Finite Group Databasebeta

← #11923089

Subgroups — #11923089

Order 1152.

772 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 81 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×C3×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C2×#64266C211yesmaximal, characteristic, complemented
3#10593660C211yesmaximal, characteristic, complemented
4#10593655C212yesmaximal, complemented
5#10593653C212yesmaximal, complemented
6#85538—33nomaximal
7#84114—31nomaximal
8C3×2−1+4⋊C3—22no—
9C2×C3×#41⋊C3C2211yescharacteristic
10C22×C3×SL(2,𝔽3)C2212yes—
11#64266—22no—
12C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
13C2×C3⋊#226C2211yescharacteristic
14#64147—41nomaximal
15C3⋊#859C2212yes—
16C2×C3×2−1+4D311yescharacteristic, complemented
17C2×2−1+4⋊C3D313yescomplemented
18C2×#985—33no—
19#5833—31no—
20C2×#950—31no—
21C2×#942—31no—
22#5426—33no—
23#5421—36no—
24#5419—36no—
25#5239—32no—
26#5238—32no—
27#5234—31no—
28#5232—31no—
29#5228—32no—
30#5227—32no—
31C2×#835—31no—
32C2×C32×D4—41no—
33C2×#638—42no—
34C2×C4×C3⋊D3—41no—
35C3×#41⋊C3D412yescomplemented
36C2×C3×SL(2,𝔽3)—22no—
37C2×C3×SL(2,𝔽3)C2311yescharacteristic
38C3⋊GL(2,𝔽3)—21no—
39C3⋊#226—21no—
40#3757—42no—
41C4⋊(C3⋊Dic3)—41no—
42#3013—91noSylow
43C3×2−1+4—22no—
44C3×2−1+4—62no—
45C2×C3×#41—31no—
46C2×C3×#41—32no—
47C2×C3×#41C2×D311yescharacteristic
48C22×C3×Q8—31no—
49C22×C3×Q8C2×D312yes—
50C2×#235—31no—
512−1+4⋊C3—26no—
52C2×#41⋊C3C2×D313yes—
53C22×SL(2,𝔽3)C2×D316yes—
54#985—66no—
55C2×GL(2,𝔽3)—33no—
56C2×#226—33no—
57#950—62no—
58#950—62no—
59#946—61no—
60C2×#216—32no—
61C2×#216—31no—
62#942—64no—
63C2×#215—32no—
64C2×#215—31no—
65#939—62no—
66#938—61no—
67#938—123no—
68C2×#214—31no—
69C2×#213—31no—
70#930—32no—
71C2×C4⋊Dic3—31no—
72#924—32no—
73C2×#208—31no—
74C22×C3⋊C8—31no—
75#859—36no—
76#835—32no—
77#835—32no—
78#835—32no—
79#832—32no—
80#830—32no—
81#830—32no—
82C23×C32—42no—
83C22×C3⋊D3—41no—
84C32×D4—82no—
85C2×C32×C4—41no—
86#638—84no—
87C2×C3⋊Dic3—41no—
88C2×C3⋊Dic3—42no—
89C4×C3⋊D3—82no—
90C3×SL(2,𝔽3)C2×D411yescharacteristic
91C2×2−1+4C3⋊D311yescharacteristic, complemented
92C2×#137—91no—
93C2×#135—91no—
94#537—91no—
95#452—92no—
96#451—92no—
97#449—92no—
98#448—92no—
99#417—91no—
100C2×#103—91no—
101#408—91no—
102C3×#41—32no—
103C3×#41—62no—
104C3×#41—62no—
105C3×#41—64no—
106C3×#41—64no—
107C3×#41#6712yescomplemented
108C2×C3×Q8—22no—
109C2×C3×Q8—31no—
110C2×C3×Q8—32no—
111C2×C3×Q8—34no—
112C2×C3×Q8—62no—
113C2×C3×Q8—62no—
114C2×C3×Q8C22×D311yescharacteristic
115C2×C3×D4—31no—
116C2×C3×D4—32no—
117C2×C3×D4—43no—
118C2×C3×D4S411yescharacteristic
