Finite Group Databasebeta

← #11922879

Subgroups — #11922879

Order 1152.

546 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 69 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×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#10593655C211yesmaximal, characteristic
3#10593653C211yesmaximal, characteristic, complemented
4#10593629C211yesmaximal, characteristic, complemented
5#10593628C211yesmaximal, characteristic, complemented
6#10593623C211yesmaximal, characteristic, complemented
7#10593622C211yesmaximal, characteristic, complemented
8#85330—33nomaximal
9#82445—31nomaximal
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
12C2×C3⋊#226C2211yescharacteristic
13#64138—41nomaximal
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C3×C4×SL(2,𝔽3)C2211yescharacteristic
16C3⋊#859C2211yescharacteristic
17C3⋊#859C2211yescharacteristic
18C2×C3×C4×Q8D311yescharacteristic, complemented
19#5479—31no—
20C2×C4×SL(2,𝔽3)D313yescomplemented
21#5421—33no—
22#5419—33no—
23#5395—33no—
24#5394—33no—
25#5389—33no—
26#5388—33no—
27#5228—31no—
28#5227—31no—
29#5030—31no—
30#5030—31no—
31#5028—31no—
32#5028—31no—
33#5026—31no—
34#5026—31no—
35#5022—31no—
36#5022—31no—
37#4970—31no—
38#4968—31no—
39#4926—31no—
40C22×C32×C4—41no—
41C2×#638—41no—
42C2×C3×SL(2,𝔽3)—21no—
43C2×C3×SL(2,𝔽3)C2311yescharacteristic
44C3⋊GL(2,𝔽3)—21no—
45C3⋊#226—21no—
46#3757—41no—
47#3752—41no—
48#3752—41no—
49#3750—41no—
50#3750—41no—
51#3117—91noSylow
52C22×C3×Q8C2×D311yescharacteristic
53C22×SL(2,𝔽3)C2×D313yes—
54C2×GL(2,𝔽3)—33no—
55C2×#226—33no—
56C3×C4×Q8—31no—
57C3×C4×Q8—31no—
58C3×C4×Q8C2×D311yescharacteristic
59C3×C4×Q8C2×D311yescharacteristic
60C2×C3×C4⋊C4—31no—
61C2×C3×C42—31no—
62C2×#216—31no—
63C2×#216—31no—
64C2×#215—31no—
65C2×#215—31no—
66#930—31no—
67#929—61no—
68#929—123no—
69C4×#67—61no—
70#924—31no—
71#876—61no—
72#875—31no—
73C4×D12—31no—
74#872—61no—
75#870—31no—
76C4×Dic6—31no—
77C4×SL(2,𝔽3)C2×D313yes—
78C4×SL(2,𝔽3)C2×D313yes—
79#859—33no—
80#859—33no—
81#835—31no—
82#835—31no—
83#835—31no—
84#835—31no—
85#830—31no—
86#830—31no—
87#810—31no—
88#810—31no—
89#809—31no—
90#809—31no—
91#808—31no—
92#808—31no—
93#807—31no—
94#807—31no—
95C4⋊(C3⋊C8)—31no—
96C4⋊(C3⋊C8)—31no—
97#803—31no—
98#803—31no—
99C23×C32—41no—
100C22×C3⋊D3—41no—
101C2×C32×C4—41no—
102C2×C32×C4—41no—
103C2×C32×C4—82no—
104#638—82no—
105C2×C3⋊Dic3—41no—
106C2×C3⋊Dic3—41no—
107C2×C3⋊Dic3—41no—
108C3×SL(2,𝔽3)#4111yescharacteristic
109#529—91no—
110C2×C4×Q8C3⋊D311yescharacteristic, complemented
111#484—91no—
112#481—91no—
113#479—91no—
114#479—91no—
115#464—91no—
116#464—91no—
117#451—91no—
118#448—91no—
119#441—91no—
120#441—91no—
121#440—91no—
122#440—91no—
123#432—91no—
124C2×C3×Q8—21no—
125C2×C3×Q8—31no—
