Finite Group Databasebeta

← #85875⋊C3 #11923209

Subgroups — #85875⋊C3 #11923209

Order 1152.

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

normal subgroups only

This group has 533 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
1C3×#519⋊C3C211yesmaximal, characteristic
2#5812⋊C3C211yesmaximal, characteristic
3C2×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic
4#5570⋊C3C211yesmaximal, characteristic, complemented
5#5569⋊C3C211yesmaximal, characteristic, complemented
6#5569⋊C3C211yesmaximal, characteristic, complemented
7SL(2,𝔽3)×Dic6C211yesmaximal, characteristic, complemented
8#85875C311yesmaximal, characteristic, complemented
9#3502⋊C3—31nomaximal
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C3×#878—41nomaximal
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13C3×C4×SL(2,𝔽3)C2211yescharacteristic
14Dic3×SL(2,𝔽3)C2211yescharacteristic
15Dic3×SL(2,𝔽3)C2211yescharacteristic
16Dic3×SL(2,𝔽3)C2211yescharacteristic
17Dic3×SL(2,𝔽3)C2211yescharacteristic
18C3×#519C2×C311yescharacteristic
19#5812C2×C311yescharacteristic
20C2×Q8×Dic3C2×C311yescharacteristic
21#5570—31no—
22#5570C2×C311yescharacteristic, complemented
23#5569—31no—
24#5569—31no—
25#5569C2×C311yescharacteristic, complemented
26#5569C2×C311yescharacteristic, complemented
27Q8×Dic6—31no—
28Q8×Dic6C2×C311yescharacteristic, complemented
29#5519—31no—
30#5519—31no—
31#5517—31no—
32#5517—31no—
33Q8×SL(2,𝔽3)—31no—
34#557⋊C3—31no—
35#557⋊C3—31no—
36#557⋊C3—31no—
37#519⋊C3—21no—
38#519⋊C3—31no—
39#519⋊C3D311yescharacteristic
40C2×C4×SL(2,𝔽3)—31no—
41C22×C3×Dic3—41no—
42C2×C3×Dic6—41no—
43C2×C3×SL(2,𝔽3)Q812yescomplemented
44C2×C3×SL(2,𝔽3)C2311yescharacteristic
45C32×C22⋊C4—41no—
46C3×#217—41no—
47C3×C4⋊Dic3—41no—
48C3×Dic3⋊C4—41no—
49C3×Dic3⋊C4—41no—
50#3502—31noSylow
51C22×C3×Q8C22×C311yescharacteristic
52C22×SL(2,𝔽3)—21no—
53C22×SL(2,𝔽3)C2×D311yescharacteristic
54C3×C22⋊Q8—121no—
55C3×C4×Q8—31no—
56C3×C4×Q8—31no—
57C3×C4×Q8C22×C311yescharacteristic
58C3×C4×Q8C22×C311yescharacteristic
59C3×#117—31no—
60C3×#117—31no—
61Q8×Dic3—31no—
62Q8×Dic3—31no—
63Q8×Dic3—31no—
64Q8×Dic3—31no—
65Q8×Dic3C22×C311yescharacteristic
66Q8×Dic3C22×C311yescharacteristic
67Q8×Dic3C22×C311yescharacteristic
68Q8×Dic3C22×C311yescharacteristic
69#926—31no—
70#926—31no—
71C2×C4⋊Dic3—31no—
72C2×C4×Dic3—31no—
73#890—32no—
74#890—31no—
75#890—31no—
76#890—31no—
77#890—31no—
78#890—31no—
79#890—31no—
80#888—31no—
81#888—31no—
82#888—32no—
83#888—31no—
84#888—31no—
