Finite Group Databasebeta

← #87019⋊C3 #11923279

Subgroups — #87019⋊C3 #11923279

Order 1152.

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

normal subgroups only

This group has 505 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
1C3×#557⋊C3C211yesmaximal, characteristic, complemented
2#5570⋊C3C211yesmaximal, characteristic, complemented
3#5569⋊C3C211yesmaximal, characteristic, complemented
4#5569⋊C3C211yesmaximal, characteristic, complemented
5#5574⋊C3C211yesmaximal, characteristic
6#5574⋊C3C211yesmaximal, characteristic
7C4×C3⋊(#41⋊C3)C211yesmaximal, characteristic
8#87019C311yesmaximal, characteristic, complemented
9#3504⋊C3—31nomaximal
10C2×C3⋊(#41⋊C3)C2211yescharacteristic
11C3×#889—41nomaximal
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13C3×C4×SL(2,𝔽3)C2211yescharacteristic
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15Dic3×SL(2,𝔽3)C2211yescharacteristic
16Dic3×SL(2,𝔽3)C2211yescharacteristic
17Dic3×SL(2,𝔽3)C2211yescharacteristic
18C3×#557C2×C311yescharacteristic
19#5730—31no—
20#5729—31no—
21#5690—31no—
22#5690—31no—
23#5689—31no—
24D3×#125—31no—
25#5579—31no—
26#5578—31no—
27#5574C2×C311yescharacteristic
28#5574C2×C311yescharacteristic
29C4×#239C2×C311yescharacteristic
30#5570C2×C311yescharacteristic
31#5569C2×C311yescharacteristic
32#5569C2×C311yescharacteristic
33#557⋊C3—21no—
34#557⋊C3—31no—
35#557⋊C3—31no—
36#557⋊C3—31no—
37#557⋊C3D311yescharacteristic, complemented
38#520⋊C3—31no—
39#520⋊C3—31no—
40C4×#41⋊C3—31no—
41C2×C3×SL(2,𝔽3)C2311yescharacteristic
42C3⋊(#41⋊C3)Q812yescomplemented
43C32×C4⋊C4—41no—
44C3×C4⋊Dic3—41no—
45C3×Dic3⋊C4—41no—
46C3×Dic3⋊C4—41no—
47C3×Dic3⋊C4—41no—
48C3×Dic3⋊C4—41no—
49C3×C4×Dic3—41no—
50#3504—31noSylow
51C2×#239C22×C311yescharacteristic
52C2×#41⋊C3—31no—
53C3×C4⋊Q8—31no—
54C3×#125—31no—
55C3×#125—31no—
56C3×#125—121no—
57C3×C4×Q8—31no—
58C3×C4×Q8C22×C311yescharacteristic
59C3×C4×Q8C22×C311yescharacteristic
60C3×C4×Q8C22×C311yescharacteristic
61#946—32no—
62#946—31no—
63Q8×Dic3C22×C311yescharacteristic
64Q8×Dic3C22×C311yescharacteristic
65Q8×Dic3C22×C311yescharacteristic
66#944—31no—
67#897—32no—
68#897—31no—
69#896—32no—
70#896—32no—
71#896—31no—
72#896—31no—
73#893—31no—
74#893—31no—
75#893—31no—
76#892—31no—
77D3×C4⋊C4—32no—
78D3×C4⋊C4—32no—
79D3×C4⋊C4—31no—
80D3×C4⋊C4—31no—
81#890—31no—
82#890—31no—
