Finite Group Databasebeta

← #10693005

Subgroups — #10693005

Order 768.

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

normal subgroups only

This group has 838 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
1#74125C211yesmaximal, characteristic
2#73567C211yesmaximal, characteristic, complemented
3#69991C211yesmaximal, characteristic, complemented
4#69967C211yesmaximal, characteristic, complemented
5#69454C211yesmaximal, characteristic, complemented
6C3×#1284C211yesmaximal, characteristic, complemented
7#67628C211yesmaximal, characteristic, complemented
8#9479—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×(C2×D3)⋊C4C2211yescharacteristic
11C2×#882—21no—
12C3×#407C2211yescharacteristic
13C3×#381C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15C4×#217C2211yescharacteristic
16#5207C2211yescharacteristic
17C2×#825—22no—
18#4958—21no—
19#4955—21no—
20#4942—21no—
21#4940—22no—
22C2×#805—22no—
23#4673C412yescomplemented
24#4671C412yescomplemented
25#4669—22no—
26#4665—22no—
27#4468—22no—
28#2404—31no—
29#2312—31no—
30#1789—31no—
31#1789—31no—
32#1462—31no—
33#1284—31no—
34#1284D311yescharacteristic, complemented
35C24×D3D411yescharacteristic
36C23×C3×C4C2311yescharacteristic
37C22×#67—21no—
38C23×Dic3D411yescharacteristic
39C2×#235—21no—
40C2×#235—21no—
41C2×#235—21no—
42C2×#235—21no—
43C2×#235—44no—
44C22×D12—22no—
45C22×D12C2×C411yescharacteristic
46C22×C4×D3—21no—
47C22×Dic6C2×C411yescharacteristic
48C2×C3×M16—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C42D412yescomplemented
52C2×#217—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—22no—
58C2×(C2×D3)⋊C4—22no—
59C2×(C2×D3)⋊C4—22no—
60C2×(C2×D3)⋊C4—22no—
61C2×(C2×D3)⋊C4C2×C412yes—
62C2×(C2×D3)⋊C4D412yes—
63C2×Dic3⋊C4—22no—
64C2×C4×Dic3D412yescomplemented
65C2×#208—21no—
66#882—44no—
67C3×C22⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×#95—21no—
70C3×#95—21no—
71C3×#95C2×C412yes—
72C3⋊#95—22no—
73C3⋊#95—21no—
74C3⋊#95—22no—
75#830—21no—
76#830—21no—
77#825—44no—
78#805—44no—
79C22×#41—31no—
80C2×#120—61no—
81C22×C22⋊C4—31no—
82C2×#104—62no—
83C2×#104—62no—
84#407—31no—
85#407C2×D311yescharacteristic
86#393—32no—
87#392—32no—
88#388—62no—
89#388—62no—
90#386—62no—
91#386—61no—
92#381C2×D311yescharacteristic
93#380—61no—
94#379—61no—
95C4×C22⋊C4—31no—
96C4×C22⋊C4C2×D311yescharacteristic
97#323—62no—
98C24×C3C2×D411yescharacteristic
99C23×D3—21no—
100C23×D3—21no—
101C23×D3—21no—
102C23×D3—42no—
103C23×D3—42no—
104C23×D3—81no—
105C23×D3C22⋊C412yes—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—42no—
109C22×C3×C4C22×C411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C22×C3×C4#4111yescharacteristic
115C2×#67—24no—
116C2×#67—42no—
117C2×#67—41no—
118C2×#67—41no—
119C22×Dic3—21no—
120C22×Dic3—21no—
121C22×Dic3—21no—
122C22×Dic3—22no—
123C22×Dic3—81no—
124C22×Dic3C22⋊C412yes—
125#235—24no—
126#235—24no—
127#235—42no—
128#235—42no—
129#235—42no—
130#235—42no—
131#235—42no—
132#235—42no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12—42no—
136C2×D12—44no—
137C2×D12—44no—