85#888—31no—
86#888—31no—
87#878—32no—
88#878—31no—
89#878—31no—
90#878—31no—
91#878A411yescharacteristic
92#870—32no—
93#870—31no—
94#870—31no—
95#869—31no—
96#869—32no—
97#869—31no—
98C4×Dic6—32no—
99C4×Dic6—32no—
100C4×Dic6—31no—
101C4×Dic6—31no—
102C4×Dic6—31no—
103C4×Dic6—31no—
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)—31no—
110C4×SL(2,𝔽3)C2×D311yescharacteristic
111C4×SL(2,𝔽3)C2×D311yescharacteristic
112C23×C32—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—
119C2×C3×Dic3—82no—
120C3×Dic6—82no—
121C3×SL(2,𝔽3)C2×Q811yescharacteristic
122Q82—31no—
123Q82—91no—
124#557—31no—
125#557—31no—
126#557—31no—
127#557—91no—
128#557—91no—
129#557—91no—
130#544—91no—
131#544—91no—
132#533—91no—
133#533—91no—
134#519—31no—
135#519C3×D311yescharacteristic
136C2×C4×Q8—31no—
137C2×C3×Q8—31no—
138C2×C3×Q8—32no—
139C2×C3×Q8—121no—
140C2×C3×Q8C3×Q812yescomplemented
141C2×C3×Q8C23×C311yescharacteristic
142C22×C3×C4—31no—
143C22×C3×C4—121no—
144C22×Dic3—31no—
145C22×Dic3C2×A411yescharacteristic
146C2×Dic6—31no—
147C2×Dic6—32no—
148C2×Dic6—31no—
149C2×Dic6—31no—
150C2×Dic6C2×A411yescharacteristic
151C2×SL(2,𝔽3)—22no—
152C2×SL(2,𝔽3)—21no—
153C2×SL(2,𝔽3)Dic612yescomplemented
154C2×SL(2,𝔽3)C22×D311yescharacteristic
155C3×C4⋊C4—31no—
156C3×C4⋊C4—31no—
157C3×C4⋊C4—31no—
158C3×C4⋊C4—31no—
159C3×C4⋊C4—121no—
160C3×C4⋊C4—121no—
161C3×C4⋊C4—121no—
162C3×C22⋊C4—31no—
163C3×C22⋊C4—41no—
164C3×C22⋊C4—81no—
165C3×C22⋊C4—121no—
166C3×C22⋊C4C2×A411yescharacteristic
167C3×C42—32no—
168C3×C42—31no—
169C3×C42—31no—
170#217—31no—
171#217C2×A411yescharacteristic
172C4⋊Dic3—32no—
173C4⋊Dic3—32no—
174C4⋊Dic3—31no—
175C4⋊Dic3—31no—
176C4⋊Dic3—31no—
177C4⋊Dic3—31no—
178C4⋊Dic3—31no—
179C4⋊Dic3—31no—
180C4⋊Dic3—31no—
181C4⋊Dic3—31no—
182C4⋊Dic3—31no—
183C4⋊Dic3C2×A411yescharacteristic
184Dic3⋊C4—32no—
185Dic3⋊C4—32no—
186Dic3⋊C4—31no—
187Dic3⋊C4—31no—
188Dic3⋊C4—31no—
189Dic3⋊C4—31no—
190Dic3⋊C4—31no—
191Dic3⋊C4—31no—
192Dic3⋊C4C2×A411yescharacteristic
193Dic3⋊C4C2×A411yescharacteristic
194C4×Dic3—32no—
195C4×Dic3—31no—
196C4×Dic3—31no—
197C4×Dic3—31no—
198C4×Dic3—31no—
199C4×Dic3—31no—
200C4×Dic3—31no—
201C22×C32—42no—
202C22×C32—41no—
203C22×C32—81no—
204C22×C32—81no—
205C32×C4—81no—
206C32×C4—81no—
207C3×Dic3—42no—
208C3×Dic3—81no—
209C3×Dic3—81no—
210C3×Dic3—81no—
211C22×Q8C2×C3×D311yescharacteristic
212C4⋊Q8—91no—
213C4⋊Q8—92no—
214C4⋊Q8—91no—
215C4⋊Q8—91no—
216C4⋊Q8—91no—
217C4⋊Q8—91no—
218C4⋊Q8—91no—
219C4⋊Q8—92no—
220C4⋊Q8—91no—
