Finite Group Databasebeta

← #11713811

Subgroups — #11713811

Order 768.

467 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 467 conjugacy classes of subgroups — too many to draw. Only its 53 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×#5388C211yesmaximal, characteristic, complemented
3C4⋊C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73065C211yesmaximal, characteristic
5#73065C211yesmaximal, characteristic
6#73061C211yesmaximal, characteristic, complemented
7#73059C211yesmaximal, characteristic, complemented
8#26185—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#5395—22no—
17#5388—22no—
18#4993—41nomaximal
19C2×#464—31no—
20C2×#441—31no—
21#2448—31no—
22Q8×C4⋊C4D311yescharacteristic, complemented
23#2094—31no—
24#2088—31no—
25#2072—31no—
26#2042—31no—
27#1936—31no—
28#1917—31no—
29#1880—31no—
30#1880—31no—
31#1878—31no—
32#1865—31no—
33#1865—31no—
34C22×SL(2,𝔽3)C2311yescharacteristic
35C2×GL(2,𝔽3)—21no—
36C2×GL(2,𝔽3)—21no—
37C2×GL(2,𝔽3)—21no—
38C2×#226—21no—
39C2×#226—21no—
40C2×#226D412yes—
41C2×C3×C4⋊C4—41no—
42C2×(C2×D3)⋊C4—41no—
43C2×(C2×D3)⋊C4—41no—
44C2×(C2×D3)⋊C4—41no—
45C2×Dic3⋊C4—41no—
46C2×Dic3⋊C4—41no—
47C4×SL(2,𝔽3)—21no—
48C4×SL(2,𝔽3)—21no—
49C4×SL(2,𝔽3)—21no—
50#859—21no—
51#859—21no—
52C3⋊#95—41no—
53C22×Q16—31no—
54C22×SD16—31no—
55C2×#124—31no—
56C2×C4×Q8—31no—
57C2×C4×Q8C2×D311yescharacteristic
58C2×C4×Q8C2×D311yescharacteristic
59C2×C4×Q8C2×D311yescharacteristic
60#464—62no—
61#464—62no—
62#441—62no—
63#441—62no—
64C2×#106—31no—
65C2×#106—31no—
66C2×C4⋊C8—31no—
67C2×#103—31no—
68C2×#103—31no—
69C2×#103—31no—
70C2×#103—31no—
71C2×#103—31no—
72C2×#102—31no—
73C2×#102—31no—
74C2×#102—31no—
75C2×#97—31no—
76#399—61no—
77#397—61no—
78#391—31no—
79#390—31no—
80#390—31no—
81#389—31no—
82#389—31no—
83#389—31no—
84#388—61no—
85#388—61no—
86#384—31no—
87C4×C4⋊C4—31no—
88#340—31no—
89#340—31no—
90#340—31no—
91#340—31no—
92#336—31no—
93#336—31no—
94#336—31no—
95#336—31no—
96C23×D3—41no—
97C22×C3×C4—41no—
98C22×C3×C4—41no—
99C22×C3×C4—41no—
100C22×Dic3—41no—
101C22×Dic3—41no—
102C22×Dic3—41no—
103C2×SL(2,𝔽3)C2×D411yescharacteristic
104C2×SL(2,𝔽3)#4111yescharacteristic
105C2×SL(2,𝔽3)#4111yescharacteristic
106GL(2,𝔽3)—41no—
107#226—22no—
108C3×C4⋊C4—82no—
109(C2×D3)⋊C4—82no—
110(C2×D3)⋊C4—82no—
111(C2×D3)⋊C4—82no—
112Dic3⋊C4—82no—
113Dic3⋊C4—82no—
114C22×Q8—31no—
115C22×Q8C22×D311yescharacteristic
116C22×D4—31no—
117C2×Q16—32no—
118C2×Q16—32no—
119C2×Q16—61no—
120C2×Q16—61no—
121C2×Q16—61no—
122C2×Q16—61no—
123C2×SD16—61no—
124C2×SD16—61no—
125C2×SD16—61no—
126C2×SD16—61no—
127C2×SD16—61no—
128C2×SD16—61no—
129C22×C8—31no—
130C22×C8—31no—
131C22×C8—31no—
132C22×C8—31no—
133#124—62no—
134#124—121no—
135C4×Q8—21no—
136C4×Q8—21no—
137C4×Q8—21no—
138C4×Q8—61no—
139C4×Q8—61no—
140C4×Q8—61no—
141C4×Q8—62no—
142C4×Q8—121no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—31no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—31no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4—61no—
152C2×C4⋊C4S411yescharacteristic
153C2×C22⋊C4—61no—
154C2×C22⋊C4—61no—
155C2×C22⋊C4—61no—
156C2×C22⋊C4—61no—
157C2×C22⋊C4—61no—
158C2×C22⋊C4—61no—
159C2×C42—31no—
160C2×C42—31no—
161C2×C42—31no—
162C2×C42—61no—
163#106—62no—
164#106—121no—
165C4⋊C8—62no—
166#103—62no—
167#103—61no—
168#103—62no—
169#103—61no—
170#103—62no—
171#103—61no—
172#103—61no—
173#102—62no—
174#102—62no—
175#102—121no—
176#97—62no—
177#95—31no—
178#95—31no—
179#95—61no—
180#95—61no—
181#95—61no—
182C23×C3—41no—
183C22×D3—42no—
184C22×D3—81no—
185C22×D3—81no—
186C22×D3—81no—
187C22×D3—81no—
188C22×D3—81no—
189C22×D3—81no—
190C2×C3×C4—42no—
191C2×C3×C4—81no—
192C2×C3×C4—81no—
193C2×C3×C4—81no—
194C2×C3×C4—81no—
