Finite Group Databasebeta

← #11712999

Subgroups — #11712999

Order 768.

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

normal subgroups only

This group has 415 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
1#85519C211yesmaximal, characteristic
2#2880⋊C3C211yesmaximal, characteristic
3#73149C211yesmaximal, characteristic
4#73123C211yesmaximal, characteristic, complemented
5#73122C211yesmaximal, characteristic, complemented
6#73122C211yesmaximal, characteristic, complemented
7#73121C211yesmaximal, characteristic, complemented
8#22588—31nomaximal, Sylow
9C22×#41⋊C3C2211yescharacteristic
10C2×#131⋊C3C2211yescharacteristic
11C2×#131⋊C3C2211yescharacteristic
12#5426C411yescharacteristic, complemented
13#5426C411yescharacteristic, complemented
14#5424C2211yescharacteristic
15#5424C2211yescharacteristic
16C2×GU(2,𝔽3)C2211yescharacteristic
17C2×GU(2,𝔽3)C2211yescharacteristic
18#5420C411yescharacteristic, complemented
19#5420C411yescharacteristic, complemented
20#4720—41nomaximal
21#2924—31no—
22#2896—31no—
23#2880D311yescharacteristic
24#1938—31no—
25#1937—31no—
26#1929—31no—
27#1929—31no—
28#1866—31no—
29#1866—31no—
30#1826—31no—
31#1811—31no—
32#1789—31no—
33#1765—31no—
34#1764—31no—
35#1753—31no—
36C2×#41⋊C3—21no—
37C2×#41⋊C3—21no—
38C2×#41⋊C3C2×C411yescharacteristic
39C2×#41⋊C3C2311yescharacteristic
40C22×SL(2,𝔽3)C2×C411yescharacteristic
41#926—41no—
42#131⋊C3—21no—
43#131⋊C3—21no—
44GU(2,𝔽3)—21no—
45GU(2,𝔽3)—21no—
46#859C2×C411yescharacteristic
47#859C2×C411yescharacteristic
48#859C2×C411yescharacteristic
49#859C2×C411yescharacteristic
50C3×C22⋊C8—41no—
51#830—41no—
52#815—41no—
53Dic3⋊C8—41no—
54Dic3⋊C8—41no—
55C8×Dic3—41no—
56C22×#41C2×D311yescharacteristic
57C2×#131C2×D311yescharacteristic
58C2×#131C2×D311yescharacteristic
59C22×M16—31no—
60C2×#117—31no—
61#433—61no—
62#433—61no—
63#432—61no—
64#431—61no—
65C2×#108—31no—
66C2×#108—31no—
67#424—31no—
68#424—31no—
69C2×#104—31no—
70C2×#104—31no—
71C2×#104—31no—
72C2×#104—31no—
73#419—31no—
74#419—31no—
75#418—31no—
76#418—31no—
77#417—31no—
78#417—31no—
79#417—31no—
80#417—31no—
81#416—31no—
82#416—31no—
83#416—31no—
84#416—31no—
85#408—31no—
86#408—31no—
87#407—31no—
88#407—31no—
89C2×C22⋊C8—31no—
90#405—31no—
91#405—31no—
92C2×#97—31no—
93C2×C4×C8—31no—
94C4×C22⋊C4—31no—
95#341—31no—
96#341—31no—
97#339—31no—
98#339—31no—
99#339—31no—
100#339—31no—
101#336—31no—
102#336—31no—
103C22×C3×C4—41no—
104#41⋊C3—21no—
105#41⋊C3#4111yescharacteristic
106#41⋊C3#4111yescharacteristic
107C2×SL(2,𝔽3)—21no—
108C2×SL(2,𝔽3)C22×C411yescharacteristic
109C2×C3×C8—41no—
110C2×C3×C8—41no—
111#217—41no—
112#217—41no—
113C4⋊Dic3—41no—
114C4⋊Dic3—41no—
115C4×Dic3—41no—
116C4×Dic3—41no—
117C2×C3⋊C8—41no—
118C2×C3⋊C8—41no—
119C2×#41—21no—
120C2×#41—21no—
121C2×#41—31no—
122C2×#41—31no—
123C2×#41—61no—
124C2×#41—61no—
125C2×#41C4×D311yescharacteristic
126C2×#41C22×D311yescharacteristic
127C22×Q8C4×D311yescharacteristic
128C22×D4—31no—
129C23×C4—31no—
130#131—21no—
131#131—21no—
132#131—61no—
133#131—61no—
134C2×M16—31no—
135C2×M16—31no—
136C2×M16—31no—
137C2×M16—31no—
138C2×M16—31no—
139C2×M16—31no—
140C2×M16—31no—
141C2×M16—31no—
142C2×M16—121no—
143C2×M16—121no—
144C22×C8—31no—
145C22×C8—31no—
146C22×C8—31no—
147C22×C8—31no—
148#117—31no—
149#117—31no—
150#117—31no—
151#117—31no—
152#117—61no—
153#117—61no—
154C2×C4⋊C4—31no—
155C2×C4⋊C4—31no—
156C2×C22⋊C4—61no—
157C2×C22⋊C4—61no—
158C2×C42—31no—
159C2×C42—31no—
160C2×C42—61no—
161#108—121no—
162#108—121no—
163C4⋊C8—61no—
164C4⋊C8—61no—
165C4⋊C8—61no—
166C4⋊C8—61no—
167#104—61no—
168#104—61no—
169#104—61no—
170#104—61no—
171#104—121no—
172#104—121no—
173#103—31no—
174#103—31no—
175#103—31no—
176#103—31no—
177#103—31no—
178#103—31no—
179#103—31no—
180#103—31no—
181#102—31no—
182#102—31no—
