Finite Group Databasebeta

← #10692990

Subgroups — #10692990

Order 768.

818 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 97 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#74063C211yesmaximal, characteristic
2#73435C211yesmaximal, characteristic, complemented
3#69991C211yesmaximal, characteristic
4#69971C211yesmaximal, characteristic, complemented
5#69454C211yesmaximal, characteristic
6C3×#1284C211yesmaximal, characteristic, complemented
7#67628C211yesmaximal, characteristic, complemented
8#9476—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×Dic3⋊C4C2211yescharacteristic
11C2×#886—21no—
12C2×#877—21no—
13C3×#407C2211yescharacteristic
14C3×#381C2211yescharacteristic
15C3×C4×C22⋊C4C2211yescharacteristic
16C4×#217C2211yescharacteristic
17#5207C2211yescharacteristic
18C2×#825—22no—
19#4958—21no—
20Dic3×C22⋊C4—21no—
21C4×Dic3⋊C4—22no—
22C2×#805—22no—
23#4672C412yescomplemented
24#4668—22no—
25#4656C412yescomplemented
26#4649—22no—
27#4469—22no—
28#2406—31no—
29#2274—31no—
30#1789—31no—
31#1789—31no—
32#1463—31no—
33#1284—31no—
34#1284D311yescharacteristic, complemented
35C23×C3×C4C2311yescharacteristic
36C22×#67—21no—
37C23×Dic3D411yescharacteristic
38C23×Dic3D411yescharacteristic
39C2×#235—21no—
40C2×#235—21no—
41C2×#235—21no—
42C2×#235—21no—
43C2×#235—44no—
44C22×D12C2×C411yescharacteristic
45C22×C4×D3—21no—
46C22×Dic6C2×C411yescharacteristic
47C2×C3×M16—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C42D412yes—
51C2×#217—21no—
52C2×(C2×D3)⋊C4—22no—
53C2×(C2×D3)⋊C4—22no—
54C2×C4⋊Dic3—22no—
55C2×Dic3⋊C4—22no—
56C2×Dic3⋊C4—21no—
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4—21no—
59C2×Dic3⋊C4—21no—
60C2×Dic3⋊C4C2×C412yes—
61C2×Dic3⋊C4D412yes—
62C2×C4×Dic3—22no—
63C2×C4×Dic3—22no—
64C2×C4×Dic3D412yes—
65C2×#208—21no—
66#886—44no—
67#877—28no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#95—21no—
71C3×#95—21no—
72C3×#95C2×C412yes—
73C3⋊#95—22no—
74C3⋊#95—21no—
75C3⋊#95—22no—
76#830—21no—
77#830—21no—
78#825—44no—
79#805—44no—
80C22×#41—31no—
81C2×#123—61no—
82C2×#117—61no—
83C22×C4⋊C4—31no—
84C2×#104—62no—
85C2×#104—62no—
86#407—31no—
87#407C2×D311yescharacteristic
88#395—32no—
89#394—32no—
90#386—61no—
91#385—62no—
92#381C2×D311yescharacteristic
93C4×C4⋊C4—62no—
94C4×C4⋊C4—62no—
95C4×C22⋊C4—31no—
96C4×C22⋊C4—61no—
97C4×C22⋊C4C2×D311yescharacteristic
98#324—62no—
99C24×C3C2×D411yescharacteristic
100C23×D3—21no—
101C22×C3×C4—22no—
102C22×C3×C4—22no—
103C22×C3×C4—42no—
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C2×#67—24no—
111C2×#67—42no—
112C2×#67—41no—
113C2×#67—41no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—24no—
117C22×Dic3—21no—
118C22×Dic3—22no—
119C22×Dic3—21no—
120C22×Dic3—21no—
121C22×Dic3—42no—
122C22×Dic3C22⋊C412yes—
123C22×Dic3C22⋊C412yes—
124#235—24no—
125#235—24no—
126#235—42no—
127#235—42no—