119C22×C3×C4—31no—
120C22×C3×C4—32no—
121C22×C3×C4—61no—
122C2×#67—61no—
123C2×#67—62no—
124C2×#67—126no—
125C22×Dic3—61no—
126#235—62no—
127#235—62no—
128C2×D12—31no—
129C2×C4×D3—61no—
130C2×C4×D3—123no—
131C2×Dic6—31no—
132#41⋊C3#6716yescomplemented
133C2×SL(2,𝔽3)—26no—
134C2×SL(2,𝔽3)C22×D313yes—
135GL(2,𝔽3)—63no—
136#226—63no—
137#217—62no—
138#217—126no—
139#216—64no—
140#216—61no—
141#216—61no—
142#215—64no—
143#215—61no—
144#215—61no—
145#214—62no—
146#213—62no—
147(C2×D3)⋊C4—62no—
148C4⋊Dic3—32no—
149C4⋊Dic3—61no—
150C4⋊Dic3—123no—
151Dic3⋊C4—62no—
152#208—62no—
153C2×C3⋊C8—32no—
154C2×C3⋊C8—31no—
155C2×C3⋊C8—31no—
156C2×C3⋊C8—62no—
157C22×C32—41no—
158C22×C32—82no—
159C22×C32—82no—
160C22×C32—82no—
161C2×C3⋊D3—42no—
162C2×C3⋊D3—81no—
163C2×C3⋊D3—81no—
164C32×C4—42no—
165C3⋊Dic3—81no—
166C3⋊Dic3—82no—
1672−1+4—22no—
1682−1+4—62no—
169C2×#41—31no—
170C2×#41—32no—
171C2×#41—91no—
172C2×#41C2×C3⋊D311yescharacteristic
173C22×Q8—31no—
174C22×Q8C2×C3⋊D312yes—
175#137—184no—
176#135—182no—
177#135—182no—
178C2×Q16—92no—
179C2×Q16—91no—
180C2×SD16—91no—
181C2×SD16—92no—
182C2×SD16—91no—
183C2×D8—91no—
184C2×M16—91no—
185C22×C8—91no—
186C22⋊Q8—92no—
187C22⋊Q8—181no—
188#121—92no—
189#121—182no—
190#121—181no—
191C2×C4⋊C4—91no—
192#103—92no—
193#103—92no—
194#103—92no—
195#102—92no—
196C22⋊C8—92no—
197C22⋊C8—92no—
198C23×C3—31no—
199C23×C3—46no—
200C23×C3C2×S412yes—
201C22×D3—61no—
202C22×D3—123no—
203C3×Q8—31no—
204C3×Q8—34no—
205C3×Q8—62no—
206C3×Q8—62no—
207C3×Q8C2×#6711yescharacteristic
208C3×D4—22no—
209C3×D4—32no—
210C3×D4—61no—
211C3×D4—64no—
212C3×D4—86no—
213C2×C3×C4—31no—
214C2×C3×C4—32no—
215C2×C3×C4—32no—
216C2×C3×C4—32no—
217C2×C3×C4—43no—
218C2×C3×C4—62no—
219C2×C3×C4—62no—
220C2×C3×C4—64no—
221C2×C3×C4C2×S411yescharacteristic
222#67—122no—
223#67—242no—
224#67—2412no—
225C2×Dic3—61no—
226C2×Dic3—62no—
227C2×Dic3—126no—
228C2×Dic3—123no—
229C2×Dic3—241no—
230D12—61no—
231D12—61no—
232C4×D3—122no—
233C4×D3—246no—
234Dic6—61no—
235Dic6—61no—
236SL(2,𝔽3)C2×#6713yes—
237C3⋊C8—61no—
238C3⋊C8—61no—
239C3⋊C8—62no—
240C2×C32—41no—
241C2×C32—41no—
242C2×C32—41no—
243C2×C32—162no—
244C3⋊D3—82no—
245#41—32no—
246#41—62no—
247#41—62no—
248#41—64no—
249#41—64no—
250#41—182no—
251#41—182no—
252#41#63812yescomplemented
253C2×Q8—22no—
254C2×Q8—31no—
255C2×Q8—32no—
256C2×Q8—34no—
257C2×Q8—62no—
258C2×Q8—62no—
259C2×Q8—91no—
260C2×Q8C22×C3⋊D311yescharacteristic
261C2×D4—31no—
262C2×D4—32no—
263C2×D4—91no—
264C2×D4—181no—
265C2×D4—182no—
266C2×D4C3⋊S411yescharacteristic
267C22×C4—31no—
268C22×C4—32no—
269C22×C4—61no—
