Finite Group Databasebeta

← #11922226

Subgroups — #11922226

Order 1152.

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

normal subgroups only

This group has 487 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
1SL(2,𝔽3)×#67C211yesmaximal, characteristic
2C2×C3⋊#859C211yesmaximal, characteristic, complemented
3C3×#5419C211yesmaximal, characteristic, complemented
4D3×#859C211yesmaximal, characteristic, complemented
5#10593210C211yesmaximal, characteristic, complemented
6#10593202C211yesmaximal, characteristic, complemented
7Dic3×#226C211yesmaximal, characteristic, complemented
8#85393—31nomaximal
9#80193—31nomaximal
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C2×D3×SL(2,𝔽3)C2211yescharacteristic
12C2×C3×#226C2211yescharacteristic
13#67⋊Dic3—41nomaximal
14Dic3×SL(2,𝔽3)C2211yescharacteristic
15C3⋊#859C2211yescharacteristic
16C3⋊#859C2211yescharacteristic
17C3×#859C2211yescharacteristic
18Q8×#67D311yescharacteristic
19#5635—31no—
20D4×SL(2,𝔽3)—31no—
21#5419—31no—
22#5419D311yescharacteristic, complemented
23C2×#859—31no—
24C2×#859—31no—
25C2×#859—61no—
26#5391—31no—
27C4×#226—31no—
28C3×#451—31no—
29C2×#835—31no—
30#5188—31no—
31#5186—31no—
32#5182—31no—
33Dic3×Q16—31no—
34#5051—31no—
35#4811—31no—
36D3×#103—31no—
37#4793—31no—
38#4792—31no—
39#67⋊C8—31no—
40#4722—31no—
41C22×C3⋊Dic3—41no—
42C2×C3×#67—41no—
43C2×C3×SL(2,𝔽3)—21no—
44C2×C3×SL(2,𝔽3)C2311yescharacteristic
45C2×D3×Dic3—41no—
46D3×SL(2,𝔽3)—21no—
47C3×#226—21no—
48C3×#217—41no—
49Dic3⋊Dic3—41no—
50(C2×D3)⋊Dic3—41no—
51Dic32—41no—
52#3299—91noSylow
53C22×C3×Q8C2×D311yescharacteristic
54C2×D3×Q8C2×D311yescharacteristic
55C22×SL(2,𝔽3)—21no—
56C22×SL(2,𝔽3)—31no—
57C22×SL(2,𝔽3)C2×D311yescharacteristic
58C2×#226C2×D311yescharacteristic
59C2×C3×Q16—31no—
60C2×C3×Q16—31no—
61C3×C22⋊Q8—31no—
62#946—31no—
63#946—31no—
64Q8×Dic3—31no—
65Q8×Dic3C2×D311yescharacteristic
66#944—31no—
67C2×#216—31no—
68D4×Dic3—121no—
69C4×#67—31no—
70C2×C4⋊Dic3—31no—
71#924—31no—
72C22×C3⋊C8—31no—
73C2×D3×C8—31no—
74#897—61no—
75D3×C4⋊C4—31no—
76#889—61no—
77#888—31no—
78#67⋊C4—61no—
79#878—61no—
80C4×SL(2,𝔽3)—31no—
81#859—31no—
82#859—31no—
83#859—61no—
84#859—61no—
85#859C2×D311yescharacteristic
86C3×#103—31no—
87C3×#103—31no—
88C3×C22⋊C8—31no—
89#835—31no—
90#835—31no—
91#835—31no—
92#835—31no—
93#835—31no—
94(C2×D3)⋊C8—31no—
95#818—31no—
96#818—31no—
97#816—31no—
98#816—31no—
99Dic3⋊C8—31no—
100C8×Dic3—31no—
101#810—31no—
102#807—31no—
103C23×C32—41no—
104C22×C3×D3—41no—
105C2×C3⋊Dic3—41no—
106C2×C3⋊Dic3—41no—
107C2×C3⋊Dic3—82no—
108C3×#67—44no—
109C2×C3×Dic3—41no—
110C2×C3×Dic3—41no—
111C2×C3×Dic3—41no—
112C3×SL(2,𝔽3)#4111yescharacteristic
113D3×Dic3—82no—
114#554—91no—
115D4×Q8—31no—
116#487—91no—
117#484—91no—
118#484—91no—
119#477—91no—
120#467—91no—
121#467—91no—
122#462—91no—
123#451—31no—
124#451—91no—
125C4×Q16—91no—
126C8×D4—91no—
127C2×#103—91no—
128C2×#103—91no—
129C2×C3×Q8—21no—
