Finite Group Databasebeta

← #11714018

Subgroups — #11714018

Order 768.

569 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 569 conjugacy classes of subgroups — too many to draw. Only its 71 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×C4×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×#5389C211yesmaximal, characteristic, complemented
3#73148C211yesmaximal, characteristic
4#73148C211yesmaximal, characteristic
5#73064C211yesmaximal, characteristic, complemented
6#73062C211yesmaximal, characteristic, complemented
7#73059C211yesmaximal, characteristic, complemented
8#32960—31nomaximal, Sylow
9C23×SL(2,𝔽3)C2211yescharacteristic
10C2×C4×SL(2,𝔽3)—22no—
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13C2×#859C2211yescharacteristic
14C2×#859C2211yescharacteristic
15C2×#859C2211yescharacteristic
16C2×#859C2211yescharacteristic
17#5389—22no—
18#5210—41nomaximal
19C22×C4×Q8D311yescharacteristic
20C2×#479—31no—
21C2×#479—31no—
22#2851—31no—
23#1865—31no—
24#1865—31no—
25#1863—31no—
26#1863—31no—
27#1853—31no—
28#1852—31no—
29#1805—31no—
30#1805—31no—
31#1797—31no—
32#1787—31no—
33#1787—31no—
34C23×C3×C4—41no—
35C22×SL(2,𝔽3)—21no—
36C22×SL(2,𝔽3)—21no—
37C22×SL(2,𝔽3)Q812yes—
38C22×SL(2,𝔽3)C2311yescharacteristic
39C2×#217—41no—
40C2×#217—41no—
41C2×Dic3⋊C4—41no—
42C2×Dic3⋊C4—41no—
43C4×SL(2,𝔽3)—21no—
44C4×SL(2,𝔽3)D412yes—
45#859—21no—
46#859—21no—
47#859—21no—
48#859—21no—
49C3⋊#95—41no—
50C3⋊#95—41no—
51C23×Q8C2×D311yescharacteristic
52C2×#125—31no—
53C2×C4×Q8—22no—
54C2×C4×Q8—31no—
55C2×C4×Q8—31no—
56C2×C4×Q8—62no—
57C2×C4×Q8C2×D311yescharacteristic
58C2×C4×Q8C2×D311yescharacteristic
59C22×C4⋊C4—31no—
60C22×C42—31no—
61#479—62no—
62#479—64no—
63#479—62no—
64C2×#107—31no—
65C2×#107—31no—
66C2×#106—31no—
67C2×#106—31no—
68C2×C4⋊C8—31no—
69C2×C4⋊C8—31no—
70C2×#103—31no—
71C2×#103—31no—
72C2×#103—31no—
73C2×#103—31no—
74C2×#103—31no—
75C2×#103—31no—
76C2×#103—31no—
77C2×#103—31no—
78C2×C22⋊C8—31no—
79C2×C22⋊C8—31no—
80#389—31no—
81#389—31no—
82#387—61no—
83#387—61no—
84#385—61no—
85#385—61no—
86#382—31no—
87#380—31no—
88#380—31no—
89#340—31no—
90#340—31no—
91#340—31no—
92#340—31no—
93#336—31no—
94#336—31no—
95C24×C3—41no—
96C22×C3×C4—41no—
97C22×C3×C4—41no—
98C22×C3×C4—82no—
99C22×C3×C4—82no—
100C22×C3×C4—82no—
101C22×Dic3—41no—
102C22×Dic3—41no—
103C22×Dic3—41no—
104C22×Dic3—41no—
105C2×SL(2,𝔽3)—22no—
106C2×SL(2,𝔽3)C2×D411yescharacteristic
107C2×SL(2,𝔽3)C2×Q811yescharacteristic
108C2×SL(2,𝔽3)#4111yescharacteristic
109#217—44no—
110#217—82no—
111Dic3⋊C4—82no—
112Dic3⋊C4—82no—
113C22×Q8—21no—
114C22×Q8—21no—
115C22×Q8—31no—
116C22×Q8—32no—
117C22×Q8—61no—
118C22×Q8—61no—
119C22×Q8Dic612yes—
120C22×Q8C22×D311yescharacteristic
121C23×C4—31no—
122C23×C4—31no—
123C23×C4S411yescharacteristic
124C22×C8—31no—
125C22×C8—31no—
126C22×C8—31no—
127C22×C8—31no—
128#125—62no—
129#125—62no—
130C4×Q8—21no—
131C4×Q8—32no—
132C4×Q8—34no—
133C4×Q8—62no—
134C4×Q8—62no—
135C4×Q8—61no—
136C4×Q8—62no—
137C4×Q8#6712yes—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4—31no—
142C2×C4⋊C4—31no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—62no—
145C2×C4⋊C4—62no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C22⋊C4—61no—
152C2×C22⋊C4—61no—
153C2×C42—31no—
154C2×C42—31no—
155C2×C42—31no—
156C2×C42—31no—
157C2×C42—62no—
158C2×C42—62no—
159#107—62no—
160#107—62no—
161#106—62no—
162#106—62no—
163C4⋊C8—62no—
164C4⋊C8—62no—
165#103—61no—
166#103—61no—
167#103—62no—
168#103—62no—
169#103—61no—
170#103—62no—
171#103—61no—
172#103—62no—
173#103—61no—
174#103—61no—
175#103—61no—
176#103—61no—
177C22⋊C8—62no—
178C22⋊C8—62no—
179#95—61no—
180#95—61no—
181#95—61no—
182#95—61no—
183#95—61no—
184#95—61no—
185C23×C3—42no—
186C23×C3—42no—
187C23×C3—42no—
188C23×C3—41no—
189C23×C3—81no—
190C23×C3—81no—
191C23×C3—81no—
