Finite Group Databasebeta

← #87274⋊C3 #11923386

Subgroups — #87274⋊C3 #11923386

Order 1152.

775 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 775 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C22×C3×#41⋊C3C211yesmaximal, characteristic, complemented
2C2×#1007⋊C3C211yesmaximal, characteristic, complemented
3#5831⋊C3C212yesmaximal, complemented
4#5834⋊C3C212yesmaximal, complemented
5#5812⋊C3C211yesmaximal, characteristic, complemented
6#87274C311yesmaximal, characteristic, complemented
7#3487⋊C3—31nomaximal
8C2×C3×#41⋊C3—22no—
9C2×C3×#41⋊C3C2212yes—
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11#1007⋊C3—22no—
12C2×D3×SL(2,𝔽3)C2211yescharacteristic, complemented
13C2×C3⋊(#41⋊C3)C2211yescharacteristic, complemented
14C3×#929—41nomaximal
15Dic3×SL(2,𝔽3)C2212yescomplemented
16C22×C3×#41C2×C311yescharacteristic, complemented
17C2×#1007C2×C311yescharacteristic
18C2×2−1+4⋊C3—31no—
19C22×#41⋊C3—21no—
20C22×#41⋊C3D311yescharacteristic, complemented
21#5834—32no—
22#5834C2×C312yescomplemented
23#5831—32no—
24#5831C2×C312yescomplemented
25#5812C2×C311yescharacteristic, complemented
26#5805—31no—
27#5805—31no—
28#5791—31no—
29#5791—31no—
30C4⋊(#41⋊C3)—32no—
31#548⋊C3—32no—
32#519⋊C3—31no—
33C22×C32×C4—41no—
34C2×C3×#67—41no—
35C3×#41⋊C3—22no—
36C2×C3×SL(2,𝔽3)D412yescomplemented
37C2×C3×SL(2,𝔽3)C2311yescharacteristic
38D3×SL(2,𝔽3)—21no—
39C3⋊(#41⋊C3)—21no—
40C3×#217—41no—
41C3×(C2×D3)⋊C4—42no—
42C3×Dic3⋊C4—42no—
43#3487—31noSylow
44C2×C3×#41—22no—
45C2×C3×#41—32no—
46C2×C3×#41—62no—
47C2×C3×#41C22×C312yes—
48C22×C3×Q8C22×C311yescharacteristic
49C22×C3×D4—31no—
50C23×C3×C4—31no—
51#1007—22no—
52#1007—62no—
53C2×#239—31no—
54C2×#239C22×C311yescharacteristic
55C2×D3×Q8—31no—
56C2×D3×Q8C22×C311yescharacteristic
57C2×#235—31no—
58C2×#235—31no—
592−1+4⋊C3—62no—
60C2×#41⋊C3—24no—
61C2×#41⋊C3—22no—
62C2×#41⋊C3—22no—
63C2×#41⋊C3—31no—
64C2×#41⋊C3C2×D312yes—
65C22×SL(2,𝔽3)—21no—
66C22×SL(2,𝔽3)—31no—
67C22×SL(2,𝔽3)C2×D311yescharacteristic
68C3×#123—121no—
69#953—32no—
70#953—31no—
71#953—31no—
72Q8×Dic3—32no—
73Q8×Dic3C22×C312yes—
74#940—32no—
75#940—32no—
76#938—34no—
77#938—32no—
78#938—32no—
79#936—32no—
80#936—32no—
81#930—32no—
82#930—32no—
83#929—31no—
84#929A411yescharacteristic
85C4×#67—34no—
86C4×#67—32no—
87C4×#67—32no—
88#926—31no—
89#926—31no—
90#924—32no—
