Finite Group Databasebeta

← #67×#226 #11922221

Subgroups — #67×#226 #11922221

Order 1152.

575 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 575 conjugacy classes of subgroups — too many to draw. Only its 63 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×#226C211yesmaximal, characteristic, complemented
2C2×D3×#226C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)×#67C211yesmaximal, characteristic
4#10593653C211yesmaximal, characteristic, complemented
5#10593210C211yesmaximal, characteristic, complemented
6#10593201C211yesmaximal, characteristic, complemented
7Dic3×#226C211yesmaximal, characteristic, complemented
8D4×#226—31nomaximal
9Q16×#67—31nomaximal
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C2×D3×SL(2,𝔽3)C2211yescharacteristic
12C2×C3⋊#226C2211yescharacteristic
13C2×C3×#226—22no—
14C2×C3×#226C2211yescharacteristic
15C2×C3×#226C2211yescharacteristic
16D3×#226—22no—
17Dic3×#67—41nomaximal
18Dic3×SL(2,𝔽3)C2211yescharacteristic
19C3⋊#859C2211yescharacteristic
20C22×#226—31no—
21C22×#226D311yescharacteristic, complemented
22C22×C3×Q16—31no—
23Q8×#67—31no—
24Q8×#67D311yescharacteristic
25C2×D3×Q16—31no—
26D4×SL(2,𝔽3)—31no—
27#5419—31no—
28#5419—31no—
29#5419—61no—
30C4⋊#226—31no—
31C4×#226—31no—
32#5228—31no—
33#5228—31no—
34#5188—31no—
35#5186—31no—
36#5186—31no—
37#5183—31no—
38#5182—31no—
39#5182—31no—
40Dic3×Q16—31no—
41#5116—31no—
42C8×#67—31no—
43C2×C3×#67—41no—
44C22×C3×Dic3—41no—
45C2×C3×SL(2,𝔽3)—21no—
46C2×C3×SL(2,𝔽3)C2311yescharacteristic
47C2×D3×Dic3—41no—
48D3×SL(2,𝔽3)—21no—
49C3⋊#226—21no—
50C3×#226—21no—
51C3×#226D412yescomplemented
52#3757—41no—
53Dic3⋊Dic3—41no—
54(C2×D3)⋊Dic3—41no—
55Dic32—41no—
56D4×Q16—91noSylow
57C22×C3×Q8—31no—
58C22×C3×Q8C2×D311yescharacteristic
59C2×D3×Q8—31no—
60C2×D3×Q8C2×D311yescharacteristic
61C22×SL(2,𝔽3)—21no—
62C22×SL(2,𝔽3)—31no—
63C22×SL(2,𝔽3)C2×D311yescharacteristic
64C2×#226—22no—
65C2×#226—31no—
66C2×#226—61no—
67C2×#226—62no—
68C2×#226C2×D311yescharacteristic
69C2×#226C2×D311yescharacteristic
70C2×C3×Q16—31no—
71C2×C3×Q16—31no—
72C2×C3×Q16—31no—
73C2×C3×Q16—31no—
74C2×C3×Q16—62no—
75C2×C3×Q16—62no—
76C22×C3×C8—31no—
77#946—31no—
78#946—31no—
79#946—61no—
80Q8×Dic3—31no—
81Q8×Dic3C2×D311yescharacteristic
82#944—31no—
83#944—31no—
84#944—61no—
85C2×#216—31no—
86C2×#216—31no—
87C2×#216—31no—
88C2×#216—31no—
89D4×Dic3—121no—
90C4×#67—31no—
91C4×#67—61no—
92#924—31no—
93#924—31no—
94#924—61no—
95D3×Q16—62no—
96D3×Q16—62no—
