Finite Group Databasebeta

← #10691254

Subgroups — #10691254

Order 768.

622 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 113 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#74032C211yesmaximal, characteristic
2#73869C211yesmaximal, characteristic, complemented
3#69944C211yesmaximal, characteristic
4C2×#4469C211yesmaximal, characteristic, complemented
5#69509C211yesmaximal, characteristic
6C3×#1289C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#9204—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C22×Dic3⋊C4C2211yescharacteristic
11C2×#886—21no—
12C2×#877—21no—
13C2×C3×C22⋊C8C2211yescharacteristic
14C3×#380C2211yescharacteristic
15C3×#380C2211yescharacteristic
16#5212C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#4994—22no—
19#4976—22no—
20#4958—21no—
21#4942—21no—
22#4672C412yescomplemented
23#4661C412yescomplemented
24#4647—22no—
25#4470—22no—
26#4469—22no—
27#4451—22no—
28#2618—31no—
29#2316—31no—
30#1519—31no—
31#1519—31no—
32C2×#324—31no—
33#1289—31no—
34#1289D311yescharacteristic, complemented
35C23×C3×C4C2311yescharacteristic
36C22×#67—21no—
37C23×Dic3D411yescharacteristic
38C23×Dic3D411yescharacteristic
39C22×D12C2×C411yescharacteristic
40C22×C3×C8—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×#217—21no—
46#930—21no—
47#930—21no—
48#930—21no—
49#930—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—22no—
52C2×(C2×D3)⋊C4—22no—
53C2×C4⋊Dic3—22no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3C2×C411yescharacteristic
56C2×Dic3⋊C4—21no—
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4—21no—
59C2×Dic3⋊C4—21no—
60C2×Dic3⋊C4—22no—
61C2×Dic3⋊C4—22no—
62C2×Dic3⋊C4C2×C412yes—
63C2×Dic3⋊C4D412yes—
64C2×C4×Dic3—22no—
65C22×C3⋊C8—21no—
66#886—42no—
67#886—42no—
68#877—42no—
69#877—42no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×#95C2×C412yes—
73C3×#95D412yes—
74C3⋊#95—22no—
75C3⋊#95—22no—
76C3⋊#95D412yes—
77#830—21no—
78#830—21no—
79C2×#123—61no—
80C2×#121—31no—
81C2×#117—61no—
82C22×C4⋊C4—31no—
83C2×C22⋊C8—31no—
84C2×C22⋊C8C2×D311yescharacteristic
85#398—32no—
86#396—62no—
87#394—32no—
88#389—62no—
89#389—62no—
90#386—61no—
91#380—31no—
92#380—61no—
93#380C2×D311yescharacteristic
94#380C2×D311yescharacteristic
95#327—62no—
96#327—62no—
97#324—62no—
98C24×C3C2×D411yescharacteristic
99C23×D3—21no—
100C22×C3×C4—22no—
101C22×C3×C4—22no—
102C22×C3×C4—81no—
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C2×#67—41no—
110C2×#67—41no—
111C2×#67—41no—
112C2×#67—41no—
113C2×#67—41no—
114C2×#67—41no—
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—21no—
121C22×Dic3—21no—
122C22×Dic3—22no—
123C22×Dic3—81no—
124C22×Dic3C22⋊C412yes—
125C22×Dic3C22⋊C412yes—
126C22×Dic3D812yes—
127C22×Dic3SD1612yes—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—81no—
131C2×C3×C8—41no—
132C2×C3×C8—81no—
133C3×C4⋊C4—81no—
134C3×C4⋊C4—81no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C22⋊C4—41no—
139#217—41no—
140#217—41no—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—41no—
143(C2×D3)⋊C4—44no—
144(C2×D3)⋊C4—82no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—44no—
148C4⋊Dic3—81no—
149Dic3⋊C4—22no—
150Dic3⋊C4—22no—
151Dic3⋊C4—22no—
152Dic3⋊C4—22no—
153Dic3⋊C4—44no—
154Dic3⋊C4—82no—
155C4×Dic3—44no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—81no—
158C22×D4—31no—
159C22×D4—61no—
160C23×C4—31no—
161C23×C4—31no—
162C23×C4C22×D311yescharacteristic
163C22×C8—21no—
164C22×C8—61no—
165#123—122no—
166#123—122no—
167#121—61no—
168#121—61no—
169#121—61no—
170#121—61no—
171#117—122no—
172#117—122no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—32no—
176C2×C4⋊C4—32no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—62no—
183C2×C4⋊C4—62no—
184C2×C4⋊C4—62no—
185C2×C4⋊C4—61no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—62no—
189C2×C22⋊C4—61no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—62no—
192C2×C42—62no—
193C22⋊C8—21no—
194C22⋊C8—21no—
195C22⋊C8—61no—
196C22⋊C8—61no—
197#95—32no—
198#95—62no—
199#95—62no—
200#95C4×D312yes—
201#95D1212yes—
202C23×C3—41no—
203C23×C3—41no—
204C23×C3C2×C22⋊C411yescharacteristic
205C23×C3#12011yescharacteristic
206C23×C3#12311yescharacteristic
207C23×C3C2×D811yescharacteristic
208C23×C3C2×SD1611yescharacteristic
209C23×C3AGL(1,ℤ/8)11yescharacteristic
210C23×C3#13711yescharacteristic
