Finite Group Databasebeta

← #10705671

Subgroups — #10705671

Order 768.

586 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 87 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#82116C211yesmaximal, characteristic
2C2×C3×#342C211yesmaximal, characteristic
3#71539C211yesmaximal, characteristic
4#67491C212yesmaximal, complemented
5#67490C212yesmaximal, complemented
6#11759—31nomaximal, Sylow
7C22×C3×C22⋊C4—21no—
8C22×C3×C22⋊C4C2211yescharacteristic
9C2×#924C411yescharacteristic, complemented
10#5247—21no—
11#5211—21no—
12C4×#217C411yescharacteristic, complemented
13#5207—21no—
14#4953C411yescharacteristic, complemented
15#4953C2211yescharacteristic
16#4947C411yescharacteristic, complemented
17#4947C2211yescharacteristic
18#4945—21no—
19#4945—21no—
20C3×#342—21no—
21C3×#342C2212yes—
22#4525—21no—
23#4525C2212yes—
24#2561—31no—
25C2×#342D311yescharacteristic
26#1462—31no—
27#1351—32no—
28#1344—32no—
29C25×C3D411yescharacteristic
30C23×C3×C4—41no—
31C23×C3×C4C2×C411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C22×Dic6C2×C411yescharacteristic, complemented
34C2×C3×C22⋊C4—24no—
35C2×C3×C22⋊C4—22no—
36C2×C3×C22⋊C4—41no—
37C2×C3×C22⋊C4—41no—
38C2×C3×C22⋊C4—41no—
39C2×C3×C22⋊C4—42no—
40C2×C3×C22⋊C4C2×C411yescharacteristic
41C2×C3×C22⋊C4D411yescharacteristic
42C2×C3×C22⋊C4D411yescharacteristic
43C2×C3×C22⋊C4C2311yescharacteristic
44C2×#217—21no—
45C2×#217—21no—
46C2×#217—21no—
47C2×#217—21no—
48C2×#217—21no—
49C2×#217—21no—
50C2×C4⋊Dic3C2×C411yescharacteristic, complemented
51#924—41no—
52#924—41no—
53C2×Dic3⋊C4—21no—
54C2×C4×Dic3—21no—
55C2×#208—41no—
56C3⋊#95—21no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95C2×C411yescharacteristic, complemented
61C3⋊#95C2×C411yescharacteristic, complemented
62#830—22no—
63#830—41no—
64C2×C22⋊Q8—31no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4C2×D311yescharacteristic
67#407—61no—
68#397—31no—
69#397—31no—
70#393—31no—
71#393—31no—
72#388—61no—
73#388—61no—
74#381—61no—
75#379—61no—
76C4×C22⋊C4—31no—
77#342—21no—
78#342C2×D312yes—
79#323—32no—
80#323—61no—
81C24×C3—21no—
82C24×C3—21no—
83C24×C3—22no—
84C24×C3—21no—
85C24×C3—21no—
86C24×C3—42no—
87C24×C3—81no—
88C24×C3C22⋊C411yescharacteristic
89C24×C3C22⋊C411yescharacteristic
90C24×C3C2×D411yescharacteristic
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—42no—
96C22×C3×C4—44no—
97C22×C3×C4—42no—
98C22×C3×C4—81no—
99C22×C3×C4—82no—
100C22×C3×C4—82no—
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—21no—
109C2×Dic6—41no—
110C2×Dic6—81no—
111C3×C22⋊C4—24no—
112C3×C22⋊C4—41no—
113C3×C22⋊C4—41no—
114C3×C22⋊C4—41no—
115C3×C22⋊C4—41no—
116C3×C22⋊C4—44no—
117C3×C22⋊C4—42no—
118#217—42no—
119#217—41no—
120#217—41no—
121#217—42no—
122#217—41no—
123#217—41no—
124#217—81no—
125#217—81no—
126C4⋊Dic3—41no—
127Dic3⋊C4—42no—
128C4×Dic3—81no—
129#208—82no—
130C2×C3⋊C8—42no—
131C25#6711yescharacteristic
132C22×Q8—31no—
133C23×C4—41no—
134C23×C4C4×D311yescharacteristic
135C23×C4D1211yescharacteristic
136C2×M16—121no—
137C22⋊Q8—121no—
138C22⋊Q8—121no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—61no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—41no—
144C2×C22⋊C4—41no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4—61no—
149C2×C22⋊C4—61no—
150C2×C22⋊C4—61no—
151C2×C22⋊C4—61no—
152C2×C22⋊C4—61no—
153C2×C22⋊C4C4×D311yescharacteristic
154C2×C22⋊C4D1211yescharacteristic
155C2×C22⋊C4#6711yescharacteristic
156C2×C22⋊C4C22×D311yescharacteristic
157C2×C42—61no—
158C22⋊C8—62no—
159C22⋊C8—121no—
160#95—31no—
161#95—31no—
162#95—61no—
163#95—61no—
164#95—61no—
165#95—61no—
166C23×C3—22no—
167C23×C3—22no—
168C23×C3—22no—
169C23×C3—22no—
170C23×C3—22no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—41no—
178C23×C3—42no—
179C23×C3—41no—
180C23×C3—41no—
181C23×C3—41no—
182C23×C3—41no—
183C23×C3—42no—
184C23×C3—81no—
185C23×C3—81no—
186C23×C3—84no—
187C23×C3—82no—
188C23×C3—82no—
189C23×C3#9911yescharacteristic
190C23×C3#9911yescharacteristic
191C23×C3#10412yes—
192C23×C3C2×C22⋊C411yescharacteristic