97C2×Dic12—31no—
98C2×D3×C8—31no—
99C4×SL(2,𝔽3)—31no—
100#859—31no—
101#859—61no—
102#835—31no—
103#835—31no—
104#835—31no—
105#835—31no—
106#830—31no—
107(C2×D3)⋊C8—31no—
108#818—31no—
109#816—31no—
110#816—31no—
111Dic3⋊C8—31no—
112C8×Dic3—31no—
113C23×C32—41no—
114C22×C3×D3—41no—
115C2×C3⋊Dic3—41no—
116C2×C3⋊Dic3—41no—
117C3×#67—44no—
118C2×C3×Dic3—41no—
119C2×C3×Dic3—41no—
120C2×C3×Dic3—41no—
121C2×C3×Dic3—82no—
122C3×SL(2,𝔽3)C2×D411yescharacteristic
123D3×Dic3—82no—
124C22×Q16—31no—
125C22×Q16—91no—
126D4×Q8—31no—
127D4×Q8—91no—
128#494—91no—
129#467—91no—
130#467—91no—
131#462—91no—
132#462—91no—
133#451—91no—
134#451—91no—
135#451—91no—
136#451—91no—
137C4×Q16—91no—
138C8×D4—91no—
139C2×C3×Q8—21no—
140C2×C3×Q8—31no—
141C2×C3×Q8—31no—
142C2×C3×Q8—31no—
143C2×C3×Q8—61no—
144C2×C3×Q8—61no—
145C2×C3×Q8—62no—
146C2×C3×Q8—121no—
147C2×C3×Q8C22×D311yescharacteristic
148C2×C3×D4—121no—
149C22×C3×C4—31no—
150C22×C3×C4—61no—
151C2×#67S411yescharacteristic
152C22×Dic3—41no—
153C22×Dic3—121no—
154D3×Q8—21no—
155D3×Q8—61no—
156D3×Q8—62no—
157D3×Q8—121no—
158C2×C4×D3—31no—
159C2×C4×D3—61no—
160C2×Dic6—31no—
161C2×Dic6—31no—
162C2×Dic6—61no—
163C2×SL(2,𝔽3)—21no—
164C2×SL(2,𝔽3)—21no—
165C2×SL(2,𝔽3)—41no—
166C2×SL(2,𝔽3)—61no—
167C2×SL(2,𝔽3)C22×D311yescharacteristic
168#226—21no—
169#226—61no—
170#226—121no—
171#226#6712yesdirect product factor
172C3×Q16—32no—
173C3×Q16—32no—
174C3×Q16—62no—
175C3×Q16—61no—
176C3×Q16—62no—
177C3×Q16—61no—
178C2×C3×C8—31no—
179C2×C3×C8—31no—
180C2×C3×C8—62no—
181#217—31no—
182#217—61no—
183#217—121no—
184#217—121no—
185#217—241no—
186#216—62no—
187#216—61no—
188#216—61no—
189#216—121no—
190#216—121no—
191(C2×D3)⋊C4—31no—
192(C2×D3)⋊C4—61no—
193C4⋊Dic3—31no—
194C4⋊Dic3—31no—
195C4⋊Dic3—61no—
196C4⋊Dic3—121no—
197Dic3⋊C4—31no—
198Dic3⋊C4—61no—
199Dic3⋊C4—61no—
200Dic3⋊C4—61no—
201Dic3⋊C4—61no—
202Dic3⋊C4—61no—
203C4×Dic3—31no—
204C4×Dic3—61no—
205C4×Dic3—121no—
206C2×C3⋊C8—31no—
207C2×C3⋊C8—31no—
208Dic12—62no—
209D3×C8—62no—
210C22×C32—42no—
211C22×C32—41no—
212C22×C32—81no—
213C22×C32—81no—
214C2×C3×D3—42no—
215C2×C3×D3—81no—
216C2×C3×D3—81no—
217C3⋊Dic3—81no—
218C3⋊Dic3—81no—
219C3×Dic3—42no—
220C3×Dic3—42no—
221C3×Dic3—81no—
222C22×Q8—31no—
223C22×Q8—31no—
224C22×Q8—91no—
225C22×Q8D3211yescharacteristic
226C2×Q16—31no—
227C2×Q16—31no—
228C2×Q16—31no—
229C2×Q16—31no—
230C2×Q16—62no—
231C2×Q16—62no—
232C2×Q16—91no—
233C2×Q16—91no—
234C2×Q16—91no—
235C2×Q16—91no—
