Finite Group Databasebeta

← #10692507

Subgroups — #10692507

Order 768.

640 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 131 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×#2972C211yesmaximal, characteristic, complemented
2#82380C211yesmaximal, characteristic
3#71619C211yesmaximal, characteristic, complemented
4#69944C212yesmaximal, complemented
5#67675C212yesmaximal, complemented
6#9384—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#939—21no—
9C2×#930C2211yescharacteristic
10C3×C8×D4—24no—
11C3×#431—21no—
12C2×C3×C22⋊C8—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#2972D311yescharacteristic, 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
38C22×C3×C8—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—42no—
49#939—42no—
50#930—22no—
51#930—22no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×C4⋊Dic3—22no—
57C2×C4⋊Dic3C2×C411yescharacteristic
58C2×Dic3⋊C4—21no—
59C3×C4⋊C8—22no—
60C3×C22⋊C8—22no—
61C3×C22⋊C8—22no—
62C3×C22⋊C8D414yescomplemented
63C3×C4×C8—22no—
64#834—42no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—22no—
69C3⋊#95C2×C411yescharacteristic
70C3⋊#95C2×C411yescharacteristic
71C2×#121—31no—
72C2×#121—61no—
73C2×C4×D4C2×D311yescharacteristic
74C8×D4—24no—
75#431—21no—
76C2×#99—61no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8C2×D312yes—
79#398—31no—
80#397—31no—
81#394—31no—
82#392—31no—
83#388—61no—
84#386—61no—
85#386—61no—
86#380—32no—
87C2×#95—31no—
88#342—61no—
89#327—64no—
90C24×C3C22⋊C411yescharacteristic
91C24×C3C2×D411yescharacteristic
92C23×D3—21no—
93C2×C3×D4—22no—
94C2×C3×D4—22no—
95C2×C3×D4D812yes—
96C2×C3×D4SD1612yes—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—81no—
100C22×C3×C4—81no—
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22⋊C411yescharacteristic
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D412yes—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4112yes—
108C2×#67—42no—
109C2×#67—42no—
110C2×#67—42no—
111C2×#67—41no—
112C2×#67—41no—
113C2×#67—42no—
114C2×#67—81no—
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3—42no—
119C22×Dic3—41no—
120C22×Dic3—41no—
121C22×Dic3—42no—
122C22×Dic3—81no—
123C2×D12—22no—
124C2×D12—81no—
125C2×C3×C8—24no—
126C2×C3×C8—82no—
127C2×C3×C8—82no—
128C3×C4⋊C4—22no—
129C3×C22⋊C4—22no—
130C3×C22⋊C4—22no—
131C3×C42—21no—
132C3×C42—21no—
133#217—42no—
134#217—42no—
135#217—42no—
136(C2×D3)⋊C4—42no—
137(C2×D3)⋊C4—42no—
138(C2×D3)⋊C4—41no—
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—81no—
141C4⋊Dic3—22no—
142C4⋊Dic3—82no—
143Dic3⋊C4—42no—
144C22×D4—31no—
145C22×D4—61no—
146C22×D4—61no—
147C22×D4D1211yescharacteristic
148C23×C4—61no—
149C23×C4#6711yescharacteristic
150C23×C4C22×D311yescharacteristic
151C22×C8—22no—
152C22×C8—22no—
153#121—62no—
154#121—62no—
155#121—122no—
156#121—122no—
157C4×D4—22no—
158C4×D4—22no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—61no—
161C2×C4⋊C4—62no—
162C2×C4⋊C4C4×D311yescharacteristic
163C2×C22⋊C4—61no—
164C2×C22⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4—62no—
168C2×C22⋊C4—61no—
169C2×C22⋊C4C4×D311yescharacteristic
170C2×C22⋊C4D1211yescharacteristic
171C2×C42#6711yescharacteristic
172C4⋊C8—22no—
173#99—122no—
174C22⋊C8—22no—
175C22⋊C8—22no—
176C22⋊C8#6714yescomplemented
177C4×C8—22no—
178#95—31no—
179#95—31no—
180#95—61no—
181#95—61no—
182#95—61no—
183#95—62no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—41no—
187C23×C3—41no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3#10212yes—
191C23×C3C2×C22⋊C411yescharacteristic
192C23×C3#12112yes—
193C23×C3C2×D811yescharacteristic
194C23×C3C2×SD1611yescharacteristic
195C23×C3#13512yes—
196C22×D3—41no—
197C22×D3—41no—
198C22×D3—41no—
199C22×D3—41no—
200C22×D3—41no—
201C22×D3—41no—
202C22×D3—41no—
203C3×D4—24no—
204C3×D4—24no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—81no—
217C2×C3×C4#10212yes—
218C2×C3×C4C4×D412yes—
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12311yescharacteristic
221#67—84no—
222#67—82no—
223#67—82no—
224C2×Dic3—22no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—82no—
