Finite Group Databasebeta

← #10692210

Subgroups — #10692210

Order 768.

534 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 103 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×#3193C211yesmaximal, characteristic
2#82639C211yesmaximal, characteristic
3#71701C211yesmaximal, characteristic, complemented
4#71689C211yesmaximal, characteristic, complemented
5#71673C211yesmaximal, characteristic
6#67685C211yesmaximal, characteristic, complemented
7#67682C211yesmaximal, characteristic, complemented
8#9341—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C3×#484—21no—
12C3×#480C412yescomplemented
13C3×#468—22no—
14C3×#450—21no—
15C3×#448C412yescomplemented
16C3×#407C2211yescharacteristic
17#5221—21no—
18#5220—21no—
19#5220—21no—
20#5212C2211yescharacteristic
21#5211C2211yescharacteristic
22C4×#217C2211yescharacteristic
23#5207C2211yescharacteristic
24#4980—21no—
25#4951—21no—
26#4950—21no—
27#4538—21no—
28#4538—21no—
29#4537—22no—
30#4526—21no—
31#4526—21no—
32#3193D311yescharacteristic
33#2377—31no—
34#1996—31no—
35#1511—31no—
36#1507—31no—
37#1305—31no—
38#1284—31no—
39C22×C3×Q8C2×C411yescharacteristic, complemented
40C22×C3×D4—21no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C23×Dic3—21no—
44C2×C3×SD16—22no—
45C2×C3×M16—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×#121C2×C412yes—
53C3×#121D412yes—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×#217—21no—
60C2×#217—21no—
61C2×#217—21no—
62C2×#217—21no—
63C2×#217—21no—
64C2×#217D411yescharacteristic
65C2×#217D411yescharacteristic
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—21no—
68C2×C4×Dic3—21no—
69C2×C4×Dic3—21no—
70C2×#208—21no—
71C3×#107—22no—
72C3×#103—22no—
73C3×#102—22no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8C2×C412yescomplemented
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95D411yescharacteristic
82C3⋊#95D411yescharacteristic
83#830—21no—
84#830—21no—
85C2×C22⋊Q8C2×D311yescharacteristic
86C2×#121C2×D311yescharacteristic
87#484—21no—
88#480Dic312yescomplemented
89#468—22no—
90#450—21no—
91#448Dic312yescomplemented
92#407—31no—
93#407C2×D311yescharacteristic
94#390—61no—
95#388—61no—
96#387—61no—
97#386—61no—
98#386—61no—
99#385—61no—
100#381—31no—
101#380—31no—
102C4×C22⋊C4—31no—
103#342—61no—
104#342—61no—
105#328—61no—
106#328—61no—
107#327—62no—
108C24×C3—21no—
109C24×C3C2×D411yescharacteristic
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—81no—
113C2×C3×D4—22no—
114C2×C3×D4—22no—
115C2×C3×D4—21no—
116C2×C3×D4—81no—
117C2×C3×D4—81no—
118C2×C3×D4—81no—
119C2×C3×D4C22⋊C412yes—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—81no—
124C22×C3×C4C22×C411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4#4111yescharacteristic
129C22×C3×C4#4111yescharacteristic
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3—41no—
135C22×Dic3—81no—
136C22×Dic3—81no—
137C22×Dic3—81no—
138C3×SD16—82no—
139C3×M16—81no—
140C2×C3×C8—22no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4C22⋊C412yes—
147C3×C22⋊C4—22no—
148C3×C22⋊C4—22no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151#217—22no—
152#217—41no—
153#217—41no—
154#217—41no—
155#217—81no—
156#217—81no—
157#217—81no—
158#217—81no—
159C4⋊Dic3—81no—
160C4⋊Dic3—81no—
161C4×Dic3—81no—
162C4×Dic3—81no—
163#208—81no—
164C2×C3⋊C8—41no—
165C22×Q8C2×Dic311yescharacteristic, complemented
166C22×D4—21no—
167C22×D4#6711yescharacteristic
168C23×C4—61no—
169C23×C4C22×D311yescharacteristic
170C2×SD16—22no—
171C2×M16—21no—
172C2×M16—61no—
173C22⋊Q8—21no—
174C22⋊Q8—21no—
175C22⋊Q8—21no—
176C22⋊Q8—21no—
177#121—21no—
178#121—21no—
179#121C2×Dic312yes—
180#121#6712yes—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4C2×Dic311yescharacteristic, complemented
185C2×C4⋊C4#6711yescharacteristic
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—31no—
189C2×C22⋊C4—31no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4—61no—
195C2×C42—61no—
196C2×C42—61no—
197#107—22no—
198#103—22no—
199#102—22no—
200C22⋊C8—21no—
201C22⋊C8—61no—
202C22⋊C8—61no—
203C22⋊C8C2×Dic312yescomplemented
204#95—31no—
205#95—31no—
206#95—61no—
207#95—61no—
208#95—61no—
209#95—61no—