130C2×C3×Q8—31no—
131C2×C3×Q8—31no—
132C2×C3×Q8—61no—
133C2×C3×Q8C22×D311yescharacteristic
134C2×C3×D4—121no—
135C22×C3×C4—31no—
136C2×#67S411yescharacteristic
137C22×Dic3—61no—
138C22×Dic3—121no—
139C22×Dic3—121no—
140C22×Dic3—241no—
141D3×Q8—21no—
142D3×Q8—61no—
143C2×C4×D3—31no—
144C2×C4×D3—61no—
145C2×Dic6—31no—
146C2×Dic6—61no—
147C2×SL(2,𝔽3)—22no—
148C2×SL(2,𝔽3)—21no—
149C2×SL(2,𝔽3)—21no—
150C2×SL(2,𝔽3)—61no—
151C2×SL(2,𝔽3)C22×D311yescharacteristic
152#226—21no—
153C3×Q16—61no—
154C3×Q16—61no—
155C3×Q16—121no—
156C2×C3×C8—31no—
157C2×C3×C8—31no—
158C3×C4⋊C4—31no—
159C3×C4⋊C4—61no—
160C3×C22⋊C4—61no—
161#217—31no—
162#217—41no—
163#217—121no—
164#216—121no—
165(C2×D3)⋊C4—31no—
166(C2×D3)⋊C4—61no—
167C4⋊Dic3—31no—
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—61no—
171C4⋊Dic3—61no—
172C4⋊Dic3—121no—
173Dic3⋊C4—31no—
174Dic3⋊C4—61no—
175Dic3⋊C4—61no—
176Dic3⋊C4—61no—
177Dic3⋊C4—61no—
178C4×Dic3—31no—
179C4×Dic3—61no—
180C4×Dic3—121no—
181C2×C3⋊C8—31no—
182C2×C3⋊C8—31no—
183C2×C3⋊C8—121no—
184D3×C8—121no—
185C22×C32—42no—
186C22×C32—41no—
187C22×C32—81no—
188C22×C32—81no—
189C2×C3×D3—42no—
190C2×C3×D3—81no—
191C2×C3×D3—81no—
192C3⋊Dic3—42no—
193C3⋊Dic3—81no—
194C3×Dic3—42no—
195C3×Dic3—81no—
196C3×Dic3—81no—
197C22×Q8—31no—
198C22×Q8D3211yescharacteristic
199C2×Q16—31no—
200C2×Q16—31no—
201C2×Q16—91no—
202C22×C8—91no—
203C22×C8—91no—
204C4⋊Q8—91no—
205C4⋊Q8—91no—
206#125—181no—
207C22⋊Q8—31no—
208C22⋊Q8—91no—
209C22⋊Q8—91no—
210C22⋊Q8—91no—
211C22⋊Q8—181no—
212C22⋊Q8—181no—
213C4×Q8—31no—
214C4×Q8—91no—
215C4×D4—91no—
216C4×D4—181no—
217C2×C4⋊C4—91no—
218C2×C4⋊C4—91no—
219#107—91no—
220#107—91no—
221#107—91no—
222C4⋊C8—91no—
223#103—31no—
224#103—31no—
225#103—91no—
226#103—91no—
227#103—91no—
228#103—91no—
229#103—91no—
230#103—91no—
231#103—91no—
232#103—91no—
233C22⋊C8—31no—
234C22⋊C8—91no—
235C4×C8—91no—
236C23×C3—41no—
237C23×C3—81no—
238C23×C3—121no—
239C23×C3C2×S411yescharacteristic
240C22×D3C2×S411yescharacteristic
241C3×Q8—31no—
242C3×Q8—61no—
243C3×Q8—61no—
244C3×Q8—61no—
245C3×Q8—61no—
246C3×Q8#23711yescharacteristic
247C3×D4—124no—
248C2×C3×C4—31no—
249C2×C3×C4—31no—
250C2×C3×C4—61no—
251C2×C3×C4—61no—
252C2×C3×C4—61no—
253C2×C3×C4—61no—
254C2×C3×C4—121no—
255#67#22614yescomplemented
256C2×Dic3—31no—
257C2×Dic3—31no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—121no—
263C2×Dic3—121no—
264C2×Dic3—241no—
265C2×Dic3—241no—
266C2×Dic3—242no—
267C2×Dic3—241no—
268C2×Dic3—242no—
269C2×Dic3—244no—
270C2×Dic3C2×S411yescharacteristic
271C4×D3—61no—
272C4×D3—61no—
273C4×D3—241no—
274Dic6—61no—
275Dic6—61no—
276Dic6—241no—
277SL(2,𝔽3)—21no—
278SL(2,𝔽3)#23711yescharacteristic
279C3×C8—61no—
280C3×C8—61no—
281C3⋊C8—61no—
282C3⋊C8—61no—
283C2×C32—41no—
284C2×C32—41no—
285C2×C32—41no—
286C2×C32—82no—
287C3×D3—82no—
288C2×Q8—21no—
289C2×Q8—31no—
290C2×Q8—31no—
291C2×Q8—61no—
292C2×Q8—61no—
293C2×Q8—91no—
294C2×Q8—181no—