138C2×D12—82no—
139C2×C4×D3—42no—
140C2×C4×D3—42no—
141C2×C4×D3—41no—
142C2×C4×D3—41no—
143C2×Dic6—21no—
144C2×Dic6—21no—
145C2×Dic6—42no—
146C3×M16—42no—
147C2×C3×C8—41no—
148C3×C22⋊C4—42no—
149C3×C22⋊C4—42no—
150C3×C42—42no—
151#217—42no—
152(C2×D3)⋊C4—22no—
153(C2×D3)⋊C4—22no—
154(C2×D3)⋊C4—22no—
155(C2×D3)⋊C4—22no—
156(C2×D3)⋊C4—44no—
157(C2×D3)⋊C4—82no—
158(C2×D3)⋊C4—82no—
159(C2×D3)⋊C4—82no—
160Dic3⋊C4—82no—
161C4×Dic3—42no—
162#208—42no—
163C2×C3⋊C8—41no—
164C25—31no—
165C2×#41—61no—
166C2×#41—61no—
167C2×#41—61no—
168C2×#41—61no—
169C2×#41—124no—
170C22×Q8—31no—
171C22×D4—31no—
172C22×D4—62no—
173C22×D4—61no—
174C23×C4—31no—
175C23×C4—61no—
176C23×C4C22×D311yescharacteristic
177C2×M16—21no—
178C2×M16—61no—
179#120—124no—
180C2×C4⋊C4—62no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—32no—
184C2×C22⋊C4—32no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—62no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—62no—
192C2×C22⋊C4—62no—
193C2×C22⋊C4—62no—
194C2×C42—32no—
195C2×C42D1212yescomplemented
196#104—124no—
197#104—124no—
198C22⋊C8—21no—
199C22⋊C8—21no—
200C22⋊C8—61no—
201C22⋊C8—61no—
202#95—21no—
203#95—21no—
204#95—62no—
205#95—62no—
206#95—61no—
207#95C4×D312yes—
208C23×C3—22no—
209C23×C3—22no—
210C23×C3—41no—
211C23×C3C2×C22⋊C411yescharacteristic
212C23×C3#12011yescharacteristic
213C23×C3#12311yescharacteristic
214C22×D3—22no—
215C22×D3—22no—
216C22×D3—22no—
217C22×D3—22no—
218C22×D3—42no—
219C22×D3—42no—
220C22×D3—42no—
221C22×D3—42no—
222C22×D3—41no—
223C22×D3—42no—
224C22×D3—42no—
225C22×D3—41no—
226C22×D3—42no—
227C22×D3—42no—
228C22×D3—81no—
229C22×D3—81no—
230C22×D3—88no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4C4×D411yescharacteristic
244C2×C3×C4C4×D411yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12111yescharacteristic
247#67—44no—
248#67—44no—
249C2×Dic3—22no—
250C2×Dic3—22no—
251C2×Dic3—42no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—81no—
259D12—22no—
260D12—22no—
261D12—41no—
262D12—41no—
263D12—88no—
264C4×D3—44no—
265C4×D3—44no—
266Dic6—22no—
267Dic6—22no—
268Dic6—41no—
269Dic6—41no—
270C3×C8—81no—
271C3⋊C8—81no—
272C24—32no—
273C24—61no—
274C24—61no—
275C24—61no—
276C24—122no—
277C24—122no—
278C24—241no—
279C24D3×D411yescharacteristic
280#41—64no—
281#41—64no—
282#41—122no—
283#41—122no—
284#41—122no—
285#41—122no—
286#41—122no—
287#41—122no—
288C2×Q8—61no—
289C2×Q8—61no—
290C2×Q8—122no—
291C2×D4—64no—
292C2×D4—61no—
293C2×D4—61no—
294C2×D4—122no—
295C2×D4—124no—
296C2×D4—124no—
297C2×D4—122no—
298C2×D4—121no—
299C2×D4—121no—
300C2×D4—242no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—32no—
304C22×C4—42no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—62no—
309C22×C4—122no—
310C22×C4—122no—
311C22×C4—121no—
312C22×C4—121no—
313C22×C4—241no—
314C22×C4C2×C4×D311yescharacteristic
315C22×C4C2×D1211yescharacteristic
316C22×C4#23511yescharacteristic
317C22×C4D3×D411yescharacteristic
318C22×C4D3×D411yescharacteristic
319C22×C4#23911yescharacteristic
320M16—42no—
321M16—122no—
322C2×C8—41no—
