Finite Group Databasebeta

← #11922140

Subgroups — #11922140

Order 1152.

437 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 437 conjugacy classes of subgroups — too many to draw. Only its 59 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
1#5812⋊C3C211yesmaximal, characteristic
2C2×C3⋊#859C211yesmaximal, characteristic
3C3×#5420C211yesmaximal, characteristic
4#10593212C211yesmaximal, characteristic, complemented
5#10593202C211yesmaximal, characteristic, complemented
6#10593199C211yesmaximal, characteristic, complemented
7#10593196C211yesmaximal, characteristic, complemented
8#85351—31nomaximal
9#78950—31nomaximal
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11#63982—41nomaximal
12Dic3×SL(2,𝔽3)C2211yescharacteristic
13Dic3×SL(2,𝔽3)C2211yescharacteristic
14C3⋊#859C2211yescharacteristic
15C3⋊#859C2211yescharacteristic
16C3×#859C2211yescharacteristic
17C3×#859C2211yescharacteristic
18#5812D311yescharacteristic
19#5519—31no—
20#519⋊C3—31no—
21#5420D311yescharacteristic
22C2×#859—31no—
23C2×#859—61no—
24#5391—31no—
25#5389—31no—
26#5389—31no—
27#5386—31no—
28C3×#419—31no—
29C2×#835—31no—
30#5121—31no—
31#5119—31no—
32#4998—31no—
33#4799—31no—
34#4799—31no—
35#4796—31no—
36#4796—31no—
37#4793—31no—
38#4793—31no—
39#4791—31no—
40#4791—31no—
41C22×C3⋊Dic3—41no—
42C2×C3×SL(2,𝔽3)Q812yescomplemented
43C2×C3×SL(2,𝔽3)C2311yescharacteristic
44C2×#627—41no—
45C3×#217—41no—
46C3×#217—41no—
47#3724—41no—
48Dic3⋊Dic3—41no—
49Dic3⋊Dic3—41no—
50#3082—91noSylow
51C22×C3×Q8C2×D311yescharacteristic
52C22×SL(2,𝔽3)—21no—
53C22×SL(2,𝔽3)C2×D311yescharacteristic
54C2×C3×M16—31no—
55C3×#117—31no—
56Q8×Dic3—31no—
57Q8×Dic3—31no—
58Q8×Dic3C2×D311yescharacteristic
59Q8×Dic3C2×D311yescharacteristic
60#926—31no—
61#926—31no—
62C2×C4⋊Dic3—31no—
63C22×C3⋊C8—31no—
64#890—31no—
65#890—31no—
66#888—31no—
67#888—31no—
68#878—61no—
69#878—61no—
70#878—121no—
71#870—61no—
72#869—61no—
73C4×SL(2,𝔽3)—31no—
74C4×SL(2,𝔽3)—31no—
75#859—31no—
76#859—31no—
77#859—61no—
78#859—61no—
79#859C2×D311yescharacteristic
80#859C2×D311yescharacteristic
81C3×#103—31no—
82C3×#103—31no—
83C3×#103—31no—
84C3×#103—31no—
85#835—31no—
86#835—31no—
87#835—31no—
88#835—31no—
89#818—31no—
90#818—31no—
91#817—31no—
92#817—31no—
93Dic3⋊C8—31no—
94Dic3⋊C8—31no—
95Dic3⋊C8—31no—
96Dic3⋊C8—31no—
97#808—31no—
98#808—31no—
99#807—31no—
100#807—31no—
101C23×C32—41no—
102C2×C3⋊Dic3—41no—
103C2×C3⋊Dic3—41no—
104C2×C3⋊Dic3—82no—
105C2×C3×Dic3—41no—
106C2×C3×Dic3—41no—
107C2×C3×Dic3—41no—
108C2×C3×Dic3—41no—
109C3×SL(2,𝔽3)C2×Q811yescharacteristic
110#627—82no—
111#544—91no—
112#519—31no—
113#479—91no—
114#479—91no—
115#479—91no—
116#479—91no—
117#477—91no—
118#477—91no—
119#475—91no—