83#890—31no—
84#890—31no—
85#889—32no—
86#889—31no—
87#889A411yescharacteristic
88#888—31no—
89#888—31no—
90#888—31no—
91C4×D12—31no—
92C4×D12—31no—
93C4×D12—31no—
94#872—31no—
95#872—31no—
96C42×D3—31no—
97#870—31no—
98#870—31no—
99#869—31no—
100C4×Dic6—31no—
101C4×Dic6—31no—
102C4×Dic6—31no—
103C4×SL(2,𝔽3)—21no—
104C4×SL(2,𝔽3)—21no—
105C4×SL(2,𝔽3)—21no—
106C4×SL(2,𝔽3)—31no—
107C4×SL(2,𝔽3)—31no—
108C4×SL(2,𝔽3)—31no—
109C4×SL(2,𝔽3)C2×D311yescharacteristic
110C4×SL(2,𝔽3)C2×D311yescharacteristic
111C4×SL(2,𝔽3)C2×D311yescharacteristic
112C2×C32×C4—41no—
113C2×C32×C4—41no—
114C2×C32×C4—41no—
115C2×C3×Dic3—41no—
116C2×C3×Dic3—41no—
117C2×C3×Dic3—41no—
118C2×C3×Dic3—41no—
119C3×SL(2,𝔽3)C2×Q811yescharacteristic
120#557—31no—
121#557—31no—
122#557—31no—
123#557C3×D311yescharacteristic
124#554—91no—
125#554—91no—
126#554—91no—
127#554—91no—
128#544—91no—
129#544—91no—
130#533—91no—
131C2×#125—91no—
132#520—31no—
133#520—31no—
134C4×#41—31no—
135C2×C3×Q8—31no—
136C2×C3×Q8C23×C311yescharacteristic
137#239—32no—
138#239C3×Q812yes—
139C2×D12—31no—
140C2×C4×D3—31no—
141C2×C4×D3—32no—
142C2×C4×D3—32no—
143C2×C4×D3—31no—
144C2×C4×D3—31no—
145C2×Dic6—31no—
146C2×Dic6—31no—
147C2×Dic6—31no—
148#41⋊C3—32no—
149C2×SL(2,𝔽3)—21no—
150C2×SL(2,𝔽3)C22×D311yescharacteristic
151C3×C4⋊C4—31no—
152C3×C4⋊C4—31no—
153C3×C4⋊C4—31no—
154C3×C4⋊C4—32no—
155C3×C4⋊C4—31no—
156C3×C4⋊C4—31no—
157C3×C4⋊C4—41no—
158C3×C4⋊C4—81no—
159C3×C4⋊C4—121no—
160C3×C4⋊C4—121no—
161C3×C4⋊C4—121no—
162C3×C4⋊C4—121no—
163C3×C4⋊C4—121no—
164C3×C4⋊C4C2×A411yescharacteristic
165C3×C42—31no—
166C3×C42—31no—
167C3×C42—31no—
168C3×C42—121no—
169(C2×D3)⋊C4—32no—
170(C2×D3)⋊C4—31no—
171(C2×D3)⋊C4—31no—
172(C2×D3)⋊C4—31no—
173(C2×D3)⋊C4—31no—
174C4⋊Dic3—32no—
175C4⋊Dic3—31no—
176C4⋊Dic3—31no—
177C4⋊Dic3—31no—
178C4⋊Dic3—31no—
179C4⋊Dic3—31no—
180C4⋊Dic3C2×A411yescharacteristic
181Dic3⋊C4—32no—
182Dic3⋊C4—32no—
183Dic3⋊C4—31no—
184Dic3⋊C4—31no—
185Dic3⋊C4—31no—
186Dic3⋊C4—31no—
187Dic3⋊C4—31no—
188Dic3⋊C4—31no—
189Dic3⋊C4C2×A411yescharacteristic
190Dic3⋊C4C2×A411yescharacteristic
191Dic3⋊C4C2×A411yescharacteristic
192Dic3⋊C4C2×A411yescharacteristic
193C4×Dic3—31no—
194C4×Dic3—31no—
195C4×Dic3—31no—
196C4×Dic3—31no—
197C4×Dic3C2×A411yescharacteristic
198C22×C32—41no—
199C32×C4—81no—
200C32×C4—81no—
201C32×C4—81no—
202C3×Dic3—42no—
203C3×Dic3—81no—