183#102—31no—
184#102—31no—
185#102—31no—
186#102—31no—
187#102—31no—
188#102—31no—
189C22⋊C8—31no—
190C22⋊C8—31no—
191C22⋊C8—31no—
192C22⋊C8—61no—
193C22⋊C8—61no—
194C22⋊C8—61no—
195C22⋊C8S411yescharacteristic
196#97—61no—
197#97—61no—
198#97—121no—
199C4×C8—61no—
200C4×C8—61no—
201C4×C8—121no—
202#95—61no—
203C23×C3—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—81no—
207C2×C3×C4—81no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212SL(2,𝔽3)#11711yescharacteristic
213C3×C8—81no—
214C3×C8—81no—
215C3⋊C8—81no—
216C3⋊C8—81no—
217C24—31no—
218#41—21no—
219#41—31no—
220#41—31no—
221#41—61no—
222#41—61no—
223#41—61no—
224#41—61no—
225#41—61no—
226#41—61no—
227#41—61no—
228#41—121no—
229#41#23711yescharacteristic
230#41#23711yescharacteristic
231C2×Q8—21no—
232C2×Q8—31no—
233C2×Q8—61no—
234C2×Q8C2×C4×D311yescharacteristic
235C2×D4—31no—
236C2×D4—31no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—121no—
241C22×C4—31no—
242C22×C4—31no—
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—31no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4C2×S411yescharacteristic
255M16—61no—
256M16—61no—
257M16—61no—
258M16—61no—
259M16—61no—
260M16—61no—
261M16—61no—
262M16—61no—
263M16—121no—
264M16—121no—
265M16—121no—
266M16—121no—
267C2×C8—31no—
268C2×C8—31no—
269C2×C8—31no—
270C2×C8—31no—
271C2×C8—31no—
272C2×C8—31no—
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×C8C2×S411yescharacteristic
284C2×C8C2×S411yescharacteristic
285C4⋊C4—31no—
286C4⋊C4—31no—
287C4⋊C4—31no—
288C4⋊C4—31no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C22⋊C4—61no—
292C22⋊C4—61no—
293C22⋊C4—61no—
294C22⋊C4—61no—
295C22⋊C4—122no—
296C42—61no—
297C42—61no—
298C42—61no—
299C42—61no—
300C42—61no—
301C42—61no—
302C42—61no—
303C42—61no—
304C22×C3—41no—
305C22×C3—81no—
306C22×C3—81no—
307C22×C3—81no—
308C3×C4—41no—
309C3×C4—41no—
310C3×C4—81no—
311Dic3—81no—
312Dic3—81no—
313Dic3—81no—
314Dic3—81no—
315C23—31no—
316C23—31no—
317C23—61no—
318C23—61no—
319C23—121no—
320C23—121no—
321C23C4×S411yescharacteristic
322Q8—31no—
323Q8—61no—
324Q8—61no—
325Q8#87711yescharacteristic
326D4—31no—
327D4—31no—
328D4—61no—
329D4—61no—
330D4—61no—
331D4—121no—
332D4—121no—
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—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—62no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—122no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—241no—
361C2×C4C4×S411yescharacteristic
362C2×C4C22×S411yescharacteristic
363C8—21no—
364C8—21no—
365C8—61no—
366C8—61no—
367C8—61no—
368C8—61no—
369C8—61no—
370C8—61no—
371C8—121no—
372C8—121no—
373C2×C3—41no—
374C2×C3—41no—
375C2×C3—41no—
376C2×C3—161no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—31no—
381C22—31no—
382C22—61no—
383C22—61no—
384C22—61no—
385C22—61no—
386C22—61no—
387C22—241no—
388C22—241no—
389C22C2×C4×S411yescharacteristic
390C4—21no—
391C4—31no—
392C4—31no—
393C4—61no—
394C4—122no—
395C4—121no—
396C4—121no—
397C4—121no—
398C4—121no—
399C4—241no—
400C4#591411yescharacteristic
401C4#591411yescharacteristic
402C3—41noSylow
403C2—41no—
404C2—61no—
405C2—61no—
406C2—121no—
407C2#8529211yescharacteristic
408C2#8544811yescharacteristic
409C2#8545311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
Fitting#2880D3
socleC22C2×C4×S4
prime radicalC22×#41⋊C3C22
cosocleC2×#41⋊C3C23
hypercenterC22⋊C8S4
hypocenterSL(2,𝔽3)#117
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
4. #73123C2semidirect
5. #73122C2semidirect
6. #73122C2semidirect
7. #73121C2semidirect
12. #5426C4semidirect
13. #5426C4semidirect
18. #5420C4semidirect
19. #5420C4semidirect

Maximal subgroups

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

subgroupindex
#855192
#2880⋊C32
#731492
#731232
#731222
#731212
#225883
#47204