Finite Group Databasebeta

← #11714052

Subgroups — #11714052

Order 768.

570 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 55 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×#519⋊C3C211yesmaximal, characteristic, complemented
2C2×#5421C211yesmaximal, characteristic
3#73148C211yesmaximal, characteristic, complemented
4#73065C211yesmaximal, characteristic, complemented
5#73065C211yesmaximal, characteristic, complemented
6#73062C211yesmaximal, characteristic, complemented
7#73059C211yesmaximal, characteristic, complemented
8#33015—31nomaximal, Sylow
9C23×SL(2,𝔽3)C2211yescharacteristic
10C22×GL(2,𝔽3)C2211yescharacteristic
11#519⋊C3—22no—
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13C2×C4×SL(2,𝔽3)C2211yescharacteristic
14#5421—22no—
15C2×#859C2211yescharacteristic
16C2×#859C2211yescharacteristic
17C2×#859C2211yescharacteristic
18#4959—41nomaximal
19C2×#519D311yescharacteristic, complemented
20C2×#482—31no—
21C2×#448—31no—
22#2624—31no—
23#1880—31no—
24#1880—31no—
25#1880—31no—
26#1880—31no—
27#1865—31no—
28#1865—31no—
29#1863—31no—
30#1863—31no—
31#1825—31no—
32#1818—31no—
33#1787—31no—
34C22×#67—41no—
35C22×SL(2,𝔽3)—21no—
36C22×SL(2,𝔽3)—21no—
37C22×SL(2,𝔽3)D412yes—
38C22×SL(2,𝔽3)C2311yescharacteristic
39C2×GL(2,𝔽3)—21no—
40C2×GL(2,𝔽3)—21no—
41C2×GL(2,𝔽3)—21no—
42C2×C3×C22⋊C4—41no—
43C2×#217—41no—
44C2×(C2×D3)⋊C4—41no—
45C2×(C2×D3)⋊C4—41no—
46C2×Dic3⋊C4—41no—
47C4×SL(2,𝔽3)—21no—
48C4×SL(2,𝔽3)—21no—
49#859—21no—
50#859—21no—
51#859—21no—
52C3⋊#95—41no—
53C23×Q8C2×D311yescharacteristic
54C22×SD16—31no—
55C22×SD16—31no—
56C2×#121—31no—
57#519—22no—
58#519—62no—
59C2×C4×Q8—31no—
60C2×C4×Q8—31no—
61C2×C4×Q8C2×D311yescharacteristic
62C2×C4×Q8C2×D311yescharacteristic
63C2×#117—31no—
64C2×#117—31no—
65#482—64no—
66#448—62no—
67#448—62no—
68C2×#107—31no—
69C2×#106—31no—
70C2×#103—31no—
71C2×#103—31no—
72C2×#103—31no—
73C2×#103—31no—
74C2×#103—31no—
75C2×#103—31no—
76C2×#102—31no—
77C2×#102—31no—
78C2×C22⋊C8—31no—
79C2×C22⋊C8—31no—
80#399—61no—
81#397—61no—
82#394—61no—
83#394—61no—
84#390—31no—
85#390—31no—
86#389—31no—
87#389—31no—
88#380—31no—
89#340—31no—
90#340—31no—
91#340—31no—
92#340—31no—
93#336—31no—
94#336—31no—
95#336—31no—
96#336—31no—
97C24×C3—41no—
98C23×D3—41no—
99C22×C3×C4—41no—
100C22×C3×C4—41no—
101C2×#67—82no—
102C2×#67—82no—
103C2×#67—82no—
104C22×Dic3—41no—
105C22×Dic3—41no—
106C22×Dic3—41no—
107C2×SL(2,𝔽3)—22no—
108C2×SL(2,𝔽3)C2×D411yescharacteristic
109C2×SL(2,𝔽3)#4111yescharacteristic
110C2×SL(2,𝔽3)#4111yescharacteristic
111GL(2,𝔽3)—41no—
112C3×C22⋊C4—82no—
113#217—82no—
114(C2×D3)⋊C4—82no—
115(C2×D3)⋊C4—82no—
116Dic3⋊C4—82no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—31no—
120C22×Q8—32no—
121C22×Q8—61no—
122C22×Q8—61no—
123C22×Q8#6712yes—
124C22×Q8C22×D311yescharacteristic