221C4⋊Q8—91no—
222#125—91no—
223#125—91no—
224#125—92no—
225#125—91no—
226#125—91no—
227#125—91no—
228#125—91no—
229#125—91no—
230#125—91no—
231#125—92no—
232C22⋊Q8—31no—
233C22⋊Q8—91no—
234C22⋊Q8—91no—
235C22⋊Q8—92no—
236C22⋊Q8—91no—
237C4×Q8—31no—
238C4×Q8—31no—
239C4×Q8—31no—
240C4×Q8—31no—
241C4×Q8—31no—
242C4×Q8—31no—
243C4×Q8—91no—
244C4×Q8—91no—
245C4×Q8—91no—
246C4×Q8—91no—
247C4×Q8—92no—
248C4×Q8—92no—
249C4×Q8—91no—
250C4×Q8—91no—
251C4×Q8—91no—
252C4×Q8—91no—
253C4×Q8C2×C3×D311yescharacteristic
254C4×Q8C2×C3×D311yescharacteristic
255#117—31no—
256#117—31no—
257#117—91no—
258#117—91no—
259C2×C4⋊C4—91no—
260C2×C42—91no—
261C23×C3—41no—
262C23×C3—81no—
263C23×C3C22×A411yescharacteristic
264C3×Q8—32no—
265C3×Q8—31no—
266C3×Q8—32no—
267C3×Q8—242no—
268C3×Q8C2×C3×Q811yescharacteristic
269C2×C3×C4—32no—
270C2×C3×C4—32no—
271C2×C3×C4—31no—
272C2×C3×C4—31no—
273C2×C3×C4—31no—
274C2×C3×C4—31no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—81no—
278C2×C3×C4—81no—
279C2×C3×C4—121no—
280C2×C3×C4—121no—
281C2×C3×C4—121no—
282C2×C3×C4—121no—
283C2×C3×C4—242no—
284C2×C3×C4C22×A411yescharacteristic
285C2×C3×C4C22×A411yescharacteristic
286C2×Dic3—22no—
287C2×Dic3—31no—
288C2×Dic3—31no—
289C2×Dic3—31no—
290C2×Dic3—31no—
291C2×Dic3—121no—
292C2×Dic3C22×A411yescharacteristic
293C2×Dic3C22×A411yescharacteristic
294C2×Dic3C22×A411yescharacteristic
295C2×Dic3C22×A411yescharacteristic
296Dic6—22no—
297Dic6—62no—
298Dic6—121no—
299Dic6—121no—
300Dic6—122no—
301SL(2,𝔽3)—21no—
302SL(2,𝔽3)C2×Dic611yescharacteristic
303C2×C32—41no—
304C2×C32—41no—
305C2×C32—41no—
306C2×C32—82no—
307C2×Q8—31no—
308C2×Q8—31no—
309C2×Q8—32no—
310C2×Q8—91no—
311C2×Q8—91no—
312C2×Q8—92no—
313C2×Q8—91no—
314C2×Q8C3×Dic612yescomplemented
315C2×Q8C22×C3×D311yescharacteristic
316C22×C4—31no—
317C22×C4—31no—
318C22×C4—91no—
319C4⋊C4—31no—
320C4⋊C4—31no—
321C4⋊C4—31no—
322C4⋊C4—31no—
323C4⋊C4—31no—
324C4⋊C4—31no—
325C4⋊C4—31no—
326C4⋊C4—91no—
327C4⋊C4—91no—
328C4⋊C4—91no—
329C4⋊C4—91no—
330C4⋊C4—91no—
331C4⋊C4—91no—
332C4⋊C4—91no—
333C4⋊C4—91no—
334C4⋊C4—91no—
335C4⋊C4—91no—
336C4⋊C4—91no—
337C4⋊C4—91no—
338C4⋊C4—92no—
339C4⋊C4—92no—
340C4⋊C4—92no—
341C4⋊C4—92no—
342C4⋊C4—91no—
343C4⋊C4—91no—
344C4⋊C4—91no—
345C22⋊C4—31no—
346C22⋊C4—31no—
347C22⋊C4—91no—
348C22⋊C4D3×A411yescharacteristic
349C42—32no—
350C42—31no—
351C42—31no—
352C42—91no—
353C42—91no—
354C42—91no—
355C42—91no—
356C42—91no—
357C42—91no—
358C42—92no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—42no—