295C2×Q8—181no—
296C2×Q8C2×D3211yescharacteristic
297C2×D4—31no—
298C22×C4—31no—
299C22×C4—91no—
300C22×C4—181no—
301C22×C4—181no—
302Q16—61no—
303Q16—61no—
304Q16—121no—
305Q16—361no—
306C2×C8—31no—
307C2×C8—31no—
308C2×C8—91no—
309C2×C8—91no—
310C2×C8—361no—
311C2×C8—361no—
312C4⋊C4—31no—
313C4⋊C4—61no—
314C4⋊C4—91no—
315C4⋊C4—91no—
316C4⋊C4—91no—
317C4⋊C4—91no—
318C4⋊C4—181no—
319C4⋊C4—181no—
320C4⋊C4—181no—
321C4⋊C4—181no—
322C4⋊C4—181no—
323C4⋊C4—181no—
324C22⋊C4—61no—
325C22⋊C4—91no—
326C22⋊C4—91no—
327C22⋊C4—181no—
328C42—91no—
329C42—181no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—41no—
333C22×C3—42no—
334C22×C3—81no—
335C22×C3—81no—
336C22×C3—82no—
337C22×C3—82no—
338C22×C3—81no—
339C22×C3—82no—
340C22×C3—122no—
341C22×C3—241no—
342C22×C3—241no—
343C22×C3C2×#22612yes—
344C22×C3C22×S411yescharacteristic
345C2×D3—21no—
346C2×D3—21no—
347C2×D3C2×#22612yes—
348C3×C4—31no—
349C3×C4—31no—
350C3×C4—61no—
351C3×C4—122no—
352C3×C4—121no—
353C3×C4—121no—
354Dic3—61no—
355Dic3—61no—
356Dic3—81no—
357Dic3—81no—
358Dic3—122no—
359Dic3—121no—
360Dic3—121no—
361Dic3—241no—
362Dic3—242no—
363Dic3—242no—
364Dic3C2×#22612yes—
365C32—41noSylow
366C23—31no—
367C23D3×S411yescharacteristic
368Q8—31no—
369Q8—61no—
370Q8—61no—
371Q8—61no—
372Q8—61no—
373Q8—181no—
374Q8—181no—
375Q8—721no—
376Q8#380511yescharacteristic
377D4—34no—
378C2×C4—31no—
379C2×C4—31no—
380C2×C4—31no—
381C2×C4—61no—
382C2×C4—61no—
383C2×C4—61no—
384C2×C4—61no—
385C2×C4—91no—
386C2×C4—91no—
387C2×C4—181no—
388C2×C4—181no—
389C2×C4—181no—
390C2×C4—181no—
391C2×C4—721no—
392C2×C4—721no—
393C8—61no—
394C8—61no—
395C8—181no—
396C8—181no—
397C2×C3—22no—
398C2×C3—41no—
399C2×C3—41no—
400C2×C3—41no—
401C2×C3—82no—
402C2×C3—84no—
403C2×C3—81no—
404C2×C3—81no—
405C2×C3—81no—
406C2×C3—242no—
407C2×C3#591411yescharacteristic
408C2×C3C22×#22611yescharacteristic
409C2×C3#592611yescharacteristic
410D3—22no—
411C22—21no—
412C22—21no—
413C22—32no—
414C22—61no—
415C22—61no—
416C22D3×#22612yes—
417C22C2×D3×S411yescharacteristic
418C4—32no—
419C4—31no—
420C4—31no—
421C4—61no—
422C4—121no—
423C4—121no—
424C4—181no—
425C4—181no—
426C4—361no—
427C4—361no—
428C3—41no—
429C3—81no—
430C3#8539311yescharacteristic, complemented
431C2—22no—
432C2—62no—
433C2C2×D3×#22611yescharacteristic
434C2#1059644611yescharacteristic
435C2#1059646811yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC22×C3×Q8C2×D3
socleC22×C3C22×S4
prime radicalSL(2,𝔽3)×#67C2
hypercenterC23D3×S4
hypocenterC3×SL(2,𝔽3)#41
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
2. C2×C3⋊#859C2semidirect
3. C3×#5419C2semidirect
4. D3×#859C2semidirect
5. #10593210C2semidirect
6. #10593202C2semidirect
7. Dic3×#226C2semidirect
22. #5419D3semidirect
255. #67#226semidirect
430. C3#85393semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×#672
C2×C3⋊#8592
C3×#54192
D3×#8592
#105932102
#105932022
Dic3×#2262
#853933
#801933
#67⋊Dic34