Finite Group Databasebeta

← #87295⋊C3 #11923404

Subgroups — #87295⋊C3 #11923404

Order 1152.

797 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 797 conjugacy classes of subgroups — too many to draw. Only its 89 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×C3×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C2×D3×#41⋊C3C211yesmaximal, characteristic, complemented
3#5827⋊C3C211yesmaximal, characteristic, complemented
4SL(2,𝔽3)×#67C212yesmaximal, complemented
5#5817⋊C3C212yesmaximal, complemented
6#87295C311yesmaximal, characteristic, complemented
7#3489⋊C3—31nomaximal
8C3×2−1+4⋊C3—22no—
9C2×C3×#41⋊C3C2211yescharacteristic
10C22×C3×SL(2,𝔽3)C2212yescomplemented
11D3×#41⋊C3—22no—
12C2×D3×SL(2,𝔽3)C2211yescharacteristic
13C2×C3⋊(#41⋊C3)C2211yescharacteristic
14C3×#938—41nomaximal
15Dic3×SL(2,𝔽3)C2212yescomplemented
16C2×C3×2−1+4C2×C311yescharacteristic, complemented
17C2×D3×#41C2×C311yescharacteristic, complemented
18C2×2−1+4⋊C3—21no—
19C2×2−1+4⋊C3D311yescharacteristic, complemented
20C22×#41⋊C3—31no—
21#5833—31no—
22#5829—32no—
23#5828—31no—
24#5827C2×C311yescharacteristic, complemented
25#5824—32no—
26#5817C2×C312yescomplemented
27#5816—31no—
28Q8×#67C2×C312yescomplemented
29C2×#946—31no—
30D4×SL(2,𝔽3)—32no—
31#548⋊C3—32no—
32#520⋊C3—31no—
33C2×C32×D4—41no—
34C2×C3×#67—42no—
35C2×C3×C4×D3—41no—
36C3×#41⋊C3D412yescomplemented
37C2×C3×SL(2,𝔽3)—22no—
38C2×C3×SL(2,𝔽3)C2311yescharacteristic
39D3×SL(2,𝔽3)—21no—
40C3⋊(#41⋊C3)—21no—
41C3×#217—42no—
42C3×C4⋊Dic3—41no—
43#3489—31noSylow
44C3×2−1+4—22no—
45C3×2−1+4—62no—
46C2×C3×#41—31no—
47C2×C3×#41—32no—
48C2×C3×#41C22×C311yescharacteristic
49C22×C3×Q8—31no—
50C22×C3×Q8C22×C312yes—
51D3×#41—22no—
52D3×#41—62no—
53C2×#239C22×C311yescharacteristic
54C2×D3×Q8C22×C311yescharacteristic
55C2×#237—31no—
56C2×D3×D4—31no—
57C2×#235—31no—
58C22×C4×D3—31no—
592−1+4⋊C3—24no—
602−1+4⋊C3—22no—
61C2×#41⋊C3—21no—
62C2×#41⋊C3—31no—
63C2×#41⋊C3—62no—
64C2×#41⋊C3C2×D311yescharacteristic
65C22×SL(2,𝔽3)—22no—
66C22×SL(2,𝔽3)—31no—
67C22×SL(2,𝔽3)C2×D312yescomplemented
68C3×#121—121no—
69#947—32no—
70#946—32no—
71#946—31no—
72#946—32no—
73#946—31no—
74#946—32no—
75Q8×Dic3C22×C312yes—
76#944—32no—
77#939—34no—
78#938—31no—
79#938A411yescharacteristic
80#937—32no—
81#935—32no—
82D4×Dic3—32no—
83#930—32no—
84#929—34no—
85C4×#67—32no—
86C4×#67—32no—
87C2×(C2×D3)⋊C4—32no—
88#926—31no—
89C2×C4⋊Dic3—31no—
90#924—32no—
91C2×Dic3⋊C4—32no—
92C4×SL(2,𝔽3)—32no—
93C23×C32—42no—
94C22×C3×D3—41no—
95C32×D4—82no—