362C22×C3—41no—
363C22×C3—81no—
364C22×C3—81no—
365C22×C3—82no—
366C22×C3—81no—
367C22×C3—161no—
368C22×C3—161no—
369C22×C3Q8×A412yes—
370C22×C3C23×A411yescharacteristic
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—31no—
374C3×C4—32no—
375C3×C4—31no—
376C3×C4—61no—
377C3×C4—61no—
378C3×C4—81no—
379C3×C4—81no—
380C3×C4—122no—
381C3×C4—161no—
382C3×C4—161no—
383C3×C4—241no—
384C3×C4—241no—
385C3×C4—241no—
386Dic3—21no—
387Dic3—21no—
388Dic3—21no—
389Dic3—61no—
390Dic3—61no—
391Dic3—61no—
392Dic3—61no—
393Dic32−1+4⋊C312yes—
394C32—41noSylow
395C23C2×D3×A411yescharacteristic
396Q8—32no—
397Q8—31no—
398Q8—32no—
399Q8—62no—
400Q8—182no—
401Q8—361no—
402Q8—361no—
403Q8—362no—
404Q8C2×C3×Dic611yescharacteristic
405C2×C4—32no—
406C2×C4—32no—
407C2×C4—31no—
408C2×C4—31no—
409C2×C4—31no—
410C2×C4—31no—
411C2×C4—31no—
412C2×C4—31no—
413C2×C4—31no—
414C2×C4—31no—
415C2×C4—62no—
416C2×C4—91no—
417C2×C4—91no—
418C2×C4—91no—
419C2×C4—91no—
420C2×C4—361no—
421C2×C4C2×D3×A411yescharacteristic
422C2×C4C2×D3×A411yescharacteristic
423C2×C3—22no—
424C2×C3—41no—
425C2×C3—41no—
426C2×C3—41no—
427C2×C3—82no—
428C2×C3—81no—
429C2×C3—81no—
430C2×C3—81no—
431C2×C3—162no—
432C2×C3C2×Q8×A411yescharacteristic
433C2×C3C2×2−1+4⋊C311yescharacteristic
434C2×C3#594511yescharacteristic
435C22—21no—
436C22—21no—
437C22A4×Dic612yes—
438C22C22×D3×A411yescharacteristic
439C4—21no—
440C4—21no—
441C4—32no—
442C4—31no—
443C4—32no—
444C4—31no—
445C4—61no—
446C4—61no—
447C4—61no—
448C4—61no—
449C4—61no—
450C4—181no—
451C4—181no—
452C4—181no—
453C4—181no—
454C3—41no—
455C3—81no—
456C3#3502⋊C311yescharacteristic, complemented
457C2—22no—
458C2C2×A4×Dic611yescharacteristic
459C2D3×2−1+4⋊C311yescharacteristic
460C2#5968⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×#519C2×C3
socleC22×C3C23×A4
prime radicalC22×C3×SL(2,𝔽3)C22
hypercenterC22⋊C4D3×A4
hypocenterC3×Q8C2×C3×Q8
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
4. #5570⋊C3C2semidirect
5. #5569⋊C3C2semidirect
6. #5569⋊C3C2semidirect
7. SL(2,𝔽3)×Dic6C2semidirect
8. #85875C3semidirect
22. #5570C2×C3semidirect
25. #5569C2×C3semidirect
26. #5569C2×C3semidirect
28. Q8×Dic6C2×C3semidirect
43. C2×C3×SL(2,𝔽3)Q8semidirect
140. C2×C3×Q8C3×Q8semidirect
153. C2×SL(2,𝔽3)Dic6semidirect
314. C2×Q8C3×Dic6semidirect
456. C3#3502⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#519⋊C32
#5812⋊C32
C2×Dic3×SL(2,𝔽3)2
#5570⋊C32
#5569⋊C32
SL(2,𝔽3)×Dic62
#858753
#3502⋊C33
C3×#8784