204C3×Dic3—81no—
205C3×Dic3—81no—
206C2×#41—31no—
207C4⋊Q8—31no—
208C4⋊Q8—91no—
209C4⋊Q8—91no—
210C4⋊Q8—91no—
211C4⋊Q8—91no—
212C4⋊Q8—91no—
213#125—31no—
214#125—31no—
215#125—31no—
216#125—91no—
217#125—91no—
218#125—91no—
219#125—91no—
220#125—91no—
221#125—91no—
222#125—91no—
223#125—92no—
224C22⋊Q8—91no—
225C22⋊Q8—91no—
226C22⋊Q8—91no—
227C22⋊Q8—92no—
228C22⋊Q8—92no—
229C22⋊Q8—92no—
230C22⋊Q8—92no—
231C22⋊Q8—91no—
232C4×Q8—31no—
233C4×Q8—31no—
234C4×Q8—31no—
235C4×Q8—31no—
236C4×Q8—91no—
237C4×Q8—91no—
238C4×Q8—91no—
239C4×Q8C2×C3×D311yescharacteristic
240C4×Q8C2×C3×D311yescharacteristic
241C4×Q8C2×C3×D311yescharacteristic
242C4×D4—91no—
243C4×D4—91no—
244C4×D4—91no—
245C4×D4—91no—
246C4×D4—91no—
247C4×D4—91no—
248#117—91no—
249#117—91no—
250#117—91no—
251C2×C4⋊C4—92no—
252C2×C4⋊C4—92no—
253C2×C4⋊C4—91no—
254C2×C4⋊C4—91no—
255C2×C42—91no—
256C22×D3—31no—
257C3×Q8—31no—
258C3×Q8—121no—
259C3×Q8C2×C3×Q811yescharacteristic
260C2×C3×C4—31no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—81no—
268C2×C3×C4—81no—
269C2×C3×C4—81no—
270C2×C3×C4—121no—
271C2×C3×C4—121no—
272C2×C3×C4—121no—
273C2×C3×C4—121no—
274C2×C3×C4C22×A411yescharacteristic
275C2×C3×C4C22×A411yescharacteristic
276C2×C3×C4C22×A411yescharacteristic
277C2×Dic3—31no—
278C2×Dic3—31no—
279C2×Dic3—31no—
280C2×Dic3C22×A411yescharacteristic
281C2×Dic3C22×A411yescharacteristic
282C2×Dic3C22×A411yescharacteristic
283C2×Dic3C22×A411yescharacteristic
284D12—32no—
285D12—32no—
286C4×D3—32no—
287C4×D3—32no—
288C4×D3—121no—
289C4×D3—121no—
290C4×D3—122no—
291C4×D3—122no—
292Dic6—121no—
293Dic6—121no—
294Dic6—121no—
295SL(2,𝔽3)—21no—
296SL(2,𝔽3)D3×Q811yescharacteristic
297C2×C32—41no—
298C2×C32—41no—
299C2×C32—41no—
300#41—32no—
301#41—92no—
302C2×Q8—31no—
303C2×Q8—91no—
304C2×Q8—91no—
305C2×Q8—91no—
306C2×Q8C22×C3×D311yescharacteristic
307C2×D4—91no—
308C22×C4—91no—
309C22×C4—91no—
310C22×C4—92no—
311C22×C4—92no—
312C22×C4—91no—
313C4⋊C4—31no—
314C4⋊C4—31no—
315C4⋊C4—31no—
316C4⋊C4—32no—
317C4⋊C4—31no—
318C4⋊C4—31no—
319C4⋊C4—31no—
320C4⋊C4—31no—
321C4⋊C4—31no—
322C4⋊C4—31no—
323C4⋊C4—31no—
324C4⋊C4—91no—
325C4⋊C4—91no—
326C4⋊C4—91no—
327C4⋊C4—91no—
328C4⋊C4—91no—
329C4⋊C4—91no—
330C4⋊C4—91no—
331C4⋊C4—91no—
332C4⋊C4—91no—
333C4⋊C4—92no—
334C4⋊C4—92no—
335C4⋊C4—92no—
336C4⋊C4—91no—
337C4⋊C4—91no—
338C4⋊C4D3×A411yescharacteristic
