Finite Group Databasebeta

← #10692517

Subgroups — #10692517

Order 768.

618 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C3×#2973C211yesmaximal, characteristic, complemented
2#82380C211yesmaximal, characteristic
3#71619C211yesmaximal, characteristic, complemented
4#69944C212yesmaximal, complemented
5#67675C212yesmaximal, complemented
6#9386—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#939—21no—
9C2×#930C2211yescharacteristic
10C3×#435—24no—
11C3×#432—21no—
12C3×#407—21no—
13C2×C3×C22⋊C8C2212yes—
14C2×#834—21no—
15#5213—21no—
16#5212C2212yes—
17C2×C3⋊#95C2211yescharacteristic
18#4991C411yescharacteristic, complemented
19#4981C411yescharacteristic, complemented
20#4960C411yescharacteristic, complemented
21#4958—21no—
22#4953C411yescharacteristic, complemented
23#4940—21no—
24#4526—21no—
25#4470—24no—
26#2973D311yescharacteristic, complemented
27#2543—31no—
28#1799—31no—
29#1519—32no—
30#1294—32no—
31C22×C3×D4D411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4C2311yescharacteristic
34C22×#67—21no—
35C22×#67—21no—
36C23×Dic3—21no—
37C22×D12C2×C411yescharacteristic
38C2×C3×M16—22no—
39C22×C3×C8—22no—
40C3×C4×D4—22no—
41C3×C4×D4—22no—
42C2×C3×C4⋊C4C2×C411yescharacteristic
43C2×C3×C22⋊C4C2×C411yescharacteristic
44C2×C3×C22⋊C4D411yescharacteristic
45C2×C3×C42D411yescharacteristic
46C2×#217—21no—
47C2×#217—22no—
48#939—44no—
49#930—22no—
50#930—22no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×C4⋊Dic3—22no—
56C2×C4⋊Dic3C2×C411yescharacteristic
57C2×Dic3⋊C4—21no—
58C3×C4⋊C8—22no—
59C3×C22⋊C8—22no—
60C3×C22⋊C8—22no—
61C3×C22⋊C8D414yescomplemented
62C3×#97—22no—
63#834—42no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—22no—
68C3⋊#95C2×C411yescharacteristic
69C3⋊#95C2×C411yescharacteristic
70C2×#121—31no—
71C2×#121—61no—
72C2×C4×D4C2×D311yescharacteristic
73#435—24no—
74#432—21no—
75C2×#99—61no—
76#407—21no—
77C2×C22⋊C8C2×D312yes—
78#398—31no—
79#397—31no—
80#394—31no—
81#392—31no—
82#388—61no—
83#386—61no—
84#386—61no—
85#380—32no—
86C2×#95—31no—
87#342—61no—
88#327—64no—
89C24×C3C22⋊C411yescharacteristic
90C24×C3C2×D411yescharacteristic
91C23×D3—21no—
92C2×C3×D4—22no—
93C2×C3×D4—21no—
94C2×C3×D4—22no—
95C2×C3×D4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—81no—
99C22×C3×C4—81no—
100C22×C3×C4C22⋊C411yescharacteristic
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D412yes—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4112yes—
107C2×#67—42no—
108C2×#67—42no—
109C2×#67—42no—
110C2×#67—42no—
111C2×#67—41no—
112C2×#67—41no—
113C2×#67—81no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—42no—
118C22×Dic3—42no—
119C22×Dic3—42no—
120C22×Dic3—81no—
121C2×D12—22no—
122C2×D12—81no—
123C3×M16—82no—
124C2×C3×C8—24no—
125C2×C3×C8—82no—
126C3×C4⋊C4—22no—
127C3×C22⋊C4—22no—
128C3×C22⋊C4—22no—
129C3×C42—21no—
130C3×C42—21no—
131#217—42no—
132#217—42no—
133#217—42no—
134(C2×D3)⋊C4—42no—
135(C2×D3)⋊C4—42no—
136(C2×D3)⋊C4—41no—
137(C2×D3)⋊C4—41no—
138(C2×D3)⋊C4—81no—
139C4⋊Dic3—22no—
140C4⋊Dic3—82no—
141Dic3⋊C4—42no—
142C22×D4—31no—
143C22×D4—61no—
144C22×D4—61no—
145C22×D4D1211yescharacteristic
146C23×C4—61no—
147C23×C4#6711yescharacteristic
148C23×C4C22×D311yescharacteristic
149C2×M16—22no—
150C22×C8—22no—
151#121—62no—
152#121—62no—
153#121—124no—
154C4×D4—22no—
155C4×D4—22no—
156C2×C4⋊C4—31no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4—62no—
159C2×C4⋊C4C4×D311yescharacteristic
160C2×C22⋊C4—61no—
161C2×C22⋊C4—61no—
162C2×C22⋊C4—61no—
163C2×C22⋊C4—61no—
164C2×C22⋊C4—62no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4C4×D311yescharacteristic
167C2×C22⋊C4D1211yescharacteristic
168C2×C42#6711yescharacteristic
169C4⋊C8—22no—
170#99—122no—
171C22⋊C8—22no—
172C22⋊C8—22no—
173C22⋊C8#6714yescomplemented
174#97—22no—
175#95—31no—
176#95—31no—
177#95—61no—
178#95—61no—
179#95—61no—
180#95—62no—
181C23×C3—21no—
182C23×C3—21no—
183C23×C3—21no—
184C23×C3—41no—
185C23×C3—41no—
186C23×C3—41no—
187C23×C3—41no—
188C23×C3C2×C22⋊C411yescharacteristic
189C23×C3#12112yes—
190C23×C3AGL(1,ℤ/8)11yescharacteristic
191C23×C3AGL(1,ℤ/8)12yes—
192C23×C3#13711yescharacteristic
193C22×D3—41no—
194C22×D3—41no—
195C22×D3—41no—
196C22×D3—41no—
197C22×D3—41no—
198C22×D3—41no—
199C22×D3—41no—
200C3×D4—42no—