236C2×Q16—91no—
237C2×Q16—182no—
238C2×Q16—182no—
239C22×C8—31no—
240C22×C8—91no—
241C4⋊Q8—91no—
242C4⋊Q8—91no—
243C4⋊Q8—181no—
244C22⋊Q8—91no—
245C22⋊Q8—91no—
246C22⋊Q8—91no—
247C22⋊Q8—91no—
248C22⋊Q8—181no—
249C22⋊Q8—181no—
250C4×Q8—31no—
251C4×Q8—91no—
252C4×D4—91no—
253C4×D4—181no—
254#107—91no—
255C4⋊C8—91no—
256#103—91no—
257#103—91no—
258#103—91no—
259#103—91no—
260#103—91no—
261#103—91no—
262C22⋊C8—91no—
263C22⋊C8—91no—
264C4×C8—91no—
265C23×C3—41no—
266C23×C3—81no—
267C23×C3—121no—
268C23×C3C2×S411yescharacteristic
269C22×D3C2×S411yescharacteristic
270C3×Q8—31no—
271C3×Q8—32no—
272C3×Q8—61no—
273C3×Q8—61no—
274C3×Q8—61no—
275C3×Q8—61no—
276C3×Q8—61no—
277C3×Q8—122no—
278C3×Q8D3×D411yescharacteristic
279C3×D4—124no—
280C2×C3×C4—31no—
281C2×C3×C4—31no—
282C2×C3×C4—61no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4—61no—
286C2×C3×C4—121no—
287C2×C3×C4—122no—
288#67#22614yesdirect product factor
289C2×Dic3—31no—
290C2×Dic3—31no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293C2×Dic3—61no—
294C2×Dic3—61no—
295C2×Dic3—82no—
296C2×Dic3—121no—
297C2×Dic3—121no—
298C2×Dic3—241no—
299C2×Dic3—241no—
300C2×Dic3—242no—
301C2×Dic3C2×S411yescharacteristic
302C4×D3—61no—
303C4×D3—61no—
304C4×D3—122no—
305Dic6—61no—
306Dic6—61no—
307Dic6—61no—
308Dic6—61no—
309Dic6—122no—
310SL(2,𝔽3)—21no—
311SL(2,𝔽3)C2×#6711yescharacteristic
312C3×C8—32no—
313C3×C8—61no—
314C3⋊C8—61no—
315C3⋊C8—61no—
316C2×C32—41no—
317C2×C32—41no—
318C2×C32—41no—
319C2×C32—82no—
320C3×D3—82no—
321C2×Q8—21no—
322C2×Q8—31no—
323C2×Q8—31no—
324C2×Q8—31no—
325C2×Q8—61no—
326C2×Q8—61no—
327C2×Q8—62no—
328C2×Q8—61no—
329C2×Q8—91no—
330C2×Q8—91no—
331C2×Q8—121no—
332C2×Q8—181no—
333C2×Q8—181no—
334C2×Q8—182no—
335C2×Q8—361no—
336C2×Q8C2×D3211yescharacteristic
337C2×D4—31no—
338C22×C4—31no—
339C22×C4—61no—
340C22×C4—91no—
341C22×C4—181no—
342Q16—32no—
343Q16—32no—
344Q16—62no—
345Q16—61no—
346Q16—62no—
347Q16—61no—
348Q16—182no—
349Q16—181no—
350Q16—182no—
351Q16—181no—
352Q16—361no—
353Q16—361no—
354C2×C8—31no—
355C2×C8—31no—
356C2×C8—62no—
357C2×C8—91no—
358C2×C8—91no—
359C2×C8—182no—
360C4⋊C4—91no—
361C4⋊C4—91no—
362C4⋊C4—91no—
363C4⋊C4—181no—
364C4⋊C4—181no—
365C4⋊C4—181no—
366C4⋊C4—181no—
367C4⋊C4—181no—
368C4⋊C4—181no—
369C22⋊C4—91no—
370C22⋊C4—91no—
371C22⋊C4—181no—
372C22⋊C4—181no—
373C42—91no—
374C42—181no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—41no—
378C22×C3—42no—
379C22×C3—81no—
380C22×C3—81no—
381C22×C3—81no—
