Finite Group Databasebeta

← #11713983

Subgroups — #11713983

Order 768.

615 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 615 conjugacy classes of subgroups — too many to draw. Only its 61 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×#520⋊C3C211yesmaximal, characteristic, complemented
2C2×#5427C211yesmaximal, characteristic
3C2×#5425C211yesmaximal, characteristic, complemented
4C2×#5395C211yesmaximal, characteristic, complemented
5#73065C211yesmaximal, characteristic, complemented
6#73061C211yesmaximal, characteristic, complemented
7C4⋊#859C211yesmaximal, characteristic, complemented
8#32919—31nomaximal, Sylow
9C22×#41⋊C3C2211yescharacteristic
10C22×GL(2,𝔽3)C2211yescharacteristic
11C22×#226C2211yescharacteristic
12#520⋊C3—22no—
13C2×C4×SL(2,𝔽3)C2211yescharacteristic
14C2×C4×SL(2,𝔽3)C2211yescharacteristic
15#5427—22no—
16#5425—22no—
17C2×#859C2211yescharacteristic
18C2×#859C2211yescharacteristic
19#5395—22no—
20#4994—41nomaximal
21C2×#520D311yescharacteristic, complemented
22C2×#464—31no—
23C2×#463—31no—
24C2×#452—31no—
25C2×#449—31no—
26#2620—31no—
27#1880—31no—
28#1879—31no—
29#1878—31no—
30#1877—31no—
31#1874—31no—
32#1872—31no—
33#1803—31no—
34#1802—31no—
35#1798—31no—
36C2×#41⋊C3—21no—
37C2×#41⋊C3—21no—
38C2×#41⋊C3D412yes—
39C22×SL(2,𝔽3)C2311yescharacteristic
40C2×GL(2,𝔽3)—21no—
41C2×GL(2,𝔽3)—21no—
42C2×GL(2,𝔽3)—21no—
43C2×#226—21no—
44C2×#226—21no—
45C2×#226—21no—
46C2×C3×C4⋊C4—41no—
47C2×(C2×D3)⋊C4—41no—
48C2×(C2×D3)⋊C4—41no—
49C2×(C2×D3)⋊C4—41no—
50C2×C4⋊Dic3—41no—
51C2×Dic3⋊C4—41no—
52C4×SL(2,𝔽3)—21no—
53C4×SL(2,𝔽3)D412yescomplemented
54#859—21no—
55#859—21no—
56C3⋊#95—41no—
57C22×#41C2×D311yescharacteristic
58C22×Q16—31no—
59C22×SD16—31no—
60C22×SD16—31no—
61C22×D8—31no—
62C2×#124—31no—
63C2×C22⋊Q8—31no—
64C2×#121—31no—
65#520—22no—
66#520—62no—
67C2×C4×Q8C2×D311yescharacteristic
68C2×C4×Q8C2×D311yescharacteristic
69C2×C4×D4—31no—
70C2×C4×D4—31no—
71C2×#117—31no—
72C22×C4⋊C4—31no—
73#464—62no—
74#464—62no—
75#463—64no—
76#452—62no—
77#452—62no—
78#449—62no—
79#449—62no—
80C2×C4⋊C8—31no—
81C2×C4⋊C8—31no—
82C2×#103—31no—
83C2×#103—31no—
84C2×#103—31no—
85C2×#103—31no—
86C2×#102—31no—
87C2×#102—31no—
88C2×#102—31no—
89C2×#102—31no—
90C2×C22⋊C8—31no—
91C2×C22⋊C8—31no—
92#396—61no—
93#396—61no—
94#394—61no—
95#394—61no—
96#391—31no—
97#390—31no—
98#384—31no—
99#340—31no—
100#340—31no—
101#340—31no—
102#340—31no—
103#336—31no—
104#336—31no—
105C23×D3—41no—
106C22×C3×C4—41no—
107C22×C3×C4—41no—
108C22×C3×C4—41no—
109C22×Dic3—41no—
110C22×Dic3—41no—
111C22×Dic3—41no—
112#41⋊C3—22no—
113C2×SL(2,𝔽3)C2×D411yescharacteristic
114C2×SL(2,𝔽3)C2×D411yescharacteristic
115C2×SL(2,𝔽3)#4111yescharacteristic
116GL(2,𝔽3)—41no—
117#226—41no—
118C3×C4⋊C4—82no—
119(C2×D3)⋊C4—82no—
120(C2×D3)⋊C4—82no—
121(C2×D3)⋊C4—82no—
122C4⋊Dic3—82no—
123Dic3⋊C4—82no—
124C2×#41—21no—
125C2×#41—21no—
126C2×#41—32no—
127C2×#41—61no—
128C2×#41—61no—
129C2×#41D1212yes—
130C22×Q8—31no—
131C22×Q8C22×D311yescharacteristic
132C22×D4—31no—
133C22×D4—31no—
134C22×D4—61no—
