Finite Group Databasebeta

← #11922964

Subgroups — #11922964

Order 1152.

504 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C3×#557⋊C3C211yesmaximal, characteristic, complemented
2#10593628C211yesmaximal, characteristic
3#10593628C211yesmaximal, characteristic
4C4×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic
5#10593623C211yesmaximal, characteristic, complemented
6#10593623C211yesmaximal, characteristic, complemented
7#10593620C211yesmaximal, characteristic, complemented
8#85423—33nomaximal
9#83695—31nomaximal
10C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
11#64105—41nomaximal
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13C3×C4×SL(2,𝔽3)C2211yescharacteristic
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C3⋊#859C2211yescharacteristic
16C3⋊#859C2211yescharacteristic
17C3⋊#859C2211yescharacteristic
18C3×#557D311yescharacteristic, complemented
19#5579—31no—
20#557⋊C3D313yescomplemented
21#5394—33no—
22#5394—33no—
23C4×GL(2,𝔽3)—33no—
24#5389—33no—
25#5389—33no—
26#5386—33no—
27#5084—31no—
28#5081—31no—
29#5066—31no—
30#5066—31no—
31#5065—31no—
32#5064—31no—
33#5026—31no—
34#5026—31no—
35C4×#215—31no—
36#5024—31no—
37#5022—31no—
38#5022—31no—
39#5020—31no—
40C2×C4×C3⋊D3—41no—
41C2×#634—41no—
42C2×C3×SL(2,𝔽3)C2311yescharacteristic
43C3⋊GL(2,𝔽3)Q812yescomplemented
44C32×C4⋊C4—41no—
45#3752—41no—
46#3752—41no—
47#3750—41no—
48#3750—41no—
49#3388—91noSylow
50C2×GL(2,𝔽3)—33no—
51C3×C4⋊Q8—31no—
52C3×#125—31no—
53C3×#125—31no—
54C3×C4×Q8—31no—
55C3×C4×Q8C2×D311yescharacteristic
56C3×C4×Q8C2×D311yescharacteristic
57C3×C4×Q8C2×D311yescharacteristic
58#946—61no—
59C2×#215—31no—
60#896—61no—
61#896—61no—
62#896—123no—
63D3×C4⋊C4—61no—
64C4×D12—31no—
65C4×D12—31no—
66C4×D12—31no—
67#870—31no—
68#870—31no—
69#869—31no—
70C4×SL(2,𝔽3)C2×D313yes—
71C4×SL(2,𝔽3)C2×D313yes—
72C4×SL(2,𝔽3)C2×D313yes—
73#859—33no—
74#859—33no—
75#859—33no—
76#835—31no—
77#835—31no—
78#835—31no—
79#809—31no—
80#809—31no—
81#809—31no—
82#808—31no—
83#808—31no—
84#808—31no—
85#808—31no—
86#808—31no—
87#807—31no—
88#807—31no—
89#807—31no—
90#807—31no—
91C4⋊(C3⋊C8)—31no—
92C4⋊(C3⋊C8)—31no—
93C4⋊(C3⋊C8)—31no—
94#803—31no—
95#803—31no—
96C4×C3⋊C8—31no—
97C22×C3⋊D3—41no—
98C2×C32×C4—41no—
99C2×C32×C4—41no—
100C2×C32×C4—41no—
101C2×C3⋊Dic3—41no—
102C2×C3⋊Dic3—41no—
103C2×C3⋊Dic3—41no—
104C4×C3⋊D3—82no—
105#634—82no—
106C3×SL(2,𝔽3)C2×Q811yescharacteristic
107#557C3⋊D311yescharacteristic, complemented
108#554—91no—
109#501—91no—
110#501—91no—
111#499—91no—