234C2×Dic3—82no—
235C2×Dic3—82no—
236C2×Dic3—82no—
237C2×Dic3—82no—
238D12—82no—
239C3×C8—44no—
240C24—61no—
241C24(C2×D3)⋊C411yescharacteristic
242C24C2×D1211yescharacteristic
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—62no—
246C2×D4—122no—
247C2×D4—122no—
248C2×D4—121no—
249C2×D4—121no—
250C2×D4—122no—
251C2×D4—122no—
252C2×D4—241no—
253C2×D4—241no—
254C2×D4#20412yes—
255C2×D4D2412yes—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—122no—
264C22×C4—121no—
265C22×C4—121no—
266C22×C4—122no—
267C22×C4—241no—
268C22×C4(C2×D3)⋊C411yescharacteristic
269C22×C4(C2×D3)⋊C411yescharacteristic
270C22×C4(C2×D3)⋊C411yescharacteristic
271C22×C4C2×C4×D311yescharacteristic
272C22×C4#23512yes—
273C22×C4C2×#6712yes—
274C22×C4C2×#6711yescharacteristic
275C2×C8—24no—
276C2×C8—82no—
277C2×C8—82no—
278C4⋊C4—22no—
279C4⋊C4—62no—
280C4⋊C4—122no—
281C4⋊C4—242no—
282C22⋊C4—22no—
283C22⋊C4—22no—
284C22⋊C4—122no—
285C22⋊C4—122no—
286C22⋊C4—122no—
287C22⋊C4—121no—
288C22⋊C4—121no—
289C22⋊C4—122no—
290C22⋊C4—122no—
291C22⋊C4—241no—
292C42—21no—
293C42—21no—
294C22×C3—21no—
295C22×C3—22no—
296C22×C3—21no—
297C22×C3—22no—
298C22×C3—22no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—42no—
302C22×C3—42no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—81no—
306C22×C3#38611yescharacteristic
307C22×C3C2×#10211yescharacteristic
308C22×C3#41611yescharacteristic
309C22×C3#46512yes—
310C22×C3#46612yes—
311C2×D3—81no—
312C2×D3—81no—
313C2×D3—81no—
314C2×D3—81no—
315C2×D3—81no—
316C2×D3—81no—
317C2×D3—81no—
318C3×C4—22no—
319C3×C4—21no—
320C3×C4—22no—
321C3×C4—21no—
322C3×C4—41no—
323C3×C4—41no—
324Dic3—42no—
325Dic3—81no—
326Dic3—81no—
327Dic3—162no—
328C23—21no—
329C23—21no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—121no—
335C23—121no—
336C23—121no—
337C23—121no—
338C23—121no—
339C23—121no—
340C23—121no—
341C23#82112yes—
342C23C2×#20411yescharacteristic
343C23C2×D2411yescharacteristic
344C23#90412yes—
345C23C2×(C2×D3)⋊C411yescharacteristic
346C23#93012yes—
347D4—24no—
348D4—24no—
349D4—242no—
350D4—244no—
351D4—242no—
352D4—242no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—62no—
365C2×C4—81no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—242no—
375C2×C4—242no—
376C2×C4—242no—
377C2×C4—242no—
378C2×C4—242no—
379C2×C4#82112yes—
380C2×C4C4×#6712yes—
381C2×C4#92911yescharacteristic
382C2×C4#95311yescharacteristic
383C8—44no—
384C2×C3—21no—
385C2×C3—22no—
386C2×C3—21no—
387C2×C3—22no—
388C2×C3—42no—
389C2×C3#188411yescharacteristic
390C2×C3#189311yescharacteristic
391C2×C3#191311yescharacteristic
392D3—161no—
393C22—21no—
394C22—22no—
395C22—21no—
396C22—22no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—42no—
401C22—42no—
402C22—41no—
403C22—41no—
404C22—81no—
405C22—241no—
406C22—241no—
407C22—241no—
408C22—241no—
409C22—241no—
410C22—241no—
411C22—241no—
412C22C2×#82111yescharacteristic
413C22#511311yescharacteristic
414C22#511412yes—
415C22#511512yes—
416C22#521311yescharacteristic
417C4—22no—
418C4—21no—
419C4—21no—
420C4—22no—
421C4—41no—
422C4—41no—
423C4—122no—
424C4—241no—
425C4—241no—
426C4—482no—
427C3#938411yescharacteristic, complemented, Sylow
428C2—21no—
429C2—22no—
430C2—21no—
431C2—22no—
432C2—42no—
433C2—481no—
434C2#7165611yescharacteristic
435C2#7197311yescharacteristic
436C2#7202111yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2972C2
socleC22×C3#386
prime radical#82380C2
cosocleC23×C3×C4C23
hypercenter#2972D3
hypocenterC3#9384
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2972

Complements

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

NH ≅ G/Ntype
1. C3×#2972C2semidirect
3. #71619C2semidirect
4. #69944C2semidirect
5. #67675C2semidirect
18. #4991C4semidirect
19. #4981C4semidirect
20. #4960C4semidirect
22. #4953C4semidirect
26. #2972D3semidirect
62. C3×C22⋊C8D4semidirect
176. C22⋊C8#67semidirect
427. C3#9384semidirect

Maximal subgroups

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

subgroupindex
C3×#29722
#823802
#716192
#699442
#676752
#93843