96C2×C32×C4—41no—
97C3×#67—84no—
98C2×C3×Dic3—42no—
99C2×C3×Dic3—41no—
100C3×C4×D3—82no—
101C3×SL(2,𝔽3)C2×D411yescharacteristic
102C2×2−1+4C3×D311yescharacteristic, complemented
103C22×#41—31no—
104D4×Q8—32no—
105#548—32no—
106#547—92no—
107#546—92no—
108#537—91no—
109#536—91no—
110#525—91no—
111C2×C22⋊Q8—91no—
112#520—31no—
113C23×D3—31no—
114C3×#41—32no—
115C3×#41—62no—
116C3×#41—62no—
117C3×#41—64no—
118C3×#41—64no—
119C3×#41C3×D412yescomplemented
120C2×C3×Q8—22no—
121C2×C3×Q8—31no—
122C2×C3×Q8—32no—
123C2×C3×Q8—62no—
124C2×C3×Q8—62no—
125C2×C3×Q8—62no—
126C2×C3×Q8C23×C311yescharacteristic
127C2×C3×D4—31no—
128C2×C3×D4—32no—
129C2×C3×D4—41no—
130C2×C3×D4—81no—
131C2×C3×D4—122no—
132C2×C3×D4C2×A411yescharacteristic
133C22×C3×C4—31no—
134C22×C3×C4—32no—
135C22×C3×C4—32no—
136C22×C3×C4—121no—
137C2×#67—32no—
138C2×#67—32no—
139C2×#67C2×A412yes—
140C22×Dic3—32no—
141C22×Dic3—31no—
142#239—21no—
143#239—61no—
144D3×Q8—21no—
145D3×Q8—61no—
146#237—62no—
147#237—62no—
148D3×D4—62no—
149D3×D4—62no—
150#235—62no—
151#235—62no—
152C2×D12—31no—
153C2×C4×D3—31no—
154C2×C4×D3—31no—
155C2×C4×D3—31no—
156C2×C4×D3—62no—
157C2×C4×D3—62no—
158C2×C4×D3C2×A411yescharacteristic
159C2×Dic6—31no—
160#41⋊C3—22no—
161#41⋊C3—61no—
162#41⋊C3#6712yescomplemented
163C2×SL(2,𝔽3)—24no—
164C2×SL(2,𝔽3)—22no—
165C2×SL(2,𝔽3)—21no—
166C2×SL(2,𝔽3)—61no—
167C2×SL(2,𝔽3)C22×D311yescharacteristic
168C3×C4⋊C4—121no—
169C3×C22⋊C4—122no—
170#217—32no—
171#217—32no—
172#217C2×A412yes—
173(C2×D3)⋊C4—34no—
174(C2×D3)⋊C4—32no—
175(C2×D3)⋊C4—32no—
176C4⋊Dic3—31no—
177C4⋊Dic3—32no—
178C4⋊Dic3C2×A411yescharacteristic
179Dic3⋊C4—34no—
180Dic3⋊C4—32no—
181Dic3⋊C4—32no—
182C4×Dic3—32no—
183C22×C32—41no—
184C22×C32—82no—
185C22×C32—82no—
186C22×C32—82no—
187C2×C3×D3—42no—
188C2×C3×D3—81no—
189C2×C3×D3—81no—
190C32×C4—42no—
191C3×Dic3—81no—
192C3×Dic3—82no—
1932−1+4—22no—
1942−1+4—62no—
195C2×#41—31no—
196C2×#41—32no—
197C2×#41—31no—
198C2×#41—62no—
199C2×#41—91no—
200C2×#41—91no—
201C2×#41—182no—
202C2×#41C2×C3×D311yescharacteristic
203C22×Q8—31no—
204C22×Q8—31no—
205C22×Q8C2×C3×D312yes—
206C22×D4—91no—
207C23×C4—91no—
208C4⋊Q8—92no—
209#124—92no—
210#123—92no—
211#123—94no—
212C22⋊Q8—91no—
213C22⋊Q8—91no—
214C22⋊Q8—92no—
215C22⋊Q8—92no—
216C22⋊Q8—92no—
217C22⋊Q8—92no—
218#121—31no—
219#121—92no—
220#121—94no—
221#121—91no—
222#120—92no—
223C4×Q8—32no—
224C4×D4—92no—
225C4×D4—92no—
226C4×D4—92no—
227#117—91no—
228C2×C4⋊C4—92no—
229C2×C4⋊C4—91no—
230C2×C22⋊C4—92no—
231C23×C3—31no—
232C23×C3—42no—