91#924—32no—
92C4×SL(2,𝔽3)—32no—
93C23×C32—41no—
94C22×C3×D3—41no—
95C2×C32×C4—42no—
96C2×C32×C4—82no—
97C3×#67—82no—
98C2×C3×Dic3—42no—
99C2×C3×Dic3—41no—
100C3×SL(2,𝔽3)C2×D411yescharacteristic
101C2×2−1+4—31no—
102C22×#41C3×D311yescharacteristic, complemented
103#550—32no—
104#550—92no—
105#548—32no—
106#548—92no—
107#535—91no—
108#535—91no—
109#525—91no—
110#525—91no—
111#519—31no—
112C24×C3—31no—
113C3×#41—22no—
114C3×#41—64no—
115C3×#41—64no—
116C3×#41—62no—
117C2×C3×Q8—31no—
118C2×C3×Q8—32no—
119C2×C3×Q8C3×D412yes—
120C2×C3×Q8C23×C311yescharacteristic
121C2×C3×D4—32no—
122C2×C3×D4—32no—
123C2×C3×D4—62no—
124C2×C3×D4—62no—
125C2×C3×D4—121no—
126C22×C3×C4—31no—
127C22×C3×C4—32no—
128C22×C3×C4—32no—
129C22×C3×C4—41no—
130C22×C3×C4—62no—
131C22×C3×C4—62no—
132C22×C3×C4—81no—
133C22×C3×C4C2×A411yescharacteristic
134C2×#67—31no—
135C2×#67—34no—
136C2×#67—32no—
137C2×#67—32no—
138C2×#67C2×A411yescharacteristic
139#239—21no—
140#239—62no—
141#239—62no—
142#239—61no—
143D3×Q8—21no—
144D3×Q8—62no—
145D3×Q8—62no—
146D3×Q8—61no—
147#235—62no—
148#235—62no—
149#235—62no—
150#235—62no—
151C2×D12—31no—
152C2×D12—31no—
153C2×C4×D3—32no—
154C2×C4×D3—31no—
155C2×C4×D3—31no—
156C2×Dic6—31no—
157C2×Dic6—31no—
158#41⋊C3—24no—
159#41⋊C3—22no—
160#41⋊C3—61no—
161C2×SL(2,𝔽3)—22no—
162C2×SL(2,𝔽3)—21no—
163C2×SL(2,𝔽3)—61no—
164C2×SL(2,𝔽3)#6712yescomplemented
165C2×SL(2,𝔽3)C22×D311yescharacteristic
166C3×C4⋊C4—122no—
167C3×C22⋊C4—122no—
168C3×C22⋊C4—121no—
169#217—34no—
170#217—31no—
171#217—32no—
172#217—32no—
173#217C2×A411yescharacteristic
174(C2×D3)⋊C4—32no—
175(C2×D3)⋊C4C2×A412yes—
176C4⋊Dic3—32no—
177C4⋊Dic3—32no—
178Dic3⋊C4—32no—
179Dic3⋊C4C2×A412yes—
180C4×Dic3—32no—
181C4×Dic3—32no—
182C22×C32—42no—
183C22×C32—41no—
184C22×C32—82no—
185C2×C3×D3—81no—
186C2×C3×D3—82no—
187C32×C4—82no—
188C3×Dic3—82no—
189C3×Dic3—81no—
1902−1+4—62no—
1912−1+4—182no—
192C2×#41—22no—
193C2×#41—32no—
194C2×#41—31no—
195C2×#41—62no—
196C2×#41—91no—
197C2×#41—91no—
198C2×#41—91no—
199C2×#41C2×C3×D312yes—
200C22×Q8—31no—
201C22×Q8—91no—
202C22×Q8C2×C3×D311yescharacteristic
203C22×D4—31no—
204C23×C4—31no—
205#127—92no—
206#127—92no—
207#124—92no—
208#124—92no—
209#123—31no—
210#123—91no—
211C22⋊Q8—92no—
212C22⋊Q8—92no—
213#121—92no—
214#121—92no—
215#121—94no—
216#121—92no—
217#121—92no—
218#120—91no—
