Finite Group Databasebeta

← #11713721

Subgroups — #11713721

Order 768.

494 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 57 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×#557⋊C3C211yesmaximal, characteristic, complemented
2C2×#5395C211yesmaximal, characteristic
3#73065C211yesmaximal, characteristic
4#73065C211yesmaximal, characteristic
5#73061C211yesmaximal, characteristic, complemented
6#73061C211yesmaximal, characteristic, complemented
7C4×#859C211yesmaximal, characteristic, complemented
8#25937—31nomaximal, Sylow
9C22×GL(2,𝔽3)C2211yescharacteristic
10C22×#226C2211yescharacteristic
11#557⋊C3—22no—
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13C2×C4×SL(2,𝔽3)C2211yescharacteristic
14C2×C4×SL(2,𝔽3)C2211yescharacteristic
15C2×#859C2211yescharacteristic
16C2×#859C2211yescharacteristic
17#5395—22no—
18#4990—41nomaximal
19C2×#557D311yescharacteristic, complemented
20C2×#488—31no—
21C2×#464—31no—
22#2592—31no—
23#2026—31no—
24#2025—31no—
25#2024—31no—
26#2023—31no—
27#1989—31no—
28#1984—31no—
29#1880—31no—
30#1880—31no—
31#1878—31no—
32#1878—31no—
33C4×#103—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×#226—21no—
41C2×C3×C4⋊C4—41no—
42C2×(C2×D3)⋊C4—41no—
43C2×(C2×D3)⋊C4—41no—
44C2×(C2×D3)⋊C4—41no—
45C2×C4×Dic3—41no—
46C4×SL(2,𝔽3)—21no—
47C4×SL(2,𝔽3)—21no—
48C4×SL(2,𝔽3)D412yes—
49#859—21no—
50#859—21no—
51C3⋊#95—41no—
52C3⋊#95—41no—
53C22×Q16—31no—
54C22×SD16—31no—
55#557—22no—
56#557—62no—
57C2×C4⋊Q8—31no—
58C2×#125—31no—
59C2×#125—31no—
60C2×#124—31no—
61C2×C4×Q8—31no—
62C2×C4×Q8C2×D311yescharacteristic
63C2×C4×Q8C2×D311yescharacteristic
64C2×C4×Q8C2×D311yescharacteristic
65#488—64no—
66#464—62no—
67#464—62no—
68C2×C4⋊C8—31no—
69C2×#103—31no—
70C2×#103—31no—
71C2×#103—31no—
72C2×#103—31no—
73C2×#103—31no—
74C2×#102—31no—
75C2×#102—31no—
76C2×#102—31no—
77C2×C4×C8—31no—
78#397—61no—
79#393—61no—
80#391—31no—
81#391—31no—
82#390—31no—
83#390—31no—
84#388—61no—
85#388—61no—
86C4×C4⋊C4—31no—
87#340—31no—
88#340—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—
97C23×D3—41no—
98C22×C3×C4—41no—
99C22×C3×C4—41no—
100C22×C3×C4—41no—
101C22×Dic3—41no—
102C22×Dic3—41no—
103C22×Dic3—41no—
104C2×SL(2,𝔽3)C2×D411yescharacteristic
105C2×SL(2,𝔽3)#4111yescharacteristic
106C2×SL(2,𝔽3)#4111yescharacteristic
107GL(2,𝔽3)—41no—
108#226—41no—
109C3×C4⋊C4—82no—
110(C2×D3)⋊C4—44no—
111(C2×D3)⋊C4—82no—
112(C2×D3)⋊C4—82no—
113C4×Dic3—82no—
114C22×Q8—31no—
115C22×Q8—31no—
116C22×Q8C22×D311yescharacteristic
117C22×D4—31no—
118C2×Q16—61no—
119C2×Q16—61no—
120C2×Q16—61no—
121C2×Q16—61no—
122C2×Q16—61no—
123C2×Q16—61no—
124C2×SD16—61no—
125C2×SD16—61no—
126C2×SD16—61no—
127C2×SD16—61no—
128C2×SD16—61no—
129C2×SD16—61no—
130C22×C8—31no—
131C22×C8—31no—
132C22×C8—31no—
133C22×C8—31no—
134C4⋊Q8—62no—
135C4⋊Q8—62no—
136#125—34no—
137#125—62no—
138#125—62no—
139#125—62no—
140#124—62no—
141#124—62no—
142C4×Q8—21no—
143C4×Q8—21no—
144C4×Q8—32no—
145C4×Q8—62no—
146C4×Q8—62no—
147C4×Q8—61no—
148C4×Q8—61no—
149C4×Q8#6712yes—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4—31no—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—31no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4S411yescharacteristic
159C2×C22⋊C4—61no—
160C2×C22⋊C4—61no—
161C2×C22⋊C4—61no—
162C2×C22⋊C4—61no—
163C2×C22⋊C4—61no—
164C2×C22⋊C4—61no—
165C2×C42—31no—
166C2×C42—31no—
167C2×C42—31no—
168C2×C42—61no—
169C4⋊C8—62no—
170#103—62no—
171#103—62no—
172#103—62no—
173#103—61no—
174#103—61no—
175#103—61no—
176#103—61no—
177#102—62no—
178#102—62no—
179#102—62no—
180C4×C8—62no—
181#95—61no—
182#95—61no—
183#95—61no—
184#95—61no—
185#95—61no—
186C23×C3—41no—
187C22×D3—42no—
188C22×D3—81no—
189C22×D3—81no—
190C22×D3—81no—