339C22⋊C4—91no—
340C22⋊C4—91no—
341C22⋊C4—92no—
342C22⋊C4—91no—
343C22⋊C4—91no—
344C42—31no—
345C42—31no—
346C42—31no—
347C42—31no—
348C42—91no—
349C42—91no—
350C42—91no—
351C42—91no—
352C22×C3—41no—
353C22×C3—81no—
354C22×C3C23×A411yescharacteristic
355C2×D3—32no—
356C2×D3—61no—
357C2×D3—61no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—31no—
362C3×C4—31no—
363C3×C4—61no—
364C3×C4—61no—
365C3×C4—61no—
366C3×C4—81no—
367C3×C4—81no—
368C3×C4—81no—
369C3×C4—82no—
370C3×C4—122no—
371C3×C4—161no—
372C3×C4—161no—
373C3×C4—241no—
374C3×C4—241no—
375C3×C4—241no—
376Dic3—21no—
377Dic3—21no—
378Dic3—21no—
379Dic3—61no—
380Dic3—61no—
381Dic3—61no—
382Dic3Q8×A412yes—
383C32—41noSylow
384C23—91no—
385Q8—31no—
386Q8—121no—
387Q8—361no—
388Q8—361no—
389Q8—361no—
390Q8C3×D3×Q811yescharacteristic
391D4—92no—
392D4—92no—
393C2×C4—31no—
394C2×C4—31no—
395C2×C4—31no—
396C2×C4—31no—
397C2×C4—31no—
398C2×C4—31no—
399C2×C4—31no—
400C2×C4—31no—
401C2×C4—92no—
402C2×C4—92no—
403C2×C4—91no—
404C2×C4—91no—
405C2×C4—91no—
406C2×C4—361no—
407C2×C4—361no—
408C2×C4—362no—
409C2×C4—362no—
410C2×C4C2×D3×A411yescharacteristic
411C2×C4C2×D3×A411yescharacteristic
412C2×C4C2×D3×A411yescharacteristic
413C2×C3—41no—
414C2×C3—41no—
415C2×C3—41no—
416C2×C3—81no—
417C2×C3—81no—
418C2×C3—81no—
419C2×C3C2×Q8×A411yescharacteristic
420C2×C3#594511yescharacteristic
421C2×C3#594511yescharacteristic
422D3—62no—
423C22—92no—
424C22—181no—
425C22—181no—
426C22C22×D3×A411yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—31no—
431C4—31no—
432C4—32no—
433C4—61no—
434C4—61no—
435C4—61no—
436C4—61no—
437C4—61no—
438C4—61no—
439C4—181no—
440C4—181no—
441C4—181no—
442C3—41no—
443C3—81no—
444C3#3504⋊C311yescharacteristic, complemented
445C2—182no—
446C2D3×Q8×A411yescharacteristic
447C2#5964⋊C311yescharacteristic
448C2#5968⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×#557C2×C3
socleC22×C3C23×A4
prime radicalC2×C3⋊(#41⋊C3)C22
hypercenterC4⋊C4D3×A4
hypocenterC3×Q8C2×C3×Q8
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#557⋊C3C2semidirect
2. #5570⋊C3C2semidirect
3. #5569⋊C3C2semidirect
4. #5569⋊C3C2semidirect
8. #87019C3semidirect
37. #557⋊C3D3semidirect
42. C3⋊(#41⋊C3)Q8semidirect
444. C3#3504⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
#5570⋊C32
#5569⋊C32
#5574⋊C32
C4×C3⋊(#41⋊C3)2
#870193
#3504⋊C33
C3×#8894