Finite Group Databasebeta

← #11712849

Subgroups — #11712849

Order 768.

545 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 545 conjugacy classes of subgroups — too many to draw. Only its 69 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×#5395C211yesmaximal, characteristic
2C2×#5394C211yesmaximal, characteristic
3C2×C4×#226C211yesmaximal, characteristic, complemented
4#73167C211yesmaximal, characteristic, complemented
5#73065C211yesmaximal, characteristic
6#73061C211yesmaximal, characteristic, complemented
7C4⋊#859C211yesmaximal, characteristic, complemented
8#21512—31nomaximal, Sylow
9C22×GL(2,𝔽3)C2211yescharacteristic
10C22×#226C2211yescharacteristic
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13C2×C4×SL(2,𝔽3)C2211yescharacteristic
14C2×#859C2211yescharacteristic
15C2×#859C2211yescharacteristic
16#5395C414yescomplemented
17#5394—22no—
18C4×#226—22no—
19#4670—41nomaximal
20C2×#464—31no—
21C2×#440—31no—
22C2×C4×Q16—31no—
23#2326—31no—
24#2276D311yescharacteristic, complemented
25#1983—31no—
26#1947—31no—
27#1928—31no—
28#1925—31no—
29#1880—31no—
30#1878—31no—
31#1803—31no—
32#1768—31no—
33#1763—31no—
34C4×#102—31no—
35C22×SL(2,𝔽3)C2311yescharacteristic
36C2×GL(2,𝔽3)—21no—
37C2×GL(2,𝔽3)—21no—
38C2×GL(2,𝔽3)C2×C412yes—
39C2×#226—21no—
40C2×#226—21no—
41C2×#226C2×C412yescomplemented
42C2×(C2×D3)⋊C4—41no—
43C2×(C2×D3)⋊C4—41no—
44C2×(C2×D3)⋊C4—41no—
45C2×C4⋊Dic3—41no—
46C2×C4×Dic3—41no—
47C4×SL(2,𝔽3)—21no—
48C4×SL(2,𝔽3)C2×C412yescomplemented
49C4×SL(2,𝔽3)D412yes—
50#859—21no—
51#859—21no—
52C3×#95—41no—
53C3⋊#95—41no—
54C22×Q16—31no—
55C22×SD16—31no—
56C2×#124—31no—
57C2×C4×Q8—31no—
58C2×C4×Q8C2×D311yescharacteristic
59C2×C4×Q8C2×D311yescharacteristic
60C2×C4×Q8C2×D311yescharacteristic
61C2×C4×D4—31no—
62#464—34no—
63#464—34no—
64#440—62no—
65#440—62no—
66C4×Q16—62no—
67C4×Q16—62no—
68C2×#107—31no—
69C2×#107—31no—
70C2×C4⋊C8—31no—
71C2×C4⋊C8—31no—
72C2×#103—31no—
73C2×#103—31no—
74C2×#103—31no—
75C2×#103—31no—
76C2×#103—31no—
77C2×#102—31no—
78C2×#102—31no—
79C2×#102—31no—
80C2×#97—31no—
81C2×C4×C8—31no—
82#391—31no—
83#390—31no—
84#388—61no—
85#388—61no—
86#385—61no—
87#385—61no—
88#384—31no—
89C4×C4⋊C4—31no—
90C4×C4⋊C4—31no—
91C4×C4⋊C4—31no—
92#376—31no—
93#340—31no—
94#340—31no—
95#340—31no—
96#340—31no—
97#336—31no—
98#336—31no—
99C23×D3—41no—
100C22×C3×C4—41no—
101C22×C3×C4—41no—
102C22×C3×C4—41no—
103C22×Dic3—41no—
104C22×Dic3—41no—
105C22×Dic3—41no—
106C2×SL(2,𝔽3)C22×C411yescharacteristic
107C2×SL(2,𝔽3)C2×D411yescharacteristic
108C2×SL(2,𝔽3)#4111yescharacteristic
109GL(2,𝔽3)—22no—
110#226—22no—
111(C2×D3)⋊C4—44no—
112(C2×D3)⋊C4—82no—