323C2×C8—121no—
324C4⋊C4—242no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—62no—
328C22⋊C4—62no—
329C22⋊C4—62no—
330C22⋊C4—62no—
331C22⋊C4—124no—
332C22⋊C4—122no—
333C22⋊C4—242no—
334C22⋊C4—242no—
335C22⋊C4—242no—
336C42—42no—
337C42—122no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—44no—
346C22×C3#38611yescharacteristic
347C22×C3#42112yes—
348C22×C3B(4,𝔽2)12yes—
349C2×D3—44no—
350C2×D3—44no—
351C2×D3—42no—
352C2×D3—42no—
353C2×D3—42no—
354C2×D3—42no—
355C2×D3—82no—
356C2×D3—84no—
357C2×D3—84no—
358C2×D3—81no—
359C2×D3—81no—
360C2×D3—82no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—82no—
366C3×C4—82no—
367Dic3—42no—
368Dic3—42no—
369Dic3—82no—
370C23—22no—
371C23—22no—
372C23—41no—
373C23—62no—
374C23—62no—
375C23—62no—
376C23—62no—
377C23—122no—
378C23—122no—
379C23—122no—
380C23—122no—
381C23—121no—
382C23—122no—
383C23—122no—
384C23—121no—
385C23—122no—
386C23—122no—
387C23—241no—
388C23—241no—
389C23—248no—
390C23D3×C22⋊C411yescharacteristic
391C23#88211yescharacteristic
392C23#89411yescharacteristic
393Q8—62no—
394Q8—62no—
395Q8—121no—
396Q8—121no—
397D4—62no—
398D4—62no—
399D4—121no—
400D4—121no—
401D4—124no—
402D4—124no—
403D4—248no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—62no—
417C2×C4—62no—
418C2×C4—124no—
419C2×C4—124no—
420C2×C4—122no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—241no—
428C2×C4C4×D1211yescharacteristic
429C2×C4#88411yescharacteristic
430C2×C4#89311yescharacteristic
431C2×C4#89511yescharacteristic
432C8—81no—
433C8—241no—
434C2×C3—22no—
435C2×C3—22no—
436C2×C3—41no—
437C2×C3#188911yescharacteristic
438C2×C3#189711yescharacteristic
439C2×C3#191111yescharacteristic
440D3—82no—
441D3—82no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—44no—
450C22—124no—
451C22—124no—
452C22—122no—
453C22—122no—
454C22—122no—
455C22—122no—
456C22—242no—
457C22—244no—
458C22—242no—
459C22—244no—
460C22—241no—
461C22—241no—
462C22#466511yescharacteristic
463C22#474112yes—
464C22#482112yes—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—82no—
470C4—82no—
471C4—122no—
472C4—122no—
473C4—242no—
474C3#947911yescharacteristic, complemented, Sylow
475C2—22no—
476C2—22no—
477C2—41no—
478C2—242no—
479C2—242no—
480C2#6870511yescharacteristic
481C2#6875711yescharacteristic
482C2#6896211yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1284C2
socleC22×C3#386
prime radical#74125C2
cosocleC23×C3×C4C23
hypercenter#1284D3
hypocenterC3#9479
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1284

Complements

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

NH ≅ G/Ntype
2. #73567C2semidirect
3. #69991C2semidirect
4. #69967C2semidirect
5. #69454C2semidirect
6. C3×#1284C2semidirect
7. #67628C2semidirect
23. #4673C4semidirect
24. #4671C4semidirect
34. #1284D3semidirect
51. C2×C3×C42D4semidirect
64. C2×C4×Dic3D4semidirect
195. C2×C42D12semidirect
474. C3#9479semidirect

Maximal subgroups

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

subgroupindex
#741252
#735672
#699912
#699672
#694542
C3×#12842
#676282
#94793