270C22×C4—181no—
271C22×C4—181no—
272Q16—181no—
273Q16—184no—
274Q16—181no—
275SD16—182no—
276SD16—181no—
277SD16—184no—
278SD16—181no—
279D8—182no—
280M16—182no—
281C2×C8—92no—
282C2×C8—91no—
283C2×C8—91no—
284C2×C8—182no—
285C4⋊C4—92no—
286C4⋊C4—182no—
287C4⋊C4—181no—
288C22⋊C4—182no—
289C22⋊C4—182no—
290C22×C3—22no—
291C22×C3—22no—
292C22×C3—22no—
293C22×C3—32no—
294C22×C3—43no—
295C22×C3—61no—
296C22×C3—61no—
297C22×C3—86no—
298C22×C3—86no—
299C22×C3—86no—
300C22×C3C22×S411yescharacteristic
301C2×D3—126no—
302C2×D3—121no—
303C2×D3—121no—
304C2×D3—121no—
305C2×D3—243no—
306C2×D3—243no—
307C3×C4—31no—
308C3×C4—31no—
309C3×C4—46no—
310C3×C4—62no—
311C3×C4GL(2,ℤ/4)12yes—
312Dic3—122no—
313Dic3—121no—
314Dic3—243no—
315Dic3—246no—
316C32—41noSylow
317C23—31no—
318C23—181no—
319C23C2×C3⋊S412yes—
320Q8—31no—
321Q8—34no—
322Q8—62no—
323Q8—62no—
324Q8—181no—
325Q8—181no—
326Q8C2×#63811yescharacteristic
327D4—22no—
328D4—32no—
329D4—61no—
330D4—64no—
331D4—181no—
332D4—181no—
333D4—362no—
334D4—722no—
335C2×C4—31no—
336C2×C4—32no—
337C2×C4—32no—
338C2×C4—32no—
339C2×C4—62no—
340C2×C4—62no—
341C2×C4—64no—
342C2×C4—182no—
343C2×C4—181no—
344C2×C4—362no—
345C2×C4—721no—
346C2×C4C2×C3⋊S411yescharacteristic
347C8—181no—
348C8—181no—
349C8—182no—
350C2×C3—42no—
351C2×C3—43no—
352C2×C3—43no—
353C2×C3—43no—
354C2×C3—62no—
355C2×C3—166no—
356C2×C3#592611yescharacteristic
357C2×C3#592711yescharacteristic
358C2×C3C2×GL(2,ℤ/4)11yescharacteristic
359D3—246no—
360D3—241no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—32no—
365C22—61no—
366C22—61no—
367C22—361no—
368C22—361no—
369C22—361no—
370C22C22×C3⋊S411yescharacteristic
371C4—31no—
372C4—31no—
373C4—62no—
374C4—362no—
375C4—361no—
376C4#6426912yes—
377C3—43no—
378C3#8553811yescharacteristic, complemented
379C2—42no—
380C2—62no—
381C2—721no—
382C2#1059658911yescharacteristic
383C2#1059659011yescharacteristic
384C2C2×#6426911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC2×C3×2−1+4D3
socleC22×C3C22×S4
hypercenterC2×D4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×2−1+4⋊C3C2semidirect
2. C2×#64266C2semidirect
3. #10593660C2semidirect
4. #10593655C2semidirect
5. #10593653C2semidirect
16. C2×C3×2−1+4D3semidirect
17. C2×2−1+4⋊C3D3semidirect
35. C3×#41⋊C3D4semidirect
91. C2×2−1+4C3⋊D3semidirect
107. C3×#41#67semidirect
132. #41⋊C3#67semidirect
252. #41#638semidirect
378. C3#85538semidirect

Maximal subgroups

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

subgroupindex
C2×C3×2−1+4⋊C32
C2×#642662
#105936602
#105936552
#105936532
#855383
#841143
#641474