195C2×C3×C4—81no—
196C2×C3×C4—81no—
197C2×C3×C4—81no—
198C2×C3×C4—81no—
199C2×Dic3—42no—
200C2×Dic3—42no—
201C2×Dic3—81no—
202C2×Dic3—81no—
203C2×Dic3—81no—
204C2×Dic3—81no—
205C2×Dic3—81no—
206C2×Dic3—81no—
207C2×Dic3—81no—
208SL(2,𝔽3)#12311yescharacteristic
209C24—61no—
210C2×Q8—31no—
211C2×Q8—31no—
212C2×Q8—31no—
213C2×Q8—32no—
214C2×Q8—61no—
215C2×Q8—61no—
216C2×Q8—61no—
217C2×Q8—61no—
218C2×Q8—61no—
219C2×Q8#23511yescharacteristic
220C2×Q8D3×D411yescharacteristic
221C2×Q8#23911yescharacteristic
222C2×D4—61no—
223C2×D4—61no—
224C2×D4—61no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228C22×C4—31no—
229C22×C4—31no—
230C22×C4—31no—
231C22×C4—31no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4C2×S411yescharacteristic
236C22×C4C2×S411yescharacteristic
237C22×C4C2×S411yescharacteristic
238Q16—62no—
239Q16—62no—
240Q16—62no—
241Q16—62no—
242SD16—121no—
243SD16—121no—
244SD16—121no—
245SD16—121no—
246C2×C8—32no—
247C2×C8—61no—
248C2×C8—61no—
249C2×C8—61no—
250C2×C8—61no—
251C2×C8—61no—
252C2×C8—61no—
253C2×C8—61no—
254C2×C8—61no—
255C2×C8—61no—
256C2×C8—61no—
257C2×C8—61no—
258C4⋊C4—22no—
259C4⋊C4—32no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—121no—
270C4⋊C4—122no—
271C4⋊C4—241no—
272C4⋊C4—241no—
273C22⋊C4—122no—
274C22⋊C4—122no—
275C22⋊C4—122no—
276C22⋊C4—122no—
277C22⋊C4—241no—
278C22⋊C4—241no—
279C42—61no—
280C42—61no—
281C42—61no—
282C42—61no—
283C42—61no—
284C42—61no—
285C42—122no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C2×D3—82no—
294C2×D3—82no—
295C2×D3—82no—
296C2×D3—161no—
297C2×D3—161no—
298C2×D3—161no—
299C2×D3—161no—
300C3×C4—82no—
301C3×C4—161no—
302C3×C4—161no—
303Dic3—82no—
304Dic3—82no—
305Dic3—161no—
306C23—121no—
307C23—121no—
308C23—121no—
309C23—121no—
310C23—121no—
311C23—121no—
312C23—121no—
313C23C22×S411yescharacteristic
314Q8—31no—
315Q8—31no—
316Q8—31no—
317Q8—61no—
318Q8—62no—
319Q8—62no—
320Q8—121no—
321Q8—121no—
322Q8#89411yescharacteristic
323D4—121no—
324D4—121no—
325D4—121no—
326D4—121no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—31no—
336C2×C4—31no—
337C2×C4—31no—
338C2×C4—31no—
339C2×C4—31no—
340C2×C4—31no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—62no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—61no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4#98312yes—
360C8—62no—
361C8—121no—
362C8—121no—
363C8—121no—
364C2×C3—41no—
365C2×C3—41no—
366C2×C3—41no—
367C2×C3—41no—
368C2×C3—41no—
369C2×C3—41no—
370C2×C3—41no—
371D3—162no—
372C22—241no—
373C22—241no—
374C22—241no—
375C22—241no—
376C22—241no—
377C22—241no—
378C22—241no—
379C22#591211yescharacteristic
380C22D4×S411yescharacteristic
381C22C2×#98311yescharacteristic
382C22#591911yescharacteristic
383C22#592311yescharacteristic
384C22#592411yescharacteristic
385C22#592611yescharacteristic
386C4—22no—
387C4—31no—
388C4—31no—
389C4—31no—
390C4—31no—
391C4—41no—
392C4—41no—
393C4—122no—
394C4—121no—
395C4—121no—
396C4—121no—
397C4—241no—
398C4—241no—
399C3—41noSylow
400C2—481no—
401C2#8530911yescharacteristic
402C2#8533011yescharacteristic
403C2#8536911yescharacteristic
404C2#8539211yescharacteristic
405C2#8541111yescharacteristic
406C2#8542111yescharacteristic
407C2#8543211yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingQ8×C4⋊C4D3
prime radicalC22×GL(2,𝔽3)C22
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#123
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
2. C2×#5388C2semidirect
3. C4⋊C4×SL(2,𝔽3)C2semidirect
6. #73061C2semidirect
7. #73059C2semidirect
22. Q8×C4⋊C4D3semidirect

Maximal subgroups

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

subgroupindex
C2×#53952
C2×#53882
C4⋊C4×SL(2,𝔽3)2
#730652
#730612
#730592
#261853
#49934