233C23×C3—82no—
234C23×C3—121no—
235C23×C3C22×A412yes—
236C22×D3—31no—
237C22×D3—62no—
238C22×D3—62no—
239C22×D3—121no—
240C22×D3C22×A411yescharacteristic
241C3×Q8—31no—
242C3×Q8—62no—
243C3×Q8—62no—
244C3×Q8—62no—
245C3×Q8C2×C3×D411yescharacteristic
246C3×D4—22no—
247C3×D4—32no—
248C3×D4—32no—
249C3×D4—64no—
250C3×D4—82no—
251C3×D4—84no—
252C3×D4—244no—
253C2×C3×C4—31no—
254C2×C3×C4—32no—
255C2×C3×C4—32no—
256C2×C3×C4—32no—
257C2×C3×C4—41no—
258C2×C3×C4—62no—
259C2×C3×C4—62no—
260C2×C3×C4—64no—
261C2×C3×C4—81no—
262C2×C3×C4—122no—
263C2×C3×C4—121no—
264C2×C3×C4—242no—
265C2×C3×C4C22×A411yescharacteristic
266#67—24no—
267#67—64no—
268#67—122no—
269C2×Dic3—32no—
270C2×Dic3—31no—
271C2×Dic3—62no—
272C2×Dic3—122no—
273C2×Dic3C22×A412yes—
274C2×Dic3C22×A411yescharacteristic
275D12—61no—
276D12—61no—
277C4×D3—22no—
278C4×D3—61no—
279C4×D3—61no—
280C4×D3—61no—
281C4×D3—61no—
282C4×D3—62no—
283Dic6—61no—
284Dic6—61no—
285SL(2,𝔽3)—21no—
286SL(2,𝔽3)C2×#6711yescharacteristic
287C2×C32—41no—
288C2×C32—41no—
289C2×C32—41no—
290C2×C32—162no—
291C3×D3—82no—
292C24—91no—
293#41—32no—
294#41—62no—
295#41—62no—
296#41—64no—
297#41—64no—
298#41—61no—
299#41—182no—
300#41—182no—
301#41—182no—
302#41—182no—
303#41—181no—
304#41C3×#6712yescomplemented
305C2×Q8—22no—
306C2×Q8—31no—
307C2×Q8—32no—
308C2×Q8—62no—
309C2×Q8—62no—
310C2×Q8—62no—
311C2×Q8—61no—
312C2×Q8—91no—
313C2×Q8—181no—
314C2×Q8C22×C3×D311yescharacteristic
315C2×D4—31no—
316C2×D4—32no—
317C2×D4—32no—
318C2×D4—91no—
319C2×D4—92no—
320C2×D4—92no—
321C2×D4—182no—
322C2×D4—182no—
323C2×D4D3×A411yescharacteristic
324C22×C4—31no—
325C22×C4—32no—
326C22×C4—32no—
327C22×C4—31no—
328C22×C4—91no—
329C22×C4—91no—
330C22×C4—91no—
331C22×C4—92no—
332C22×C4—91no—
333C22×C4—182no—
334C22×C4—182no—
335C4⋊C4—31no—
336C4⋊C4—92no—
337C4⋊C4—92no—
338C4⋊C4—92no—
339C4⋊C4—94no—
340C4⋊C4—91no—
341C22⋊C4—32no—
342C22⋊C4—92no—
343C22⋊C4—92no—
344C22⋊C4—92no—
345C22⋊C4—94no—
346C22⋊C4—92no—
347C42—92no—
348C22×C3—22no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—32no—
352C22×C3—41no—
353C22×C3—61no—
354C22×C3—61no—
355C22×C3—82no—
356C22×C3—82no—
357C22×C3—82no—
358C22×C3—84no—
359C22×C3—81no—
360C22×C3—122no—
361C22×C3—162no—
362C22×C3—162no—
363C22×C3—241no—
364C22×C3—241no—
365C22×C3C23×A411yescharacteristic
366C2×D3—21no—
367C2×D3—21no—
368C2×D3—61no—
369C2×D3—61no—
370C2×D3—61no—
371C2×D3—124no—
372C2×D32−1+4⋊C312yescomplemented
373C3×C4—31no—
374C3×C4—31no—
375C3×C4—42no—
376C3×C4—62no—
377C3×C4—82no—
378C3×C4—241no—