219#120—91no—
220#120—92no—
221C4×Q8—32no—
222C4×Q8—92no—
223C4×D4—92no—
224C4×D4—92no—
225C4×D4—94no—
226#117—91no—
227#117—91no—
228C23×C3—32no—
229C23×C3—41no—
230C23×C3—62no—
231C23×C3—81no—
232C23×C3—121no—
233C23×C3—122no—
234C23×C3C22×A411yescharacteristic
235C22×D3—31no—
236C22×D3C22×A411yescharacteristic
237C3×Q8—32no—
238C3×Q8—31no—
239C3×Q8—32no—
240C3×Q8C2×C3×D411yescharacteristic
241C3×D4—62no—
242C3×D4—62no—
243C3×D4—64no—
244C3×D4—242no—
245C2×C3×C4—22no—
246C2×C3×C4—32no—
247C2×C3×C4—32no—
248C2×C3×C4—31no—
249C2×C3×C4—31no—
250C2×C3×C4—42no—
251C2×C3×C4—62no—
252C2×C3×C4—64no—
253C2×C3×C4—62no—
254C2×C3×C4—82no—
255C2×C3×C4—84no—
256C2×C3×C4—82no—
257C2×C3×C4—122no—
258C2×C3×C4—121no—
259C2×C3×C4C22×A412yes—
260#67—22no—
261#67—62no—
262#67—122no—
263#67—122no—
264#67—124no—
265C2×Dic3—32no—
266C2×Dic3—31no—
267C2×Dic3C22×A412yes—
268C2×Dic3C22×A411yescharacteristic
269D12—62no—
270D12—61no—
271D12—61no—
272C4×D3—61no—
273C4×D3—61no—
274C4×D3—61no—
275C4×D3—61no—
276C4×D3—64no—
277Dic6—62no—
278Dic6—61no—
279Dic6—61no—
280SL(2,𝔽3)—21no—
281SL(2,𝔽3)C2×#6711yescharacteristic
282C2×C32—41no—
283C2×C32—42no—
284C2×C32—82no—
285C3×D3—161no—
286C24—31no—
287#41—22no—
288#41—64no—
289#41—64no—
290#41—62no—
291#41—61no—
292#41—182no—
293#41—182no—
294#41—182no—
295#41—182no—
296#41—182no—
297#41—182no—
298#41—181no—
299C2×Q8—31no—
300C2×Q8—32no—
301C2×Q8—61no—
302C2×Q8—91no—
303C2×Q8—91no—
304C2×Q8—182no—
305C2×Q8—182no—
306C2×Q8—181no—
307C2×Q8C3×#6712yes—
308C2×Q8C22×C3×D311yescharacteristic
309C2×D4—32no—
310C2×D4—32no—
311C2×D4—31no—
312C2×D4—62no—
313C2×D4—62no—
314C2×D4—91no—
315C2×D4—91no—
316C2×D4—92no—
317C2×D4—91no—
318C2×D4—92no—
319C2×D4—94no—
320C22×C4—31no—
321C22×C4—32no—
322C22×C4—32no—
323C22×C4—62no—
324C22×C4—62no—
325C22×C4—91no—
326C22×C4—91no—
327C22×C4—92no—
328C22×C4D3×A411yescharacteristic
329C4⋊C4—32no—
330C4⋊C4—92no—
331C4⋊C4—92no—
332C4⋊C4—92no—
333C22⋊C4—31no—
334C22⋊C4—32no—
335C22⋊C4—92no—
336C22⋊C4—92no—
337C22⋊C4—94no—
338C22⋊C4—91no—
339C22⋊C4—92no—
340C42—92no—
341C42—92no—
342C22×C3—22no—
343C22×C3—42no—
344C22×C3—41no—
345C22×C3—62no—
346C22×C3—62no—
347C22×C3—62no—
348C22×C3—82no—
349C22×C3—84no—
350C22×C3—82no—
351C22×C3—81no—
352C22×C3—124no—
353C22×C3—241no—
354C22×C3—242no—
355C22×C3D4×A412yes—
356C22×C3C23×A411yescharacteristic