191C22×D3—81no—
192C22×D3—81no—
193C22×D3—81no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4—81no—
197C2×C3×C4—81no—
198C2×C3×C4—81no—
199C2×C3×C4—81no—
200C2×C3×C4—81no—
201C2×C3×C4—81no—
202C2×C3×C4—81no—
203C2×Dic3—42no—
204C2×Dic3—42no—
205C2×Dic3—81no—
206C2×Dic3—81no—
207C2×Dic3—81no—
208C2×Dic3—81no—
209C2×Dic3—81no—
210C2×Dic3—81no—
211C2×Dic3—81no—
212SL(2,𝔽3)#12411yescharacteristic
213C24—61no—
214C2×Q8—31no—
215C2×Q8—31no—
216C2×Q8—31no—
217C2×Q8—62no—
218C2×Q8—61no—
219C2×Q8—61no—
220C2×Q8—61no—
221C2×Q8—61no—
222C2×Q8—61no—
223C2×Q8—61no—
224C2×Q8—62no—
225C2×Q8—121no—
226C2×Q8#23911yescharacteristic
227C2×Q8#23911yescharacteristic
228C2×Q8C2×#6711yescharacteristic
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—61no—
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—31no—
238C22×C4—31no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4C2×S411yescharacteristic
243C22×C4C2×S411yescharacteristic
244C22×C4C2×S411yescharacteristic
245Q16—121no—
246Q16—121no—
247Q16—121no—
248Q16—121no—
249SD16—121no—
250SD16—121no—
251SD16—121no—
252SD16—121no—
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—22no—
266C4⋊C4—32no—
267C4⋊C4—32no—
268C4⋊C4—32no—
269C4⋊C4—62no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—64no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—62no—
279C22⋊C4—122no—
280C22⋊C4—122no—
281C22⋊C4—122no—
282C22⋊C4—122no—
283C22⋊C4—241no—
284C22⋊C4—241no—
285C42—32no—
286C42—32no—
287C42—61no—
288C42—61no—
289C42—61no—
290C42—61no—
291C42—241no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—41no—
299C2×D3—82no—
300C2×D3—82no—
301C2×D3—82no—
302C2×D3—161no—
303C2×D3—161no—
304C2×D3—161no—
305C2×D3—161no—
306C3×C4—82no—
307C3×C4—82no—
308C3×C4—161no—
309Dic3—82no—
310Dic3—82no—
311Dic3—161no—
312C23—121no—
313C23—121no—
314C23—121no—
315C23—121no—
316C23—121no—
317C23—121no—
318C23—121no—
319C23C22×S411yescharacteristic
320Q8—31no—
321Q8—31no—
322Q8—31no—
323Q8—61no—
324Q8—124no—
325Q8—121no—
326Q8—121no—
327Q8—121no—
328Q8—121no—
329Q8#94711yescharacteristic
330D4—121no—
331D4—121no—
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—61no—
349C2×C4—61no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—61no—
355C2×C4—61no—
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—121no—
365C2×C4#98512yes—
366C2×C4GL(2,ℤ/4)12yes—
367C8—121no—
368C8—121no—
369C8—121no—
370C8—121no—
371C2×C3—41no—
372C2×C3—41no—
373C2×C3—41no—
374C2×C3—41no—
375C2×C3—41no—
376C2×C3—41no—
377C2×C3—41no—
378D3—162no—
379C22—241no—
380C22—241no—
381C22—241no—
382C22—241no—
383C22—241no—
384C22—241no—
385C22—241no—
386C22#591911yescharacteristic
387C22#591911yescharacteristic
388C22C2×#98511yescharacteristic
389C22#592311yescharacteristic
390C22#592611yescharacteristic
391C22#592711yescharacteristic
392C22C2×GL(2,ℤ/4)11yescharacteristic
393C4—22no—
394C4—22no—
395C4—31no—
396C4—31no—
397C4—31no—
398C4—31no—
399C4—41no—
400C4—62no—
401C4—121no—
402C4—121no—
403C4—241no—
404C4—241no—
405C4—241no—
406C3—41noSylow
407C2—481no—
408C2#8541911yescharacteristic
409C2#8542011yescharacteristic
410C2#8542111yescharacteristic
411C2#8542511yescharacteristic
412C2#8550811yescharacteristic
413C2#8552111yescharacteristic
414C2#8553911yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#557D3
prime radicalC22×GL(2,𝔽3)C22
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#124
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×#557⋊C3C2semidirect
5. #73061C2semidirect
6. #73061C2semidirect
7. C4×#859C2semidirect
19. C2×#557D3semidirect

Maximal subgroups

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

subgroupindex
C2×#557⋊C32
C2×#53952
#730652
#730612
C4×#8592
#259373
#49904