Finite Group Databasebeta

← #10691518

Subgroups — #10691518

Order 768.

478 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 93 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×#5035C211yesmaximal, characteristic
2#78584C211yesmaximal, characteristic, complemented
3C3×#1471C211yesmaximal, characteristic
4#71673C211yesmaximal, characteristic
5#69975C211yesmaximal, characteristic, complemented
6#67687C211yesmaximal, characteristic, complemented
7#67678C211yesmaximal, characteristic, complemented
8#9242—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#926C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C2×#832C411yescharacteristic, complemented
14C2×#832C411yescharacteristic, complemented
15C4×#217C2211yescharacteristic
16#5207C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#5102—21no—
19#5101—21no—
20#5035—21no—
21#5035—21no—
22#5035—21no—
23#5035—21no—
24C2×#808C411yescharacteristic, complemented
25C2×#807C411yescharacteristic, complemented
26#4720—22no—
27#4719—21no—
28#4718—21no—
29C3×#323—21no—
30C3×#323—21no—
31#4555—21no—
32#4555—21no—
33#4537—22no—
34C2×#482—31no—
35#2924—31no—
36#1826—31no—
37#1507—31no—
38#1471D311yescharacteristic
39#1293—31no—
40#1282—31no—
41C22×C3×D4—21no—
42C22×C3×D4C2×C411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C2×C3×M16—21no—
45C22×C3×C8—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121—21no—
50C2×C3×C4⋊C4C2×C411yescharacteristic
51C2×C3×C22⋊C4—21no—
52C2×#217—21no—
53C2×#217—21no—
54#926—21no—
55#926—21no—
56#926D411yescharacteristic
57#926D411yescharacteristic
58#926D411yescharacteristic
59#926D411yescharacteristic
60C2×C4⋊Dic3C2×C411yescharacteristic
61C2×C4⋊Dic3C2×C411yescharacteristic
62C2×C4×Dic3—21no—
63C2×C4×Dic3—21no—
64C2×#208—21no—
65C22×C3⋊C8C2×C411yescharacteristic
66C22×C3⋊C8C2×C411yescharacteristic
67C3×C22⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8D411yescharacteristic
71C3×C22⋊C8D411yescharacteristic
72#832—21no—
73#832—21no—
74#832—21no—
75#832—21no—
76C3⋊#95—21no—
77#830—21no—
78#830—21no—
79#830—21no—
80#830D412yes—
81#815—22no—
82Dic3⋊C8—22no—
83Dic3⋊C8—22no—
84C8×Dic3—22no—
85#808—21no—
86#808—21no—
87#807—21no—
88#807—21no—
89C2×#121C2×D311yescharacteristic
90C2×#117—31no—
91#482—61no—
92#482—61no—
93#482—61no—
94#482—61no—
95#433—62no—
96#432—61no—
97#431—61no—
98C2×#107—31no—
99C2×#106—31no—
100#424—61no—
101C2×#102—31no—
102C2×#102—31no—
103#407—31no—
104#407C2×D311yescharacteristic
105C2×C22⋊C8—31no—
106C2×C22⋊C8C2×D311yescharacteristic
107#405—61no—
108C4×C22⋊C4—31no—
109#339—61no—
110#339—61no—
111#327—62no—
112#323—21no—
113#323—21no—
114C24×C3—21no—
115C24×C3#4111yescharacteristic
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—82no—
123C22×C3×C4—21no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4C22×C411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3—41no—
135C3×M16—41no—
136C3×M16—41no—
137C2×C3×C8—21no—
138C2×C3×C8—21no—
139C2×C3×C8—41no—
140C2×C3×C8—41no—
141C2×C3×C8—41no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—41no—
146#217—21no—
147#217—21no—
148#217—21no—
149#217—21no—
150#217—22no—
151#217—41no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3C22⋊C411yescharacteristic
155C4⋊Dic3C22⋊C411yescharacteristic
156C4⋊Dic3C22⋊C411yescharacteristic
157C4⋊Dic3C22⋊C411yescharacteristic
158C4×Dic3—21no—
159C4×Dic3—21no—
160C4×Dic3—21no—
161C4×Dic3—21no—
162C4×Dic3—41no—
163C4×Dic3—41no—
164#208—41no—
165#208—41no—
166C2×C3⋊C8—21no—
167C2×C3⋊C8—21no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—41no—
170C2×C3⋊C8—41no—
171C22×D4—21no—
172C22×D4C4×D311yescharacteristic
173C23×C4C22×D311yescharacteristic
174C2×M16—21no—
175C2×M16—61no—
176C22×C8—21no—
177C22×C8—31no—
178C22×C8—31no—
179#121—21no—
180#121—21no—
181#121—21no—
182#121—21no—
183#117—31no—
184#117—31no—
185#117—31no—
186#117—31no—
187#117—61no—
188#117—61no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4C4×D311yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4—61no—
195C2×C42—61no—
196C2×C42—61no—
197#107—61no—
198#107—61no—
199#106—61no—
200#106—61no—
201C4⋊C8—62no—
202C4⋊C8—62no—
203#102—61no—
204#102—61no—
205#102—61no—
206#102—61no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8—21no—