357C2×D3—21no—
358C2×D3—22no—
359C2×D3—61no—
360C2×D3—62no—
361C3×C4—22no—
362C3×C4—32no—
363C3×C4—31no—
364C3×C4—31no—
365C3×C4—82no—
366C3×C4—84no—
367C3×C4—241no—
368C3×C4—242no—
369Dic3—21no—
370Dic3—22no—
371Dic3—61no—
372Dic3—62no—
373C32—41noSylow
374C23—31no—
375C23—32no—
376C23—62no—
377C23—91no—
378C23—122no—
379C23C2×D3×A411yescharacteristic
380Q8—32no—
381Q8—31no—
382Q8—32no—
383Q8—182no—
384Q8—181no—
385Q8—181no—
386Q8C2×C3×#6711yescharacteristic
387D4—62no—
388D4—62no—
389D4—64no—
390D4—62no—
391D4—182no—
392D4—181no—
393D4—181no—
394D4—182no—
395D4—362no—
396D4—362no—
397D4—364no—
398C2×C4—22no—
399C2×C4—32no—
400C2×C4—32no—
401C2×C4—31no—
402C2×C4—31no—
403C2×C4—32no—
404C2×C4—31no—
405C2×C4—62no—
406C2×C4—62no—
407C2×C4—64no—
408C2×C4—92no—
409C2×C4—91no—
410C2×C4—181no—
411C2×C4—181no—
412C2×C4—181no—
413C2×C4—181no—
414C2×C4—184no—
415C2×C4C2×D3×A412yes—
416C2×C3—22no—
417C2×C3—42no—
418C2×C3—41no—
419C2×C3—82no—
420C2×C3—84no—
421C2×C3—82no—
422C2×C3—81no—
423C2×C3—122no—
424C2×C3—481no—
425C2×C3C2×D4×A411yescharacteristic
426C2×C3#594512yes—
427D3—41no—
428D3—121no—
429C22—22no—
430C22—62no—
431C22—62no—
432C22—62no—
433C22—62no—
434C22—61no—
435C22—124no—
436C22—182no—
437C22—181no—
438C22A4×#6712yes—
439C22C22×D3×A411yescharacteristic
440C4—22no—
441C4—31no—
442C4—32no—
443C4—31no—
444C4—62no—
445C4—61no—
446C4—182no—
447C4—181no—
448C3—41no—
449C3—81no—
450C3#3487⋊C311yescharacteristic, complemented
451C2—22no—
452C2—122no—
453C2—121no—
454C2—361no—
455C2#5964⋊C312yes—
456C2C2×A4×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC22×C3×#41C2×C3
socleC22×C3C23×A4
hypercenterC22×C4D3×A4
hypocenterC3×Q8C2×C3×D4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#41⋊C3C2semidirect
2. C2×#1007⋊C3C2semidirect
3. #5831⋊C3C2semidirect
4. #5834⋊C3C2semidirect
5. #5812⋊C3C2semidirect
6. #87274C3semidirect
12. C2×D3×SL(2,𝔽3)C22semidirect
13. C2×C3⋊(#41⋊C3)C22semidirect
15. Dic3×SL(2,𝔽3)C22semidirect
16. C22×C3×#41C2×C3semidirect
20. C22×#41⋊C3D3semidirect
22. #5834C2×C3semidirect
24. #5831C2×C3semidirect
25. #5812C2×C3semidirect
36. C2×C3×SL(2,𝔽3)D4semidirect
102. C22×#41C3×D3semidirect
164. C2×SL(2,𝔽3)#67semidirect
450. C3#3487⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C2×#1007⋊C32
#5831⋊C32
#5834⋊C32
#5812⋊C32
#872743
#3487⋊C33
C3×#9294