211C22×D3—42no—
212C22×D3—42no—
213C22×D3—41no—
214C22×D3—41no—
215C22×D3—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4C4×D411yescharacteristic
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12111yescharacteristic
232#67—81no—
233#67—81no—
234#67—81no—
235#67—81no—
236C2×Dic3—22no—
237C2×Dic3—22no—
238C2×Dic3—22no—
239C2×Dic3—22no—
240C2×Dic3—22no—
241C2×Dic3—22no—
242C2×Dic3—22no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—42no—
251C2×Dic3—81no—
252C2×Dic3#10214yes—
253D12—81no—
254D12—81no—
255C3×C8—81no—
256C3⋊C8—81no—
257C24—61no—
258C24D3×D411yescharacteristic
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—121no—
262C2×D4—121no—
263C2×D4—121no—
264C2×D4—121no—
265C2×D4—121no—
266C2×D4—121no—
267C2×D4—241no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—32no—
271C22×C4—32no—
272C22×C4—32no—
273C22×C4—32no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—62no—
282C22×C4—81no—
283C22×C4—241no—
284C22×C4C2×C4×D311yescharacteristic
285C22×C4C2×D1211yescharacteristic
286C22×C4#23511yescharacteristic
287C22×C4D3×D411yescharacteristic
288C22×C4D3×D411yescharacteristic
289C22×C4#23911yescharacteristic
290C2×C8—41no—
291C2×C8—81no—
292C2×C8—121no—
293C2×C8—241no—
294C4⋊C4—62no—
295C4⋊C4—62no—
296C4⋊C4—62no—
297C4⋊C4—62no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C4⋊C4—81no—
301C4⋊C4—81no—
302C4⋊C4—124no—
303C4⋊C4—124no—
304C4⋊C4—242no—
305C4⋊C4—241no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—124no—
311C22⋊C4—121no—
312C22⋊C4—121no—
313C22⋊C4—121no—
314C22⋊C4—121no—
315C22⋊C4—242no—
316C42—124no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3#38611yescharacteristic
328C22×C3C2×#10211yescharacteristic
329C22×C3#41711yescharacteristic
330C22×C3#44711yescharacteristic
331C22×C3#44811yescharacteristic
332C22×C3#48011yescharacteristic
333C22×C3#48111yescharacteristic
334C2×D3—82no—
335C2×D3—82no—
336C2×D3—81no—
337C2×D3—81no—
338C2×D3—81no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—82no—
344C3×C4—82no—
345Dic3—24no—
346Dic3—42no—
347Dic3—82no—
348Dic3—81no—
349C23—41no—
350C23—41no—
351C23—122no—
352C23—122no—
353C23—121no—
354C23—121no—
355C23—121no—
356C23D3×C22⋊C411yescharacteristic
357C23#88211yescharacteristic
358C23#89411yescharacteristic
359C23D3×D811yescharacteristic
360C23D3×SD1611yescharacteristic
361C23#91411yescharacteristic
362C23#91811yescharacteristic
363D4—241no—
364D4—241no—
365D4—241no—
366D4—241no—
367D4—241no—
368D4—241no—
369C2×C4—34no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—62no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4—62no—
386C2×C4—62no—
387C2×C4—62no—
388C2×C4—62no—
389C2×C4—122no—
390C2×C4—122no—
391C2×C4—122no—
392C2×C4—122no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—122no—
396C2×C4—121no—
397C2×C4—241no—
398C2×C4C4×D1211yescharacteristic
399C2×C4#88411yescharacteristic
400C2×C4#89311yescharacteristic
401C2×C4#89511yescharacteristic
402C8—81no—
403C8—241no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3#187411yescharacteristic
411C2×C3#191211yescharacteristic
412C2×C3#191311yescharacteristic
413D3—161no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—242no—
425C22—242no—
426C22—241no—
427C22—241no—
428C22—241no—
429C22#466511yescharacteristic
430C22D3×#10211yescharacteristic
431C22#477311yescharacteristic
432C22#480311yescharacteristic
433C22#480611yescharacteristic
434C22#486311yescharacteristic
435C22#488211yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—64no—
441C4—82no—
442C4—82no—
443C4—122no—
444C4—242no—
445C4—241no—
446C3#920411yescharacteristic, complemented, Sylow
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—21no—
452C2—21no—
453C2—481no—
454C2#6870811yescharacteristic
455C2#6888011yescharacteristic
456C2#6896411yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1289C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1289D3
hypocenterC3#9204
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
2. #73869C2semidirect
4. C2×#4469C2semidirect
6. C3×#1289C2semidirect
7. #67625C2semidirect
22. #4672C4semidirect
23. #4661C4semidirect
34. #1289D3semidirect
446. C3#9204semidirect

Maximal subgroups

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

subgroupindex
#740322
#738692
#699442
C2×#44692
#695092
C3×#12892
#676252
#92043