210C22⋊C8—32no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C22⋊C8#6711yescharacteristic
215C22⋊C8#6711yescharacteristic
216#97—62no—
217C4×C8—62no—
218#95—61no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—81no—
227C23×C3—81no—
228C23×C3C4×D411yescharacteristic
229C23×C3#12111yescharacteristic
230C23×C3#12311yescharacteristic
231C3×D4—41no—
232C3×D4—41no—
233C3×D4—81no—
234C3×D4—82no—
235C3×D4—82no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—81no—
247C2×C3×C4C2×C22⋊C411yescharacteristic
248C2×C3×C4C4×D411yescharacteristic
249C2×C3×C4#12011yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×Dic3—21no—
252C2×Dic3—21no—
253C2×Dic3—21no—
254C2×Dic3—21no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—42no—
258C2×Dic3—81no—
259C2×Dic3—81no—
260C2×Dic3—81no—
261C3×C8—41no—
262C3×C8—41no—
263C3×C8—81no—
264C3⋊C8—42no—
265C3⋊C8—41no—
266C3⋊C8—41no—
267C24—21no—
268C24#23511yescharacteristic
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—82no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—121no—
282C22×C4C2×C4×D311yescharacteristic
283C22×C4D3×D411yescharacteristic
284C22×C4D3×D411yescharacteristic
285C22×C4D3×D411yescharacteristic
286C22×C4#23711yescharacteristic
287C22×C4C2×#6711yescharacteristic
288M16—41no—
289M16—41no—
290M16—121no—
291M16—121no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—61no—
298C2×C8—61no—
299C2×C8—62no—
300C2×C8—121no—
301C2×C8—121no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—31no—
305C4⋊C4—31no—
306C4⋊C4—31no—
307C4⋊C4—31no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—61no—
313C22⋊C4—61no—
314C22⋊C4—61no—
315C22⋊C4—61no—
316C22⋊C4—62no—
317C22⋊C4—121no—
318C42—61no—
319C42—61no—
320C42—61no—
321C42—61no—
322C42—121no—
323C42—121no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3—81no—
335C22×C3—81no—
336C22×C3—81no—
337C22×C3—161no—
338C22×C3—161no—
339C22×C3#38611yescharacteristic
340C22×C3#44311yescharacteristic
341C22×C3#44411yescharacteristic
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—81no—
346C3×C4#41011yescharacteristic
347C3×C4#41011yescharacteristic
348Dic3—41no—
349Dic3—41no—
350Dic3—41no—
351Dic3—41no—
352Dic3—82no—
353C23—21no—
354C23—21no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—81no—
361C23—81no—
362C23#88311yescharacteristic
363C23#88411yescharacteristic
364C23C4×#6711yescharacteristic
365D4—41no—
366D4—41no—
367D4—81no—
368D4—82no—
369D4—82no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—61no—
383C2×C4—61no—
384C2×C4—81no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—122no—
388C2×C4—241no—
389C2×C4—241no—
390C2×C4—241no—
391C2×C4D3×C22⋊C411yescharacteristic
392C2×C4#67⋊C411yescharacteristic
393C2×C4#93711yescharacteristic
394C2×C4#93811yescharacteristic
395C8—41no—
396C8—41no—
397C8—81no—
398C8—122no—
399C8—121no—
400C8—121no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—41no—
404C2×C3—41no—
405C2×C3—161no—
406C2×C3#187611yescharacteristic
407C2×C3#189911yescharacteristic
408C2×C3#190311yescharacteristic
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—161no—
423C22—161no—
424C22#495611yescharacteristic
425C22#514411yescharacteristic
426C22#514511yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—81no—
431C4—121no—
432C4—121no—
433C4—121no—
434C4—121no—
435C4—242no—
436C4#474011yescharacteristic
437C4#474011yescharacteristic
438C3#924211yescharacteristic, complemented, Sylow
439C2—21no—
440C2—21no—
441C2—41no—
442C2—41no—
443C2—161no—
444C2#7001111yescharacteristic
445C2#7002911yescharacteristic
446C2#7008711yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1471C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#1471D3
hypocenterC3#9242
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1471

Complements

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

NH ≅ G/Ntype
2. #78584C2semidirect
5. #69975C2semidirect
6. #67687C2semidirect
7. #67678C2semidirect
13. C2×#832C4semidirect
14. C2×#832C4semidirect
24. C2×#808C4semidirect
25. C2×#807C4semidirect
438. C3#9242semidirect

Maximal subgroups

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

subgroupindex
C2×#50352
#785842
C3×#14712
#716732
#699752
#676872
#676782
#92423