201C3×D4—42no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—42no—
208C2×C3×C4—42no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—42no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—81no—
215C2×C3×C4C4×D412yes—
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12311yescharacteristic
218#67—84no—
219#67—82no—
220#67—82no—
221C2×Dic3—22no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—84no—
231C2×Dic3—82no—
232C2×Dic3—82no—
233C2×Dic3—82no—
234D12—82no—
235C3×C8—44no—
236C24—61no—
237C24(C2×D3)⋊C411yescharacteristic
238C24C2×D1211yescharacteristic
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—21no—
242C2×D4—22no—
243C2×D4—62no—
244C2×D4—122no—
245C2×D4—122no—
246C2×D4—122no—
247C2×D4—121no—
248C2×D4—121no—
249C2×D4—122no—
250C2×D4—241no—
251C2×D4—241no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—61no—
255C22×C4—61no—
256C22×C4—61no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—122no—
260C22×C4—122no—
261C22×C4—122no—
262C22×C4—241no—
263C22×C4(C2×D3)⋊C411yescharacteristic
264C22×C4(C2×D3)⋊C411yescharacteristic
265C22×C4(C2×D3)⋊C411yescharacteristic
266C22×C4C2×C4×D311yescharacteristic
267C22×C4#23512yes—
268C22×C4C2×#6712yes—
269C22×C4C2×#6711yescharacteristic
270M16—82no—
271C2×C8—24no—
272C2×C8—82no—
273C4⋊C4—22no—
274C4⋊C4—62no—
275C4⋊C4—122no—
276C4⋊C4—242no—
277C22⋊C4—22no—
278C22⋊C4—22no—
279C22⋊C4—122no—
280C22⋊C4—122no—
281C22⋊C4—122no—
282C22⋊C4—121no—
283C22⋊C4—121no—
284C22⋊C4—122no—
285C22⋊C4—122no—
286C22⋊C4—241no—
287C42—21no—
288C42—21no—
289C22×C3—21no—
290C22×C3—22no—
291C22×C3—21no—
292C22×C3—22no—
293C22×C3—42no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—42no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—81no—
301C22×C3#38611yescharacteristic
302C22×C3#41711yescharacteristic
303C22×C3#41811yescharacteristic
304C22×C3#46812yes—
305C22×C3#46912yes—
306C2×D3—81no—
307C2×D3—81no—
308C2×D3—81no—
309C2×D3—81no—
310C2×D3—81no—
311C2×D3—81no—
312C2×D3—81no—
313C3×C4—21no—
314C3×C4—22no—
315C3×C4—21no—
316C3×C4—41no—
317C3×C4—41no—
318C3×C4—41no—
319Dic3—42no—
320Dic3—81no—
321Dic3—81no—
322Dic3—162no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—121no—
331C23—121no—
332C23—121no—
333C23—121no—
334C23—121no—
335C23—121no—
336C23—121no—
337C23#90812yes—
338C23#90811yescharacteristic
339C23#90911yescharacteristic
340C23C2×(C2×D3)⋊C411yescharacteristic
341C23#93012yes—
342D4—42no—
343D4—42no—
344D4—242no—
345D4—244no—
346D4—242no—
347D4—242no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—62no—
361C2×C4—81no—
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—244no—
371C2×C4—242no—
372C2×C4—242no—
373C2×C4—242no—
374C2×C4C4×#6712yes—
375C2×C4#92911yescharacteristic
376C2×C4#95311yescharacteristic
377C8—44no—
378C2×C3—21no—
379C2×C3—22no—
380C2×C3—21no—
381C2×C3—22no—
382C2×C3—81no—
383C2×C3#188411yescharacteristic
384C2×C3#189511yescharacteristic
385C2×C3#191311yescharacteristic
386D3—161no—
387C22—21no—
388C22—22no—
389C22—21no—
390C22—22no—
391C22—42no—
392C22—41no—
393C22—41no—
394C22—42no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—81no—
399C22—241no—
400C22—241no—
401C22—241no—
402C22—241no—
403C22—241no—
404C22—241no—
405C22—241no—
406C22#513111yescharacteristic
407C22#513311yescharacteristic
408C22#513412yes—
409C22#513512yes—
410C22#521311yescharacteristic
411C4—21no—
412C4—21no—
413C4—22no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—122no—
418C4—241no—
419C4—241no—
420C4—482no—
421C3#938611yescharacteristic, complemented, Sylow
422C2—21no—
423C2—22no—
424C2—21no—
425C2—22no—
426C2—81no—
427C2—481no—
428C2#7165611yescharacteristic
429C2#7199011yescharacteristic
430C2#7202111yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2973C2
socleC22×C3#386
prime radical#82380C2
cosocleC23×C3×C4C23
hypercenter#2973D3
hypocenterC3#9386
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
1. C3×#2973C2semidirect
3. #71619C2semidirect
4. #69944C2semidirect
5. #67675C2semidirect
18. #4991C4semidirect
19. #4981C4semidirect
20. #4960C4semidirect
22. #4953C4semidirect
26. #2973D3semidirect
61. C3×C22⋊C8D4semidirect
173. C22⋊C8#67semidirect
421. C3#9386semidirect

Maximal subgroups

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

subgroupindex
C3×#29732
#823802
#716192
#699442
#676752
#93863