135C23×C4—31no—
136C23×C4—61no—
137C2×Q16—61no—
138C2×Q16—61no—
139C2×Q16—61no—
140C2×Q16—61no—
141C2×Q16—61no—
142C2×Q16—61no—
143C2×SD16—62no—
144C2×SD16—62no—
145C2×SD16—62no—
146C2×SD16—61no—
147C2×SD16—61no—
148C2×SD16—61no—
149C2×SD16—61no—
150C2×SD16—61no—
151C2×SD16—61no—
152C2×D8—62no—
153C2×D8—62no—
154C2×D8—62no—
155C22×C8—31no—
156C22×C8—31no—
157C22×C8—31no—
158C22×C8—31no—
159#124—62no—
160#124—62no—
161C22⋊Q8—62no—
162C22⋊Q8—62no—
163#121—62no—
164#121—62no—
165C4×Q8—21no—
166C4×Q8—32no—
167C4×Q8—61no—
168C4×Q8#6712yes—
169C4×D4—34no—
170C4×D4—62no—
171C4×D4—62no—
172C4×D4—62no—
173#117—62no—
174#117—62no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—62no—
181C2×C4⋊C4—62no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4S411yescharacteristic
186C2×C22⋊C4—31no—
187C2×C22⋊C4—31no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—61no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—61no—
194C2×C42—31no—
195C2×C42—31no—
196C4⋊C8—62no—
197C4⋊C8—62no—
198#103—62no—
199#103—62no—
200#103—61no—
201#103—61no—
202#103—61no—
203#103—61no—
204#102—62no—
205#102—62no—
206#102—62no—
207#102—62no—
208C22⋊C8—62no—
209C22⋊C8—62no—
210#95—61no—
211#95—61no—
212C23×C3—41no—
213C22×D3—42no—
214C22×D3—81no—
215C22×D3—81no—
216C22×D3—81no—
217C22×D3—81no—
218C22×D3—81no—
219C22×D3—81no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—81no—
223C2×C3×C4—81no—
224C2×C3×C4—81no—
225C2×C3×C4—81no—
226C2×C3×C4—81no—
227C2×C3×C4—81no—
228C2×C3×C4—81no—
229C2×Dic3—42no—
230C2×Dic3—81no—
231C2×Dic3—81no—
232C2×Dic3—81no—
233C2×Dic3—81no—
234C2×Dic3—81no—
235C2×Dic3—81no—
236C2×Dic3—81no—
237C2×Dic3—81no—
238SL(2,𝔽3)#12111yescharacteristic
239C24—31no—
240C24—61no—
241#41—22no—
242#41—64no—
243#41—62no—
244#41—62no—
245#41—62no—
246C2×Q8—31no—
247C2×Q8—31no—
248C2×Q8—31no—
249C2×Q8—61no—
250C2×Q8—61no—
251C2×Q8—61no—
252C2×Q8—61no—
253C2×Q8—61no—
254C2×Q8—61no—
255C2×Q8C2×D1211yescharacteristic
256C2×Q8#23511yescharacteristic
257C2×Q8C2×#6711yescharacteristic
258C2×D4—32no—
259C2×D4—32no—
260C2×D4—32no—
261C2×D4—61no—
262C2×D4—61no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—61no—
266C2×D4—61no—
267C2×D4—61no—
268C2×D4—61no—
269C2×D4—61no—
270C2×D4—122no—
271C2×D4—241no—
272C2×D4—241no—
273C22×C4—31no—
274C22×C4—31no—
275C22×C4—32no—
276C22×C4—32no—
277C22×C4—31no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4—64no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—241no—
287C22×C4—241no—
288C22×C4C2×S411yescharacteristic
289C22×C4C2×S411yescharacteristic
290C22×C4C2×S411yescharacteristic
291Q16—121no—
292Q16—121no—
293Q16—121no—
294Q16—121no—
295SD16—124no—
296SD16—121no—
297SD16—121no—
298SD16—121no—
299SD16—121no—
300D8—124no—
301C2×C8—61no—
302C2×C8—61no—
303C2×C8—61no—
304C2×C8—61no—
305C2×C8—61no—
306C2×C8—61no—
307C2×C8—61no—
308C2×C8—61no—
309C2×C8—61no—
310C2×C8—61no—
311C2×C8—61no—
312C2×C8—61no—
313C4⋊C4—22no—
314C4⋊C4—32no—
315C4⋊C4—32no—
316C4⋊C4—62no—
317C4⋊C4—61no—
318C4⋊C4—61no—
319C4⋊C4—61no—
320C4⋊C4—61no—