192C23×C3—81no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4—81no—
196C2×C3×C4—81no—
197C2×C3×C4—81no—
198C2×C3×C4—81no—
199C2×C3×C4—88no—
200C2×Dic3—42no—
201C2×Dic3—42no—
202C2×Dic3—81no—
203C2×Dic3—81no—
204C2×Dic3—81no—
205C2×Dic3—81no—
206C2×Dic3—81no—
207C2×Dic3—81no—
208C2×Dic3—81no—
209C2×Dic3—81no—
210C2×Dic3—81no—
211C2×Dic3—81no—
212SL(2,𝔽3)C22⋊Q811yescharacteristic
213C24C2×S411yescharacteristic
214C2×Q8—22no—
215C2×Q8—31no—
216C2×Q8—31no—
217C2×Q8—32no—
218C2×Q8—31no—
219C2×Q8—32no—
220C2×Q8—32no—
221C2×Q8—61no—
222C2×Q8—61no—
223C2×Q8—61no—
224C2×Q8—64no—
225C2×Q8—62no—
226C2×Q8—62no—
227C2×Q8—62no—
228C2×Q8C2×Dic611yescharacteristic
229C2×Q8#23511yescharacteristic
230C2×Q8C2×#6711yescharacteristic
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—31no—
235C22×C4—32no—
236C22×C4—32no—
237C22×C4—31no—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4—61no—
243C22×C4—61no—
244C22×C4—62no—
245C22×C4—62no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4—121no—
251C22×C4C2×S411yescharacteristic
252C22×C4C2×S411yescharacteristic
253C2×C8—61no—
254C2×C8—61no—
255C2×C8—61no—
256C2×C8—61no—
257C2×C8—61no—
258C2×C8—61no—
259C2×C8—61no—
260C2×C8—61no—
261C2×C8—61no—
262C2×C8—61no—
263C2×C8—61no—
264C2×C8—61no—
265C4⋊C4—32no—
266C4⋊C4—32no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—64no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—122no—
279C4⋊C4—122no—
280C4⋊C4—122no—
281C4⋊C4—122no—
282C22⋊C4—122no—
283C22⋊C4—122no—
284C42—32no—
285C42—32no—
286C42—61no—
287C42—61no—
288C42—62no—
289C42—62no—
290C22×C3—44no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—82no—
299C22×C3—82no—
300C22×C3—82no—
301C22×C3—82no—
302C22×C3—82no—
303C22×C3—82no—
304C3×C4—82no—
305C3×C4—82no—
306Dic3—82no—
307Dic3—82no—
308Dic3—161no—
309Dic3—161no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23A4⋊Q812yes—
315C23#98312yes—
316C23#98512yes—
317C23C22×S411yescharacteristic
318Q8—31no—
319Q8—31no—
320Q8—31no—
321Q8—34no—
322Q8—61no—
323Q8—62no—
324Q8—62no—
325Q8—62no—
326Q8#92411yescharacteristic
327C2×C4—28no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—31no—
333C2×C4—31no—
334C2×C4—31no—
335C2×C4—31no—
336C2×C4—31no—
337C2×C4—31no—
338C2×C4—32no—
339C2×C4—32no—
340C2×C4—62no—
341C2×C4—62no—
342C2×C4—62no—
343C2×C4—62no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—61no—
351C2×C4—124no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4#98312yes—
365C2×C4GL(2,ℤ/4)12yes—
366C8—121no—
367C8—121no—
368C8—121no—
369C8—121no—
370C2×C3—41no—
371C2×C3—41no—
372C2×C3—41no—
373C2×C3—41no—
374C2×C3—41no—
375C2×C3—41no—
376C2×C3—41no—
377C2×C3—84no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22#538914yescomplemented
385C22C2×A4⋊Q811yescharacteristic
386C22#591211yescharacteristic
387C22C2×#98311yescharacteristic
388C22C2×#98311yescharacteristic
389C22C2×#98511yescharacteristic
390C22#592311yescharacteristic
391C22C2×GL(2,ℤ/4)11yescharacteristic
392C4—22no—
393C4—22no—
394C4—31no—
395C4—31no—
396C4—31no—
397C4—31no—
398C4—62no—
399C4—62no—
400C4—121no—
401C4—241no—
402C4—241no—
403C4—241no—
404C4—241no—
405C3—41noSylow
406C2—24no—
407C2C2×#538911yescharacteristic
408C2#8531811yescharacteristic
409C2#8532911yescharacteristic
410C2#8533011yescharacteristic
411C2#8548011yescharacteristic
412C2#8550711yescharacteristic
413C2#8552111yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC22×C4×Q8D3
prime radicalC23×SL(2,𝔽3)C22
hypercenterC23×C4S4
hypocenterSL(2,𝔽3)C22⋊Q8
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C2×#5389C2semidirect
5. #73064C2semidirect
6. #73062C2semidirect
7. #73059C2semidirect
384. C22#5389semidirect

Maximal subgroups

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

subgroupindex
C22×C4×SL(2,𝔽3)2
C2×#53892
#731482
#730642
#730622
#730592
#329603
#52104