113(C2×D3)⋊C4—82no—
114C4⋊Dic3—82no—
115C4×Dic3—82no—
116C22×Q8—31no—
117C22×Q8C22×D311yescharacteristic
118C22×D4—31no—
119C23×C4—61no—
120C2×Q16—32no—
121C2×Q16—32no—
122C2×Q16—61no—
123C2×Q16—61no—
124C2×Q16—61no—
125C2×Q16—61no—
126C2×SD16—32no—
127C2×SD16—32no—
128C2×SD16—61no—
129C2×SD16—61no—
130C2×SD16—61no—
131C2×SD16—61no—
132C22×C8—31no—
133C22×C8—31no—
134C22×C8—31no—
135C22×C8—31no—
136#124—34no—
137#124—62no—
138C4×Q8—21no—
139C4×Q8—32no—
140C4×Q8—32no—
141C4×Q8—62no—
142C4×Q8—62no—
143C4×Q8—61no—
144C4×Q8C4×D312yes—
145C4×Q8D1212yes—
146C4×D4—62no—
147C4×D4—62no—
148C2×C4⋊C4—31no—
149C2×C4⋊C4—31no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—61no—
154C2×C4⋊C4—61no—
155C2×C4⋊C4—61no—
156C2×C22⋊C4—61no—
157C2×C22⋊C4—61no—
158C2×C22⋊C4—61no—
159C2×C22⋊C4—61no—
160C2×C22⋊C4—61no—
161C2×C42—31no—
162C2×C42—31no—
163C2×C42—31no—
164C2×C42—61no—
165C2×C42—61no—
166#107—62no—
167#107—62no—
168C4⋊C8—34no—
169C4⋊C8—62no—
170#103—34no—
171#103—62no—
172#103—62no—
173#103—61no—
174#103—61no—
175#103—61no—
176#103—61no—
177#102—34no—
178#102—62no—
179#102—62no—
180#97—62no—
181C4×C8—62no—
182#95—31no—
183#95—31no—
184#95—31no—
185#95—61no—
186#95—61no—
187#95S411yescharacteristic
188C23×C3—41no—
189C22×D3—42no—
190C22×D3—81no—
191C22×D3—81no—
192C22×D3—81no—
193C22×D3—81no—
194C22×D3—81no—
195C22×D3—81no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4—81no—
199C2×C3×C4—81no—
200C2×C3×C4—81no—
201C2×C3×C4—81no—
202C2×C3×C4—81no—
203C2×C3×C4—81no—
204C2×C3×C4—81no—
205C2×Dic3—42no—
206C2×Dic3—42no—
207C2×Dic3—81no—
208C2×Dic3—81no—
209C2×Dic3—81no—
210C2×Dic3—81no—
211C2×Dic3—81no—
212C2×Dic3—81no—
213C2×Dic3—81no—
214SL(2,𝔽3)C4×D411yescharacteristic
215C24—61no—
216C2×Q8—31no—
217C2×Q8—31no—
218C2×Q8—31no—
219C2×Q8—32no—
220C2×Q8—61no—
221C2×Q8—61no—
222C2×Q8—61no—
223C2×Q8—61no—
224C2×Q8—61no—
225C2×Q8C2×C4×D311yescharacteristic
226C2×Q8C2×D1211yescharacteristic
227C2×Q8#23511yescharacteristic
228C2×D4—32no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—61no—
234C22×C4—31no—
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—31no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—122no—
242C22×C4—241no—
243C22×C4—241no—
244C22×C4C2×S411yescharacteristic
245C22×C4C2×S411yescharacteristic
246C22×C4C2×S411yescharacteristic
247Q16—62no—
248Q16—62no—
249Q16—62no—
250Q16—62no—
251SD16—62no—
252SD16—62no—
253SD16—62no—
254SD16—62no—
255C2×C8—32no—
256C2×C8—32no—
257C2×C8—61no—
258C2×C8—61no—
259C2×C8—61no—
260C2×C8—61no—
261C2×C8—61no—
262C2×C8—61no—
263C2×C8—61no—
264C2×C8—61no—
265C2×C8—61no—
266C2×C8—61no—
267C4⋊C4—32no—
268C4⋊C4—32no—