321C4⋊C4—61no—
322C4⋊C4—61no—
323C4⋊C4—122no—
324C4⋊C4—241no—
325C4⋊C4—241no—
326C22⋊C4—62no—
327C22⋊C4—62no—
328C22⋊C4—122no—
329C22⋊C4—122no—
330C22⋊C4—122no—
331C22⋊C4—122no—
332C22⋊C4—241no—
333C22⋊C4—241no—
334C42—32no—
335C42—32no—
336C42—61no—
337C42—61no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—41no—
345C2×D3—82no—
346C2×D3—82no—
347C2×D3—82no—
348C2×D3—161no—
349C2×D3—161no—
350C2×D3—161no—
351C2×D3—161no—
352C3×C4—82no—
353C3×C4—82no—
354C3×C4—161no—
355Dic3—82no—
356Dic3—161no—
357Dic3—161no—
358C23—32no—
359C23—61no—
360C23—61no—
361C23—61no—
362C23—61no—
363C23—61no—
364C23—61no—
365C23—121no—
366C23—121no—
367C23—121no—
368C23—121no—
369C23—121no—
370C23—121no—
371C23—121no—
372C23C22×S411yescharacteristic
373Q8—31no—
374Q8—31no—
375Q8—31no—
376Q8—61no—
377Q8—121no—
378Q8—121no—
379Q8—121no—
380Q8—121no—
381Q8#93011yescharacteristic
382D4—34no—
383D4—62no—
384D4—62no—
385D4—62no—
386D4—121no—
387D4—121no—
388D4—121no—
389D4—121no—
390D4—244no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—31no—
399C2×C4—31no—
400C2×C4—31no—
401C2×C4—31no—
402C2×C4—31no—
403C2×C4—31no—
404C2×C4—32no—
405C2×C4—62no—
406C2×C4—62no—
407C2×C4—62no—
408C2×C4—62no—
409C2×C4—61no—
410C2×C4—61no—
411C2×C4—61no—
412C2×C4—61no—
413C2×C4—61no—
414C2×C4—121no—
415C2×C4—121no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—121no—
419C2×C4—121no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4—244no—
424C2×C4C4⋊S412yes—
425C2×C4GL(2,ℤ/4)12yes—
426C8—121no—
427C8—121no—
428C8—121no—
429C8—121no—
430C2×C3—41no—
431C2×C3—41no—
432C2×C3—41no—
433C2×C3—41no—
434C2×C3—41no—
435C2×C3—41no—
436C2×C3—41no—
437D3—162no—
438C22—62no—
439C22—62no—
440C22—62no—
441C22—121no—
442C22—121no—
443C22—121no—
444C22—121no—
445C22—241no—
446C22—241no—
447C22—241no—
448C22—241no—
449C22—241no—
450C22—241no—
451C22—241no—
452C22C2×C4⋊S411yescharacteristic
453C22#591211yescharacteristic
454C22#592411yescharacteristic
455C22#592511yescharacteristic
456C22#592611yescharacteristic
457C22#592711yescharacteristic
458C22C2×GL(2,ℤ/4)11yescharacteristic
459C4—22no—
460C4—22no—
461C4—31no—
462C4—31no—
463C4—31no—
464C4—31no—
465C4—41no—
466C4—62no—
467C4—121no—
468C4—241no—
469C4—241no—
470C4—241no—
471C3—41noSylow
472C2—122no—
473C2—481no—
474C2#8536811yescharacteristic
475C2#8536911yescharacteristic
476C2#8537011yescharacteristic
477C2#8537111yescharacteristic
478C2#8548611yescharacteristic
479C2#8552911yescharacteristic
480C2#8553911yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#520D3
prime radicalC2×#5427C2
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#121
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#520⋊C3C2semidirect
3. C2×#5425C2semidirect
4. C2×#5395C2semidirect
5. #73065C2semidirect
6. #73061C2semidirect
7. C4⋊#859C2semidirect
21. C2×#520D3semidirect
53. C4×SL(2,𝔽3)D4semidirect

Maximal subgroups

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

subgroupindex
C2×#520⋊C32
C2×#54272
C2×#54252
C2×#53952
#730652
#730612
C4⋊#8592
#329193
#49944