210#95—61no—
211C23×C3—21no—
212C23×C3—21no—
213C23×C3—22no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3C4×D411yescharacteristic
223C23×C3#12011yescharacteristic
224C23×C3#12111yescharacteristic
225C3×Q8—42no—
226C3×Q8—81no—
227C3×D4—42no—
228C3×D4—81no—
229C3×D4—82no—
230C3×D4—82no—
231C2×C3×C4—22no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—81no—
243C2×C3×C4—81no—
244C2×C3×C4C2×C22⋊C411yescharacteristic
245C2×C3×C4C4×D411yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12311yescharacteristic
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—42no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—81no—
262C2×Dic3—81no—
263C2×Dic3—84no—
264C3×C8—42no—
265C3⋊C8—81no—
266C24—21no—
267C24D3×D411yescharacteristic
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—81no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—21no—
274C2×D4—81no—
275C2×D4—81no—
276C2×D4—81no—
277C2×D4#21712yes—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—81no—
286C22×C4—121no—
287C22×C4—241no—
288C22×C4—241no—
289C22×C4—241no—
290C22×C4D3×D411yescharacteristic
291C22×C4#23711yescharacteristic
292C22×C4#23711yescharacteristic
293C22×C4C22×Dic311yescharacteristic
294C22×C4C2×#6711yescharacteristic
295C22×C4C2×#6711yescharacteristic
296SD16—82no—
297M16—81no—
298M16—241no—
299C2×C8—22no—
300C2×C8—121no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—241no—
307C4⋊C4—241no—
308C4⋊C4#21712yes—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—62no—
314C22⋊C4—121no—
315C22⋊C4—121no—
316C22⋊C4—121no—
317C22⋊C4—241no—
318C22⋊C4—241no—
319C22⋊C4—241no—
320C22⋊C4—241no—
321C42—241no—
322C42—241no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—42no—
332C22×C3—42no—
333C22×C3—42no—
334C22×C3—81no—
335C22×C3—81no—
336C22×C3—81no—
337C22×C3—81no—
338C22×C3—81no—
339C22×C3—81no—
340C22×C3#38611yescharacteristic
341C22×C3#45511yescharacteristic
342C22×C3#45611yescharacteristic
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—42no—
348C3×C4—81no—
349C3×C4—81no—
350Dic3—81no—
351Dic3—81no—
352Dic3—81no—
353Dic3—81no—
354Dic3—82no—
355C23—21no—
356C23—21no—
357C23—22no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23D4×Dic311yescharacteristic
367C23#93711yescharacteristic
368C23#93911yescharacteristic
369Q8—42no—
370Q8—81no—
371D4—42no—
372D4—81no—
373D4—82no—
374D4—82no—
375C2×C4—22no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—122no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—241no—
402C2×C4—241no—
403C2×C4—244no—
404C2×C4D4×Dic311yescharacteristic
405C2×C4#93511yescharacteristic
406C2×C4#93911yescharacteristic
407C2×C4C2×#21711yescharacteristic
408C8—42no—
409C8—241no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—41no—
413C2×C3—41no—
414C2×C3—82no—
415C2×C3#188211yescharacteristic
416C2×C3#190411yescharacteristic
417C2×C3#191311yescharacteristic
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22#520211yescharacteristic
436C22#520511yescharacteristic
437C22#522011yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—42no—
443C4—81no—
444C4—81no—
445C4—241no—
446C4—241no—
447C4—241no—
448C4—241no—
449C4—242no—
450C3#934111yescharacteristic, complemented, Sylow
451C2—21no—
452C2—21no—
453C2—41no—
454C2—41no—
455C2—82no—
456C2#7184811yescharacteristic
457C2#7208911yescharacteristic
458C2#7210011yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#3193C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#3193D3
hypocenterC3#9341
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3193

Complements

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

NH ≅ G/Ntype
3. #71701C2semidirect
4. #71689C2semidirect
6. #67685C2semidirect
7. #67682C2semidirect
12. C3×#480C4semidirect
15. C3×#448C4semidirect
39. C22×C3×Q8C2×C4semidirect
55. C2×C3×C4⋊C4C2×C4semidirect
75. C3×C22⋊C8C2×C4semidirect
88. #480Dic3semidirect
91. #448Dic3semidirect
165. C22×Q8C2×Dic3semidirect
184. C2×C4⋊C4C2×Dic3semidirect
203. C22⋊C8C2×Dic3semidirect
450. C3#9341semidirect

Maximal subgroups

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

subgroupindex
C3×#31932
#826392
#717012
#716892
#716732
#676852
#676822
#93413