125C22×D4—31no—
126C22×D4—61no—
127C23×C4—31no—
128C2×SD16—61no—
129C2×SD16—61no—
130C2×SD16—62no—
131C2×SD16—62no—
132C2×SD16—62no—
133C2×SD16—61no—
134C2×SD16—61no—
135C2×SD16—61no—
136C2×SD16—61no—
137C22×C8—31no—
138C22×C8—31no—
139C22×C8—31no—
140C22×C8—31no—
141#121—62no—
142#121—62no—
143C4×Q8—21no—
144C4×Q8—21no—
145C4×Q8—62no—
146C4×Q8—62no—
147C4×Q8—62no—
148C4×Q8—62no—
149C4×Q8—61no—
150C4×Q8—61no—
151#117—34no—
152#117—62no—
153#117—62no—
154#117—62no—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—31no—
157C2×C4⋊C4—31no—
158C2×C4⋊C4—31no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—61no—
164C2×C22⋊C4—31no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4—61no—
168C2×C22⋊C4—61no—
169C2×C22⋊C4—61no—
170C2×C22⋊C4—61no—
171C2×C22⋊C4S411yescharacteristic
172C2×C42—31no—
173C2×C42—31no—
174C2×C42—61no—
175#107—62no—
176#106—62no—
177#103—61no—
178#103—61no—
179#103—62no—
180#103—62no—
181#103—62no—
182#103—61no—
183#103—61no—
184#103—61no—
185#103—61no—
186#102—62no—
187#102—62no—
188C22⋊C8—62no—
189C22⋊C8—62no—
190#95—61no—
191#95—61no—
192#95—61no—
193C23×C3—42no—
194C23×C3—41no—
195C23×C3—81no—
196C23×C3—81no—
197C23×C3—81no—
198C23×C3—81no—
199C23×C3—81no—
200C23×C3—81no—
201C22×D3—42no—
202C22×D3—81no—
203C22×D3—81no—
204C22×D3—81no—
205C22×D3—81no—
206C22×D3—81no—
207C22×D3—81no—
208C2×C3×C4—81no—
209C2×C3×C4—81no—
210C2×C3×C4—81no—
211C2×C3×C4—81no—
212C2×C3×C4—81no—
213C2×C3×C4—81no—
214#67—88no—
215C2×Dic3—42no—
216C2×Dic3—42no—
217C2×Dic3—81no—
218C2×Dic3—81no—
219C2×Dic3—81no—
220C2×Dic3—81no—
221C2×Dic3—81no—
222C2×Dic3—81no—
223C2×Dic3—81no—
224SL(2,𝔽3)#12311yescharacteristic
225C24—61no—
226C24C2×S411yescharacteristic
227C2×Q8—22no—
228C2×Q8—32no—
229C2×Q8—31no—
230C2×Q8—31no—
231C2×Q8—31no—
232C2×Q8—32no—
233C2×Q8—32no—
234C2×Q8—61no—
235C2×Q8—61no—
236C2×Q8—61no—
237C2×Q8—64no—
238C2×Q8—62no—
239C2×Q8—62no—
240C2×Q8—62no—
241C2×Q8#23511yescharacteristic
242C2×Q8#23511yescharacteristic
243C2×Q8C2×#6711yescharacteristic
244C2×D4—61no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—122no—
251C2×D4—241no—
252C2×D4—241no—
253C22×C4—31no—
254C22×C4—32no—
255C22×C4—32no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—31no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4—61no—
263C22×C4—61no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4C2×S411yescharacteristic
267C22×C4C2×S411yescharacteristic
268SD16—121no—
269SD16—121no—
270SD16—121no—
271SD16—124no—
272SD16—121no—
273C2×C8—61no—
274C2×C8—61no—
275C2×C8—61no—
276C2×C8—61no—
277C2×C8—61no—
278C2×C8—61no—
279C2×C8—61no—
280C2×C8—61no—
281C2×C8—61no—
282C2×C8—61no—
283C2×C8—61no—
284C2×C8—61no—
285C4⋊C4—32no—
286C4⋊C4—32no—
287C4⋊C4—62no—
288C4⋊C4—62no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—241no—
298C4⋊C4—241no—