112#479—91no—
113#479—91no—
114#478—91no—
115#478—91no—
116#476—91no—
117#475—91no—
118#446—91no—
119#440—91no—
120#440—91no—
121C4×SD16—91no—
122C2×C3×Q8—31no—
123C2×C3×Q8C22×D311yescharacteristic
124C2×D12—31no—
125C2×C4×D3—61no—
126C2×C4×D3—61no—
127C2×C4×D3—61no—
128C2×C4×D3—123no—
129C2×Dic6—61no—
130C2×Dic6—123no—
131C2×SL(2,𝔽3)C22×D313yes—
132GL(2,𝔽3)—36no—
133C3×C4⋊C4—31no—
134C3×C4⋊C4—31no—
135C3×C4⋊C4—31no—
136C3×C4⋊C4—31no—
137C3×C4⋊C4—31no—
138C3×C4⋊C4—43no—
139C3×C4⋊C4—61no—
140C3×C4⋊C4S411yescharacteristic
141C3×C42—31no—
142C3×C42—31no—
143C3×C42—31no—
144#215—32no—
145#215—32no—
146(C2×D3)⋊C4—61no—
147(C2×D3)⋊C4—61no—
148(C2×D3)⋊C4—61no—
149(C2×D3)⋊C4—123no—
150(C2×D3)⋊C4—123no—
151C4⋊Dic3—31no—
152C4⋊Dic3—31no—
153C4⋊Dic3—31no—
154C4⋊Dic3—61no—
155Dic3⋊C4—61no—
156Dic3⋊C4—61no—
157Dic3⋊C4—61no—
158Dic3⋊C4—61no—
159Dic3⋊C4—123no—
160Dic3⋊C4—123no—
161C2×C3⋊C8—31no—
162C2×C3⋊C8—31no—
163C2×C3⋊C8—31no—
164C2×C3⋊C8—31no—
165C22×C32—41no—
166C2×C3⋊D3—42no—
167C2×C3⋊D3—81no—
168C2×C3⋊D3—81no—
169C32×C4—81no—
170C32×C4—81no—
171C32×C4—81no—
172C3⋊Dic3—42no—
173C3⋊Dic3—81no—
174C3⋊Dic3—81no—
175C2×SD16—91no—
176C4⋊Q8—31no—
177C4⋊Q8—91no—
178#125—31no—
179#125—31no—
180#125—91no—
181#125—91no—
182C22⋊Q8—181no—
183C22⋊Q8—181no—
184C22⋊Q8—181no—
185C4×Q8—31no—
186C4×Q8C2×C3⋊D311yescharacteristic
187C4×Q8C2×C3⋊D311yescharacteristic
188C4×Q8C2×C3⋊D311yescharacteristic
189C4×D4—91no—
190C4×D4—91no—
191C4×D4—91no—
192C2×C4⋊C4—181no—
193#107—91no—
194#107—91no—
195#107—91no—
196#107—91no—
197#106—91no—
198#106—91no—
199#106—91no—
200#106—91no—
201#106—91no—
202C4⋊C8—91no—
203C4⋊C8—91no—
204C4⋊C8—91no—
205#103—91no—
206#103—91no—
207#103—91no—
208#102—91no—
209#102—91no—
210#102—91no—
211#97—91no—
212#97—91no—
213C4×C8—91no—
214C22×D3—61no—
215C22×D3—123no—
216C3×Q8—31no—
217C3×Q8—121no—
218C3×Q8D3×Q811yescharacteristic
219C2×C3×C4—31no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—31no—
223C2×C3×C4—43no—
224C2×C3×C4—43no—
225C2×C3×C4—43no—
226C2×C3×C4C2×S411yescharacteristic
227C2×C3×C4C2×S411yescharacteristic
228C2×C3×C4C2×S411yescharacteristic
229C2×Dic3—61no—
230C2×Dic3—61no—
231C2×Dic3—61no—
232C2×Dic3—123no—
233C2×Dic3—123no—
234C2×Dic3—123no—
235D12—32no—
236D12—61no—
237C4×D3—122no—
238C4×D3—241no—
239C4×D3—241no—
240C4×D3—246no—
241Dic6—241no—
242Dic6—246no—
243SL(2,𝔽3)D3×Q813yes—
244C3⋊C8—32no—
245C3⋊C8—61no—
246C3⋊C8—61no—
247C3⋊C8—61no—
248C2×C32—41no—
249C2×C32—41no—
250C2×C32—41no—
251C3⋊D3—82no—
252C2×Q8—31no—
253C2×Q8—181no—