126C2×C3×Q8—61no—
127C2×C3×Q8C22×D311yescharacteristic
128C22×C3×C4—31no—
129C22×C3×C4—31no—
130C22×C3×C4—43no—
131C22×C3×C4S411yescharacteristic
132C2×#67—61no—
133C2×#67—123no—
134C2×D12—31no—
135C2×C4×D3—61no—
136C2×Dic6—31no—
137C2×SL(2,𝔽3)—23no—
138C2×SL(2,𝔽3)C22×D313yes—
139GL(2,𝔽3)—63no—
140#226—63no—
141C3×C4⋊C4—31no—
142C3×C4⋊C4—31no—
143C3×C4⋊C4—61no—
144C3×C42—31no—
145C3×C42—31no—
146C3×C42—31no—
147C3×C42—31no—
148#217—61no—
149#217—123no—
150#216—61no—
151#216—62no—
152#216—61no—
153#215—61no—
154#215—62no—
155#215—61no—
156(C2×D3)⋊C4—61no—
157(C2×D3)⋊C4—61no—
158(C2×D3)⋊C4—61no—
159(C2×D3)⋊C4—61no—
160(C2×D3)⋊C4—123no—
161(C2×D3)⋊C4—123no—
162C4⋊Dic3—31no—
163C4⋊Dic3—31no—
164Dic3⋊C4—61no—
165Dic3⋊C4—61no—
166Dic3⋊C4—61no—
167Dic3⋊C4—61no—
168Dic3⋊C4—123no—
169Dic3⋊C4—123no—
170C4×Dic3—61no—
171C2×C3⋊C8—31no—
172C2×C3⋊C8—31no—
173C2×C3⋊C8—31no—
174C2×C3⋊C8—31no—
175C22×C32—42no—
176C22×C32—41no—
177C22×C32—81no—
178C22×C32—81no—
179C2×C3⋊D3—81no—
180C2×C3⋊D3—81no—
181C2×C3⋊D3—81no—
182C32×C4—81no—
183C32×C4—81no—
184C3⋊Dic3—81no—
185C3⋊Dic3—81no—
186C3⋊Dic3—81no—
187C22×Q8C2×C3⋊D311yescharacteristic
188C2×Q16—91no—
189C2×Q16—91no—
190C2×SD16—91no—
191C2×SD16—91no—
192#126—181no—
193#125—91no—
194#124—91no—
195#123—181no—
196C22⋊Q8—91no—
197#121—91no—
198C4×Q8—31no—
199C4×Q8—31no—
200C4×Q8—91no—
201C4×Q8C2×C3⋊D311yescharacteristic
202C4×Q8C2×C3⋊D311yescharacteristic
203C4×D4—91no—
204C4×D4—181no—
205#117—181no—
206C2×C4⋊C4—31no—
207C2×C42—31no—
208#107—91no—
209#107—91no—
210#106—91no—
211#106—91no—
212C4⋊C8—91no—
213C4⋊C8—91no—
214#103—91no—
215#103—91no—
216#103—91no—
217#103—91no—
218#103—91no—
219#103—91no—
220#102—91no—
221#102—91no—
222C22⋊C8—91no—
223C22⋊C8—91no—
224#97—91no—
225#97—91no—
226C23×C3—43no—
227C23×C3C2×S411yescharacteristic
228C22×D3—61no—
229C22×D3—123no—
230C3×Q8—31no—
231C3×Q8—32no—
232C3×Q8—61no—
233C3×Q8#23511yescharacteristic
234C2×C3×C4—22no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—43no—
240C2×C3×C4—43no—
241C2×C3×C4—61no—
242C2×C3×C4—61no—
243C2×C3×C4—86no—
244C2×C3×C4—121no—
245C2×C3×C4C2×S411yescharacteristic
246C2×C3×C4C2×S411yescharacteristic
247#67—122no—
248#67—246no—
249C2×Dic3—61no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252C2×Dic3—123no—
253C2×Dic3—123no—
254C2×Dic3—123no—
255D12—61no—
256D12—61no—
257C4×D3—122no—
258Dic6—61no—
259Dic6—61no—
260SL(2,𝔽3)#23513yes—
261C3⋊C8—61no—
262C3⋊C8—61no—
263C3⋊C8—61no—
264C3⋊C8—61no—
265C2×C32—41no—
266C2×C32—41no—
267C2×C32—41no—
268C2×C32—82no—
269C3⋊D3—161no—
270C2×Q8—21no—
271C2×Q8—31no—
272C2×Q8—61no—
273C2×Q8—91no—
274C2×Q8C22×C3⋊D311yescharacteristic
275C2×D4—91no—
276C2×D4—181no—
277C22×C4—31no—
278C22×C4—31no—
279C22×C4—181no—
280C22×C4C3⋊S411yescharacteristic