299C22⋊C4—22no—
300C22⋊C4—62no—
301C22⋊C4—122no—
302C22⋊C4—241no—
303C22⋊C4—241no—
304C22⋊C4—241no—
305C22⋊C4—241no—
306C22⋊C4—241no—
307C42—61no—
308C42—61no—
309C42—61no—
310C42—61no—
311C42—64no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—82no—
320C22×C3—82no—
321C22×C3—82no—
322C22×C3—161no—
323C22×C3—161no—
324C22×C3—161no—
325C22×C3—161no—
326C2×D3—82no—
327C2×D3—82no—
328C2×D3—82no—
329C2×D3—161no—
330C2×D3—161no—
331C2×D3—161no—
332C2×D3—161no—
333C3×C4—161no—
334C3×C4—161no—
335Dic3—82no—
336Dic3—82no—
337Dic3—161no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—121no—
345C23—121no—
346C23—121no—
347C23—121no—
348C23—121no—
349C23—121no—
350C23—121no—
351C23GL(2,ℤ/4)12yes—
352C23C22×S411yescharacteristic
353Q8—31no—
354Q8—31no—
355Q8—31no—
356Q8—34no—
357Q8—61no—
358Q8—62no—
359Q8—62no—
360Q8—62no—
361Q8#92911yescharacteristic
362D4—121no—
363D4—121no—
364D4—121no—
365D4—121no—
366D4—244no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—31no—
374C2×C4—31no—
375C2×C4—31no—
376C2×C4—31no—
377C2×C4—31no—
378C2×C4—31no—
379C2×C4—32no—
380C2×C4—62no—
381C2×C4—62no—
382C2×C4—62no—
383C2×C4—62no—
384C2×C4—61no—
385C2×C4—61no—
386C2×C4—61no—
387C2×C4—61no—
388C2×C4—61no—
389C2×C4—61no—
390C2×C4—61no—
391C2×C4—61no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4—121no—
400C2×C4—121no—
401C8—121no—
402C8—121no—
403C8—121no—
404C8—121no—
405C2×C3—41no—
406C2×C3—41no—
407C2×C3—41no—
408C2×C3—41no—
409C2×C3—41no—
410C2×C3—41no—
411C2×C3—41no—
412C2×C3—162no—
413D3—162no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—241no—
422C22—241no—
423C22—241no—
424C22—241no—
425C22—241no—
426C22—241no—
427C22—241no—
428C22#591211yescharacteristic
429C22#591211yescharacteristic
430C22#592611yescharacteristic
431C22#592611yescharacteristic
432C22#592611yescharacteristic
433C22#592711yescharacteristic
434C22C2×GL(2,ℤ/4)11yescharacteristic
435C4—31no—
436C4—31no—
437C4—31no—
438C4—31no—
439C4—41no—
440C4—41no—
441C4—62no—
442C4—121no—
443C4—121no—
444C4—241no—
445C4—241no—
446C4—241no—
447C3—41noSylow
448C2—42no—
449C2—481no—
450C2#8535611yescharacteristic
451C2#8535711yescharacteristic
452C2#8535811yescharacteristic
453C2#8535811yescharacteristic
454C2#8548511yescharacteristic
455C2#8553711yescharacteristic
456C2#8553911yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#519D3
prime radicalC2×#5421C2
hypercenterC2×C22⋊C4S4
hypocenterSL(2,𝔽3)#123
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#519⋊C3C2semidirect
3. #73148C2semidirect
4. #73065C2semidirect
5. #73065C2semidirect
6. #73062C2semidirect
7. #73059C2semidirect
19. C2×#519D3semidirect

Maximal subgroups

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

subgroupindex
C2×#519⋊C32
C2×#54212
#731482
#730652
#730622
#730592
#330153
#49594