269C4⋊C4—32no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—122no—
278C4⋊C4—241no—
279C4⋊C4—241no—
280C22⋊C4—64no—
281C22⋊C4—64no—
282C22⋊C4—122no—
283C22⋊C4—241no—
284C22⋊C4—241no—
285C42—32no—
286C42—32no—
287C42—32no—
288C42—32no—
289C42—61no—
290C42—61no—
291C42—122no—
292C42—241no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C2×D3—82no—
301C2×D3—82no—
302C2×D3—82no—
303C2×D3—161no—
304C2×D3—161no—
305C2×D3—161no—
306C2×D3—161no—
307C3×C4—82no—
308C3×C4—82no—
309C3×C4—161no—
310Dic3—82no—
311Dic3—82no—
312Dic3—161no—
313C23—62no—
314C23—121no—
315C23—121no—
316C23—121no—
317C23—121no—
318C23—121no—
319C23—121no—
320C23C22×S411yescharacteristic
321Q8—31no—
322Q8—31no—
323Q8—31no—
324Q8—61no—
325Q8—62no—
326Q8—62no—
327Q8—121no—
328Q8—121no—
329Q8C4×D1211yescharacteristic
330D4—62no—
331D4—62no—
332D4—121no—
333D4—121no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—31no—
342C2×C4—31no—
343C2×C4—31no—
344C2×C4—31no—
345C2×C4—31no—
346C2×C4—31no—
347C2×C4—32no—
348C2×C4—32no—
349C2×C4—61no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—62no—
354C2×C4—61no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—244no—
366C2×C4C4×S412yes—
367C2×C4C4⋊S412yes—
368C8—62no—
369C8—62no—
370C8—121no—
371C8—121no—
372C2×C3—41no—
373C2×C3—41no—
374C2×C3—41no—
375C2×C3—41no—
376C2×C3—41no—
377C2×C3—41no—
378C2×C3—41no—
379D3—162no—
380C22—122no—
381C22—122no—
382C22—122no—
383C22—241no—
384C22—241no—
385C22—241no—
386C22—241no—
387C22C2×C4×S411yescharacteristic
388C22C2×C4⋊S411yescharacteristic
389C22#591211yescharacteristic
390C22#592511yescharacteristic
391C22#592611yescharacteristic
392C22#592711yescharacteristic
393C22#592711yescharacteristic
394C4—22no—
395C4—22no—
396C4—31no—
397C4—31no—
398C4—31no—
399C4—31no—
400C4—41no—
401C4—62no—
402C4—62no—
403C4—122no—
404C4—121no—
405C4—241no—
406C4—241no—
407C3—41noSylow
408C2—242no—
409C2C4×C4⋊S411yescharacteristic
410C2#8535411yescharacteristic
411C2#8535711yescharacteristic
412C2#8535911yescharacteristic
413C2#8536511yescharacteristic
414C2#8536811yescharacteristic
415C2#8537111yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
Fitting#2276D3
prime radicalC22×GL(2,𝔽3)C22
cosocleC22×SL(2,𝔽3)C23
hypercenter#95S4
hypocenterSL(2,𝔽3)C4×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
3. C2×C4×#226C2semidirect
4. #73167C2semidirect
6. #73061C2semidirect
7. C4⋊#859C2semidirect
16. #5395C4semidirect
24. #2276D3semidirect
41. C2×#226C2×C4semidirect
48. C4×SL(2,𝔽3)C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#53952
C2×#53942
C2×C4×#2262
#731672
#730652
#730612
C4⋊#8592
#215123
#46704