128#235—42no—
129#235—42no—
130#235—42no—
131#235—42no—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12—42no—
135C2×C4×D3—42no—
136C2×C4×D3—42no—
137C2×C4×D3—41no—
138C2×C4×D3—41no—
139C2×Dic6—21no—
140C2×Dic6—21no—
141C2×Dic6—42no—
142C3×M16—42no—
143C2×C3×C8—41no—
144C3×C22⋊C4—24no—
145C3×C22⋊C4—42no—
146C3×C42—42no—
147#217—24no—
148(C2×D3)⋊C4—44no—
149(C2×D3)⋊C4—44no—
150C4⋊Dic3—44no—
151Dic3⋊C4—28no—
152Dic3⋊C4—22no—
153Dic3⋊C4—22no—
154Dic3⋊C4—22no—
155Dic3⋊C4—22no—
156C4×Dic3—28no—
157C4×Dic3—42no—
158C4×Dic3—82no—
159#208—42no—
160C2×C3⋊C8—41no—
161C2×#41—61no—
162C2×#41—61no—
163C2×#41—61no—
164C2×#41—61no—
165C2×#41—124no—
166C22×Q8—31no—
167C22×D4—31no—
168C22×D4—61no—
169C23×C4—31no—
170C23×C4—31no—
171C23×C4—61no—
172C23×C4C22×D311yescharacteristic
173C2×M16—21no—
174C2×M16—61no—
175#123—124no—
176#117—68no—
177C2×C4⋊C4—32no—
178C2×C4⋊C4—32no—
179C2×C4⋊C4—62no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—62no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—62no—
189C2×C22⋊C4—62no—
190C2×C42—32no—
191C2×C42—62no—
192C2×C42—62no—
193C2×C42D1212yes—
194#104—124no—
195#104—124no—
196C22⋊C8—21no—
197C22⋊C8—21no—
198C22⋊C8—61no—
199C22⋊C8—61no—
200#95—21no—
201#95—21no—
202#95—62no—
203#95—62no—
204#95—61no—
205#95C4×D312yes—
206C23×C3—22no—
207C23×C3—22no—
208C23×C3—41no—
209C23×C3C2×C22⋊C411yescharacteristic
210C23×C3#12011yescharacteristic
211C23×C3#12311yescharacteristic
212C22×D3—22no—
213C22×D3—41no—
214C22×D3—42no—
215C22×D3—41no—
216C22×D3—42no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—24no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4C4×D411yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233#67—44no—
234#67—44no—
235C2×Dic3—22no—
236C2×Dic3—22no—
237C2×Dic3—24no—
238C2×Dic3—22no—
239C2×Dic3—22no—
240C2×Dic3—24no—
241C2×Dic3—22no—
242C2×Dic3—22no—
243C2×Dic3—41no—
244C2×Dic3—42no—
245C2×Dic3—41no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—81no—
250C2×Dic3#10414yescomplemented
251D12—22no—
252D12—22no—
253D12—41no—
254D12—41no—
255C4×D3—44no—
256C4×D3—44no—
257Dic6—22no—
258Dic6—22no—
259Dic6—41no—
260Dic6—41no—
261C3×C8—81no—
262C3⋊C8—81no—
263C24—61no—
264C24D3×D411yescharacteristic
265#41—64no—
266#41—64no—
267#41—122no—
268#41—122no—
269#41—122no—
270#41—122no—
271#41—122no—
272#41—122no—
273C2×Q8—61no—
274C2×Q8—61no—
275C2×Q8—122no—
276C2×D4—64no—
277C2×D4—61no—
278C2×D4—61no—
279C2×D4—122no—
280C2×D4—122no—
281C2×D4—121no—
282C2×D4—121no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—32no—
286C22×C4—32no—
287C22×C4—42no—
288C22×C4—64no—
289C22×C4—61no—
290C22×C4—61no—
291C22×C4—61no—
292C22×C4—62no—
293C22×C4—61no—
294C22×C4—61no—
295C22×C4—122no—
296C22×C4—122no—
297C22×C4—121no—
298C22×C4—121no—
299C22×C4—122no—
300C22×C4C2×C4×D311yescharacteristic
301C22×C4C2×D1211yescharacteristic
