirect
372. C3C2×#63728semidirect
375. C2#11722782direct (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×#37882
C2×#888092
#117227822
C2×#637283
C2×#8249