382C22×C3—82no—
383C22×C3—122no—
384C22×C3—161no—
385C22×C3—161no—
386C22×C3—241no—
387C22×C3—241no—
388C22×C3C2×#22612yescomplemented
389C22×C3C22×S411yescharacteristic
390C2×D3—21no—
391C2×D3—21no—
392C2×D3C2×#22612yescomplemented
393C3×C4—31no—
394C3×C4—31no—
395C3×C4—62no—
396C3×C4—61no—
397C3×C4—122no—
398C3×C4—121no—
399Dic3—42no—
400Dic3—61no—
401Dic3—61no—
402Dic3—81no—
403Dic3—121no—
404Dic3—121no—
405Dic3—241no—
406Dic3—241no—
407Dic3—481no—
408Dic3—481no—
409Dic3C2×#22612yescomplemented
410C32—41noSylow
411C23—31no—
412C23D3×S411yescharacteristic
413Q8—31no—
414Q8—32no—
415Q8—61no—
416Q8—61no—
417Q8—61no—
418Q8—61no—
419Q8—61no—
420Q8—122no—
421Q8—181no—
422Q8—181no—
423Q8—181no—
424Q8—181no—
425Q8—362no—
426Q8D3×#6711yescharacteristic
427D4—34no—
428C2×C4—31no—
429C2×C4—31no—
430C2×C4—31no—
431C2×C4—61no—
432C2×C4—61no—
433C2×C4—61no—
434C2×C4—61no—
435C2×C4—91no—
436C2×C4—91no—
437C2×C4—122no—
438C2×C4—181no—
439C2×C4—181no—
440C2×C4—181no—
441C2×C4—181no—
442C2×C4—362no—
443C8—32no—
444C8—61no—
445C8—181no—
446C8—181no—
447C2×C3—22no—
448C2×C3—41no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3—82no—
452C2×C3—81no—
453C2×C3—81no—
454C2×C3—81no—
455C2×C3—162no—
456C2×C3—242no—
457C2×C3D4×S411yescharacteristic
458C2×C3C22×#22611yescharacteristic
459C2×C3#592711yescharacteristic
460D3—22no—
461C22—21no—
462C22—21no—
463C22—32no—
464C22—61no—
465C22—61no—
466C22D3×#22612yescomplemented
467C22C2×D3×S411yescharacteristic
468C4—32no—
469C4—31no—
470C4—31no—
471C4—61no—
472C4—62no—
473C4—121no—
474C4—181no—
475C4—181no—
476C4—361no—
477C4—361no—
478C3—41no—
479C3—81no—
480C3D4×#22611yescharacteristic, complemented
481C2—22no—
482C2—62no—
483C2C2×D3×#22611yescharacteristic
484C2#1059643911yescharacteristic
485C2#67×S411yescharacteristic
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)C2×D4
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
1. C22×C3×#226C2semidirect
2. C2×D3×#226C2semidirect
4. #10593653C2semidirect
5. #10593210C2semidirect
6. #10593201C2semidirect
7. Dic3×#226C2semidirect
21. C22×#226D3semidirect
51. C3×#226D4semidirect
171. #226#67direct (N × H)
288. #67#226direct (N × H)
388. C22×C3C2×#226semidirect
392. C2×D3C2×#226semidirect
409. Dic3C2×#226semidirect
466. C22D3×#226semidirect
480. C3D4×#226semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#2262
C2×D3×#2262
SL(2,𝔽3)×#672
#105936532
#105932102
#105932012
Dic3×#2262
D4×#2263
Q16×#673
Dic3×#674