302C22×C4#23511yescharacteristic
303C22×C4D3×D411yescharacteristic
304C22×C4D3×D411yescharacteristic
305C22×C4#23911yescharacteristic
306M16—42no—
307M16—122no—
308C2×C8—41no—
309C2×C8—121no—
310C4⋊C4—68no—
311C4⋊C4—62no—
312C4⋊C4—62no—
313C4⋊C4—62no—
314C4⋊C4—62no—
315C4⋊C4—124no—
316C22⋊C4—24no—
317C22⋊C4—42no—
318C22⋊C4—64no—
319C22⋊C4—124no—
320C22⋊C4—124no—
321C42—42no—
322C42—68no—
323C42—122no—
324C42—242no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—44no—
333C22×C3#38611yescharacteristic
334C22×C3C2×#10412yes—
335C22×C3B(4,𝔽2)12yes—
336C2×D3—44no—
337C2×D3—42no—
338C2×D3—82no—
339C2×D3—81no—
340C2×D3—81no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—44no—
346C3×C4—82no—
347Dic3—24no—
348Dic3—44no—
349Dic3—42no—
350Dic3—42no—
351C23—22no—
352C23—22no—
353C23—41no—
354C23—62no—
355C23—121no—
356C23—122no—
357C23—121no—
358C23—122no—
359C23D3×C22⋊C411yescharacteristic
360C23#88211yescharacteristic
361C23#89411yescharacteristic
362Q8—62no—
363Q8—62no—
364Q8—121no—
365Q8—121no—
366D4—62no—
367D4—62no—
368D4—121no—
369D4—121no—
370D4—124no—
371D4—124no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—34no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—62no—
386C2×C4—62no—
387C2×C4—64no—
388C2×C4—62no—
389C2×C4—62no—
390C2×C4—64no—
391C2×C4—62no—
392C2×C4—62no—
393C2×C4—124no—
394C2×C4—124no—
395C2×C4—121no—
396C2×C4—122no—
397C2×C4—121no—
398C2×C4—122no—
399C2×C4—122no—
400C2×C4—122no—
401C2×C4—241no—
402C2×C4C4×D1211yescharacteristic
403C2×C4#88411yescharacteristic
404C2×C4#89311yescharacteristic
405C2×C4#89511yescharacteristic
406C8—81no—
407C8—241no—
408C2×C3—22no—
409C2×C3—22no—
410C2×C3—41no—
411C2×C3#188911yescharacteristic
412C2×C3#189611yescharacteristic
413C2×C3#191211yescharacteristic
414D3—82no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—44no—
423C22—124no—
424C22—122no—
425C22—242no—
426C22—241no—
427C22—241no—
428C22#466511yescharacteristic
429C22#474112yes—
430C22D3×#10412yes—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—44no—
436C4—64no—
437C4—82no—
438C4—124no—
439C4—122no—
440C4—122no—
441C3#947611yescharacteristic, complemented, Sylow
442C2—22no—
443C2—22no—
444C2—41no—
445C2—242no—
446C2#6870811yescharacteristic
447C2#6875611yescharacteristic
448C2#6896211yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1284C2
socleC22×C3#386
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenter#1284D3
hypocenterC3#9476
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. #73435C2semidirect
4. #69971C2semidirect
6. C3×#1284C2semidirect
7. #67628C2semidirect
23. #4672C4semidirect
25. #4656C4semidirect
34. #1284D3semidirect
250. C2×Dic3#104semidirect
441. C3#9476semidirect

Maximal subgroups

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

subgroupindex
#740632
#734352
#699912
#699712
#694542
C3×#12842
#676282
#94763