120#475—91no—
121#428—91no—
122#427—91no—
123#424—91no—
124#419—31no—
125C2×#103—91no—
126C2×C3×Q8—31no—
127C2×C3×Q8—32no—
128C2×C3×Q8Dic612yescomplemented
129C2×C3×Q8C22×D311yescharacteristic
130C22×C3×C4—31no—
131C22×Dic3—61no—
132C22×Dic3—121no—
133C22×Dic3—241no—
134C2×Dic6—61no—
135C2×Dic6—61no—
136C2×Dic6—121no—
137C2×SL(2,𝔽3)—22no—
138C2×SL(2,𝔽3)—21no—
139C2×SL(2,𝔽3)Dic612yescomplemented
140C2×SL(2,𝔽3)C22×D311yescharacteristic
141C3×M16—62no—
142C2×C3×C8—31no—
143C2×C3×C8—31no—
144C3×C4⋊C4—31no—
145C3×C4⋊C4—31no—
146C3×C22⋊C4—61no—
147C3×C22⋊C4—121no—
148C3×C42—61no—
149#217—31no—
150#217—41no—
151#217S411yescharacteristic
152C4⋊Dic3—31no—
153C4⋊Dic3—31no—
154C4⋊Dic3—31no—
155C4⋊Dic3—31no—
156C4⋊Dic3—31no—
157C4⋊Dic3—31no—
158C4⋊Dic3—61no—
159C4⋊Dic3—61no—
160C4⋊Dic3—61no—
161C4⋊Dic3—121no—
162Dic3⋊C4—61no—
163Dic3⋊C4—61no—
164Dic3⋊C4—61no—
165Dic3⋊C4—61no—
166Dic3⋊C4—121no—
167Dic3⋊C4—121no—
168C4×Dic3—31no—
169C4×Dic3—31no—
170C4×Dic3—61no—
171C2×C3⋊C8—31no—
172C2×C3⋊C8—31no—
173C2×C3⋊C8—62no—
174C22×C32—42no—
175C22×C32—41no—
176C22×C32—81no—
177C22×C32—81no—
178C3⋊Dic3—42no—
179C3⋊Dic3—81no—
180C3×Dic3—81no—
181C3×Dic3—81no—
182C3×Dic3—81no—
183C3×Dic3—81no—
184C22×Q8D3211yescharacteristic
185C2×M16—31no—
186C22×C8—91no—
187C4⋊Q8—91no—
188C4⋊Q8—91no—
189C4⋊Q8—181no—
190#125—91no—
191#125—91no—
192#125—181no—
193C22⋊Q8—181no—
194C22⋊Q8—181no—
195C4×Q8—31no—
196C4×Q8—31no—
197C4×Q8—91no—
198C4×Q8—91no—
199#117—31no—
200#117—91no—
201#117—91no—
202C2×C4⋊C4—91no—
203#107—91no—
204#107—91no—
205#107—91no—
206#107—91no—
207#106—91no—
208#106—91no—
209#106—91no—
210#106—91no—
211C4⋊C8—91no—
212C4⋊C8—91no—
213C4⋊C8—91no—
214C4⋊C8—91no—
215#103—31no—
216#103—31no—
217#103—31no—
218#103—31no—
219#103—91no—
220#103—91no—
221#103—91no—
222#103—91no—
223C23×C3—41no—
224C23×C3—81no—
225C23×C3C2×S411yescharacteristic
226C3×Q8—32no—
227C3×Q8—31no—
228C3×Q8—61no—
229C3×Q8C2×Dic611yescharacteristic
230C2×C3×C4—32no—
231C2×C3×C4—32no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—121no—
237C2×C3×C4—121no—
238C2×Dic3—31no—
239C2×Dic3—31no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—61no—
243C2×Dic3—61no—
244C2×Dic3—121no—
245C2×Dic3—121no—
246C2×Dic3—241no—
247C2×Dic3—241no—
248C2×Dic3—241no—
249C2×Dic3—242no—
250C2×Dic3—244no—
251C2×Dic3C2×S411yescharacteristic
252C2×Dic3C2×S411yescharacteristic
253Dic6—241no—
254Dic6—241no—
255Dic6—242no—
256SL(2,𝔽3)—21no—
257SL(2,𝔽3)C2×Dic611yescharacteristic
258C3×C8—61no—
259C3×C8—61no—
260C3⋊C8—61no—
261C3⋊C8—61no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—82no—
266C2×Q8—31no—
267C2×Q8—32no—
268C2×Q8—181no—
269C2×Q8—181no—
270C2×Q8#62712yescomplemented