254C2×Q8C22×C3⋊D311yescharacteristic
255C2×D4—91no—
256C22×C4—181no—
257C22×C4—181no—
258C22×C4—181no—
259SD16—92no—
260SD16—92no—
261C2×C8—91no—
262C2×C8—91no—
263C2×C8—91no—
264C2×C8—91no—
265C4⋊C4—31no—
266C4⋊C4—31no—
267C4⋊C4—31no—
268C4⋊C4—31no—
269C4⋊C4—31no—
270C4⋊C4—61no—
271C4⋊C4—91no—
272C4⋊C4—91no—
273C4⋊C4—91no—
274C4⋊C4—181no—
275C4⋊C4—181no—
276C4⋊C4—181no—
277C4⋊C4—181no—
278C4⋊C4—181no—
279C4⋊C4C3⋊S411yescharacteristic
280C22⋊C4—181no—
281C22⋊C4—181no—
282C22⋊C4—181no—
283C42—31no—
284C42—31no—
285C42—31no—
286C22×C3—43no—
287C22×C3C22×S411yescharacteristic
288C2×D3—62no—
289C2×D3—126no—
290C2×D3—121no—
291C2×D3—121no—
292C2×D3—243no—
293C2×D3—243no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—31no—
298C3×C4—31no—
299C3×C4—61no—
300C3×C4—61no—
301C3×C4—61no—
302C3×C4—83no—
303C3×C4—83no—
304C3×C4—83no—
305Dic3—126no—
306Dic3—121no—
307Dic3—121no—
308Dic3—121no—
309Dic3—243no—
310Dic3—243no—
311C32—41noSylow
312C23—181no—
313Q8—31no—
314Q8—121no—
315Q8—721no—
316Q8Q8×C3⋊D311yescharacteristic
317D4—92no—
318D4—181no—
319C2×C4—31no—
320C2×C4—31no—
321C2×C4—31no—
322C2×C4—31no—
323C2×C4—181no—
324C2×C4—181no—
325C2×C4—181no—
326C2×C4—362no—
327C2×C4—721no—
328C2×C4—721no—
329C2×C4C2×C3⋊S411yescharacteristic
330C2×C4C2×C3⋊S411yescharacteristic
331C2×C4C2×C3⋊S411yescharacteristic
332C8—92no—
333C8—181no—
334C8—181no—
335C8—181no—
336C2×C3—43no—
337C2×C3—43no—
338C2×C3—43no—
339C2×C3Q8×S411yescharacteristic
340C2×C3#592311yescharacteristic
341C2×C3#592611yescharacteristic
342D3—122no—
343D3—246no—
344C22—182no—
345C22—361no—
346C22—361no—
347C22C22×C3⋊S411yescharacteristic
348C4—21no—
349C4—21no—
350C4—21no—
351C4—31no—
352C4—31no—
353C4—61no—
354C4—61no—
355C4—61no—
356C4—361no—
357C4—361no—
358C4—361no—
359C3—43no—
360C3#8542311yescharacteristic, complemented
361C2—362no—
362C2Q8×C3⋊S411yescharacteristic
363C2#1059658611yescharacteristic
364C2#1059658911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×#557D3
socleC22×C3C22×S4
prime radicalC2×C3⋊GL(2,𝔽3)C22
hypercenterC4⋊C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×Q8
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#557⋊C3C2semidirect
5. #10593623C2semidirect
6. #10593623C2semidirect
7. #10593620C2semidirect
18. C3×#557D3semidirect
20. #557⋊C3D3semidirect
43. C3⋊GL(2,𝔽3)Q8semidirect
107. #557C3⋊D3semidirect
360. C3#85423semidirect

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
#105936282
C4×C3⋊GL(2,𝔽3)2
#105936232
#105936202
#854233
#836953
#641054