193C23×C3AGL(1,ℤ/8)11yescharacteristic
194C23×C3#13711yescharacteristic
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—42no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—44no—
205C2×C3×C4—81no—
206C2×C3×C4—81no—
207C2×C3×C4—81no—
208C2×C3×C4—88no—
209C2×C3×C4—82no—
210C2×C3×C4—82no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223Dic6—82no—
224C3⋊C8—82no—
225C24—21no—
226C24—22no—
227C24—21no—
228C24—21no—
229C24—21no—
230C24—42no—
231C24—81no—
232C24(C2×D3)⋊C411yescharacteristic
233C24(C2×D3)⋊C411yescharacteristic
234C24C2×#6711yescharacteristic
235C2×Q8—121no—
236C2×Q8—241no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—61no—
245C22×C4—61no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—81no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4(C2×D3)⋊C411yescharacteristic
252C22×C4(C2×D3)⋊C411yescharacteristic
253C22×C4C2×C4×D311yescharacteristic
254C22×C4C2×D1211yescharacteristic
255M16—242no—
256C2×C8—122no—
257C4⋊C4—121no—
258C4⋊C4—122no—
259C22⋊C4—24no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—44no—
265C22⋊C4—42no—
266C22⋊C4—122no—
267C22⋊C4—122no—
268C22⋊C4—121no—
269C22⋊C4—121no—
270C22⋊C4—121no—
271C22⋊C4—121no—
272C22⋊C4—241no—
273C22⋊C4—241no—
274C42—241no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—22no—
284C22×C3—21no—
285C22×C3—42no—
286C22×C3—42no—
287C22×C3—42no—
288C22×C3—41no—
289C22×C3—41no—
290C22×C3—44no—
291C22×C3—44no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—81no—
297C22×C3—81no—
298C22×C3—82no—
299C22×C3—82no—
300C22×C3#35412yes—
301C22×C3C2×#9911yescharacteristic
302C22×C3#41711yescharacteristic
303C22×C3C2×#10411yescharacteristic
304C3×C4—41no—
305C3×C4—41no—
306C3×C4—41no—
307C3×C4—41no—
308C3×C4—84no—
309Dic3—81no—
310Dic3—81no—
311Dic3—81no—
312Dic3—81no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—21no—
323C23—21no—
324C23—41no—
325C23—42no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—42no—
331C23—81no—
332C23—81no—
333C23—84no—
334C23—82no—
335C23—82no—
336C23#80512yes—
337C23#80611yescharacteristic
338C23#80611yescharacteristic
339C23C2×(C2×D3)⋊C411yescharacteristic
340C23#93211yescharacteristic
341C23#94211yescharacteristic
342Q8—242no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—44no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—88no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—121no—
371C8—242no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—22no—
375C2×C3—41no—
376C2×C3—41no—
377C2×C3—42no—
378C2×C3—41no—
379C2×C3#150411yescharacteristic
380C2×C3#212111yescharacteristic
381C2×C3C2×#35411yescharacteristic
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—22no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—41no—
396C22—41no—
397C22—44no—
398C22—44no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—81no—
404C22—81no—
405C22—82no—
406C22—82no—
407C22#447312yes—
408C22C2×#80511yescharacteristic
409C22C2×#80611yescharacteristic
410C22#496611yescharacteristic
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—84no—
416C4—241no—
417C4—241no—
418C4—241no—
419C4—241no—
420C3#1175911yescharacteristic, complemented, Sylow
421C2—21no—
422C2—21no—
423C2—22no—
424C2—41no—
425C2—41no—
426C2—42no—
427C2—41no—
428C2#6939811yescharacteristic
429C2#7001511yescharacteristic
430C2C2×#447311yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC2×C3×#342C2
socleC22×C3C2×#99
prime radicalC25×C3D4
cosocleC2×C3×C22⋊C4C23
hypercenterC2×#342D3
hypocenterC3#11759
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#342

Complements

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

NH ≅ G/Ntype
4. #67491C2semidirect
5. #67490C2semidirect
9. C2×#924C4semidirect
12. C4×#217C4semidirect
14. #4953C4semidirect
16. #4947C4semidirect
33. C22×Dic6C2×C4semidirect
50. C2×C4⋊Dic3C2×C4semidirect
60. C3⋊#95C2×C4semidirect
61. C3⋊#95C2×C4semidirect
420. C3#11759semidirect

Maximal subgroups

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

subgroupindex
#821162
C2×C3×#3422
#715392
#674912
#674902
#117593