281Q16—181no—
282Q16—182no—
283Q16—181no—
284SD16—181no—
285SD16—182no—
286SD16—181no—
287C2×C8—91no—
288C2×C8—91no—
289C2×C8—91no—
290C2×C8—91no—
291C4⋊C4—31no—
292C4⋊C4—31no—
293C4⋊C4—61no—
294C4⋊C4—91no—
295C4⋊C4—91no—
296C4⋊C4—181no—
297C4⋊C4—181no—
298C4⋊C4—181no—
299C4⋊C4—181no—
300C22⋊C4—181no—
301C22⋊C4—181no—
302C22⋊C4—181no—
303C22⋊C4—181no—
304C22⋊C4—181no—
305C42—31no—
306C42—31no—
307C42—31no—
308C42—31no—
309C42—181no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—46no—
313C22×C3—43no—
314C22×C3—83no—
315C22×C3—83no—
316C22×C3#98312yes—
317C22×C3C22×S411yescharacteristic
318C2×D3—121no—
319C2×D3—121no—
320C2×D3—121no—
321C2×D3—243no—
322C2×D3—243no—
323C2×D3—243no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—31no—
327C3×C4—31no—
328C3×C4—61no—
329C3×C4—61no—
330C3×C4—61no—
331C3×C4—83no—
332C3×C4—83no—
333Dic3—121no—
334Dic3—121no—
335Dic3—121no—
336Dic3—243no—
337Dic3—243no—
338Dic3—243no—
339C32—41noSylow
340C23—181no—
341C23C2×C3⋊S411yescharacteristic
342Q8—31no—
343Q8—32no—
344Q8—61no—
345Q8—181no—
346Q8—181no—
347Q8#382811yescharacteristic
348D4—181no—
349D4—181no—
350D4—362no—
351C2×C4—22no—
352C2×C4—31no—
353C2×C4—31no—
354C2×C4—31no—
355C2×C4—31no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—121no—
359C2×C4—181no—
360C2×C4—181no—
361C2×C4—181no—
362C2×C4—362no—
363C2×C4C2×C3⋊S411yescharacteristic
364C2×C4C2×C3⋊S411yescharacteristic
365C8—181no—
366C8—181no—
367C8—181no—
368C8—181no—
369C2×C3—22no—
370C2×C3—43no—
371C2×C3—43no—
372C2×C3—43no—
373C2×C3—86no—
374C2×C3#591211yescharacteristic
375C2×C3C2×#98311yescharacteristic
376C2×C3#592311yescharacteristic
377D3—241no—
378D3—483no—
379C22—21no—
380C22—21no—
381C22—361no—
382C22—361no—
383C22—361no—
384C22#6426412yes—
385C22C22×C3⋊S411yescharacteristic
386C4—21no—
387C4—21no—
388C4—31no—
389C4—31no—
390C4—61no—
391C4—61no—
392C4—61no—
393C4—361no—
394C4—361no—
395C4—361no—
396C3—43no—
397C3#8533011yescharacteristic, complemented
398C2—22no—
399C2—721no—
400C2#1059657511yescharacteristic
401C2C2×#6426411yescharacteristic
402C2#1059658611yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC2×C3×C4×Q8D3
socleC22×C3C22×S4
prime radical#10593655C2
hypercenterC22×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)#41
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×SL(2,𝔽3)C2semidirect
3. #10593653C2semidirect
4. #10593629C2semidirect
5. #10593628C2semidirect
6. #10593623C2semidirect
7. #10593622C2semidirect
18. C2×C3×C4×Q8D3semidirect
20. C2×C4×SL(2,𝔽3)D3semidirect
110. C2×C4×Q8C3⋊D3semidirect
397. C3#85330semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×SL(2,𝔽3)2
#105936552
#105936532
#105936292
#105936282
#105936232
#105936222
#853303
#824453
#641384