271C2×Q8C2×D3211yescharacteristic
272C22×C4—31no—
273C22×C4—181no—
274M16—62no—
275C2×C8—31no—
276C2×C8—31no—
277C2×C8—91no—
278C2×C8—91no—
279C2×C8—182no—
280C4⋊C4—31no—
281C4⋊C4—31no—
282C4⋊C4—91no—
283C4⋊C4—91no—
284C4⋊C4—91no—
285C4⋊C4—91no—
286C4⋊C4—91no—
287C4⋊C4—91no—
288C4⋊C4—181no—
289C4⋊C4—181no—
290C4⋊C4—181no—
291C4⋊C4—181no—
292C4⋊C4—181no—
293C4⋊C4—181no—
294C4⋊C4—181no—
295C22⋊C4—31no—
296C22⋊C4—61no—
297C22⋊C4—91no—
298C42—61no—
299C42—91no—
300C42—91no—
301C42—181no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—41no—
305C22×C3—42no—
306C22×C3—81no—
307C22×C3—81no—
308C22×C3—82no—
309C22×C3—82no—
310C22×C3—81no—
311C22×C3—82no—
312C22×C3A4⋊Q812yes—
313C22×C3C22×S411yescharacteristic
314C3×C4—31no—
315C3×C4—31no—
316C3×C4—32no—
317C3×C4—121no—
318C3×C4—121no—
319C3×C4—241no—
320C3×C4—241no—
321Dic3—21no—
322Dic3—21no—
323Dic3—61no—
324Dic3—61no—
325Dic3—81no—
326Dic3—81no—
327Dic3—122no—
328Dic3—121no—
329Dic3—121no—
330Dic3—241no—
331Dic3—242no—
332Dic3—242no—
333C32—41noSylow
334C23D3×S411yescharacteristic
335Q8—32no—
336Q8—31no—
337Q8—61no—
338Q8—721no—
339Q8—721no—
340Q8C2×#62711yescharacteristic
341C2×C4—31no—
342C2×C4—31no—
343C2×C4—32no—
344C2×C4—32no—
345C2×C4—31no—
346C2×C4—31no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—91no—
350C2×C4—91no—
351C2×C4—181no—
352C2×C4—181no—
353C2×C4—721no—
354C8—61no—
355C8—61no—
356C8—181no—
357C8—181no—
358C2×C3—22no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3—41no—
362C2×C3—82no—
363C2×C3—84no—
364C2×C3—81no—
365C2×C3—81no—
366C2×C3—81no—
367C2×C3C2×A4⋊Q811yescharacteristic
368C2×C3#592611yescharacteristic
369C2×C3#592711yescharacteristic
370C22—21no—
371C22—21no—
372C22#6420412yes—
373C22C2×D3×S411yescharacteristic
374C4—31no—
375C4—31no—
376C4—32no—
377C4—61no—
378C4—61no—
379C4—121no—
380C4—121no—
381C4—181no—
382C4—181no—
383C4—361no—
384C4—361no—
385C3—41no—
386C3—81no—
387C3#8535111yescharacteristic, complemented
388C2—22no—
389C2#1059644011yescharacteristic
390C2#1059644611yescharacteristic
391C2C2×#6420411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC22×C3×Q8C2×D3
socleC22×C3C22×S4
prime radicalC22×C3×SL(2,𝔽3)C22
hypercenterC23D3×S4
hypocenterC3×SL(2,𝔽3)C2×Q8
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
4. #10593212C2semidirect
5. #10593202C2semidirect
6. #10593199C2semidirect
7. #10593196C2semidirect
42. C2×C3×SL(2,𝔽3)Q8semidirect
128. C2×C3×Q8Dic6semidirect
139. C2×SL(2,𝔽3)Dic6semidirect
270. C2×Q8#627semidirect
387. C3#85351semidirect

Maximal subgroups

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

subgroupindex
#5812⋊C32
C2×C3⋊#8592
C3×#54202
#105932122
#105932022
#105931992
#105931962
#853513
#789503
#639824