379C3×C4—242no—
380C3×C4D4×A412yes—
381Dic3—21no—
382Dic3—22no—
383Dic3—61no—
384Dic3—62no—
385C32—41noSylow
386C23—31no—
387C23—31no—
388C23—91no—
389C23—182no—
390C23—182no—
391C23—361no—
392C23C2×D3×A412yes—
393Q8—31no—
394Q8—62no—
395Q8—62no—
396Q8—62no—
397Q8—181no—
398Q8—181no—
399Q8C2×C3×#6711yescharacteristic
400D4—22no—
401D4—32no—
402D4—32no—
403D4—64no—
404D4—64no—
405D4—181no—
406D4—181no—
407D4—184no—
408D4—362no—
409C2×C4—31no—
410C2×C4—32no—
411C2×C4—32no—
412C2×C4—32no—
413C2×C4—31no—
414C2×C4—32no—
415C2×C4—62no—
416C2×C4—62no—
417C2×C4—64no—
418C2×C4—62no—
419C2×C4—92no—
420C2×C4—91no—
421C2×C4—181no—
422C2×C4—181no—
423C2×C4—181no—
424C2×C4—181no—
425C2×C4—182no—
426C2×C4—182no—
427C2×C4—362no—
428C2×C4C2×D3×A411yescharacteristic
429C2×C3—42no—
430C2×C3—41no—
431C2×C3—41no—
432C2×C3—41no—
433C2×C3—62no—
434C2×C3—81no—
435C2×C3—81no—
436C2×C3—81no—
437C2×C3—162no—
438C2×C3—164no—
439C2×C3—242no—
440C2×C3C2×D4×A411yescharacteristic
441C2×C3C2×2−1+4⋊C311yescharacteristic
442C2×C3#594511yescharacteristic
443D3—22no—
444D3—121no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—32no—
449C22—32no—
450C22—61no—
451C22—61no—
452C22—61no—
453C22—61no—
454C22—181no—
455C22—181no—
456C22—181no—
457C22—364no—
458C22C22×D3×A411yescharacteristic
459C4—31no—
460C4—31no—
461C4—62no—
462C4—62no—
463C4—61no—
464C4—182no—
465C4—181no—
466C4A4×#6712yes—
467C3—41no—
468C3—81no—
469C3#3489⋊C311yescharacteristic, complemented
470C2—42no—
471C2—62no—
472C2—62no—
473C2—361no—
474C2D3×2−1+4⋊C311yescharacteristic
475C2#5968⋊C311yescharacteristic
476C2C2×A4×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC2×C3×2−1+4C2×C3
socleC22×C3C23×A4
hypercenterC2×D4D3×A4
hypocenterC3×Q8C2×C3×D4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×2−1+4⋊C3C2semidirect
2. C2×D3×#41⋊C3C2semidirect
3. #5827⋊C3C2semidirect
4. SL(2,𝔽3)×#67C2semidirect
5. #5817⋊C3C2semidirect
6. #87295C3semidirect
10. C22×C3×SL(2,𝔽3)C22semidirect
15. Dic3×SL(2,𝔽3)C22semidirect
16. C2×C3×2−1+4C2×C3semidirect
17. C2×D3×#41C2×C3semidirect
19. C2×2−1+4⋊C3D3semidirect
24. #5827C2×C3semidirect
26. #5817C2×C3semidirect
28. Q8×#67C2×C3semidirect
36. C3×#41⋊C3D4semidirect
67. C22×SL(2,𝔽3)C2×D3semidirect
102. C2×2−1+4C3×D3semidirect
119. C3×#41C3×D4semidirect
162. #41⋊C3#67semidirect
304. #41C3×#67semidirect
372. C2×D32−1+4⋊C3semidirect
469. C3#3489⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×2−1+4⋊C32
C2×D3×#41⋊C32
#5827⋊C32
SL(2,𝔽3)×#672
#5817⋊C32
#872953
#3489⋊C33
C3×#9384