Finite Group Databasebeta

← #10692206

Subgroups — #10692206

Order 768.

534 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 95 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#82639C211yesmaximal, characteristic
2#82631C211yesmaximal, characteristic
3C3×#1507C211yesmaximal, characteristic
4#69981C211yesmaximal, characteristic, complemented
5#69457C211yesmaximal, characteristic, complemented
6#67684C211yesmaximal, characteristic, complemented
7#67616C211yesmaximal, characteristic, complemented
8#9341—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×C4×C22⋊C4C2211yescharacteristic
13#5221—21no—
14#5220—21no—
15#5220—21no—
16#5218—21no—
17#5212C2211yescharacteristic
18#5211C2211yescharacteristic
19#5207C2211yescharacteristic
20#5042—22no—
21#5040C412yescomplemented
22#5033C412yescomplemented
23#4980—21no—
24#4970—21no—
25#4951—21no—
26#4950—21no—
27C3×#327—22no—
28#4527—21no—
29#4527—21no—
30#4467—21no—
31#4467—21no—
32#3193—31no—
33#2377—31no—
34#1996—31no—
35#1511—31no—
36#1507D311yescharacteristic
37#1305—31no—
38#1284—31no—
39C22×C3×D4—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C23×Dic3—21no—
43C22×Dic6C2×C411yescharacteristic, complemented
44C2×C3×M16—21no—
45C3×#121—21no—
46C3×#121—21no—
47C3×#121C2×C412yes—
48C3×#121D412yes—
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C42—21no—
53C2×#217—21no—
54C2×#217—21no—
55C2×#217—21no—
56C2×#217—21no—
57C2×#217—21no—
58C2×#217D411yescharacteristic
59C2×#217D411yescharacteristic
60C2×#214—22no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3C2×C411yescharacteristic, complemented
64#924—21no—
65#924—21no—
66#924—21no—
67#924—21no—
68C2×Dic3⋊C4—21no—
69C2×C4×Dic3—21no—
70C2×#208—21no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#95—21no—
74#832—22no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95D411yescharacteristic
80C3⋊#95D411yescharacteristic
81#830—21no—
82#830C2×C412yescomplemented
83#810—22no—
84#807—22no—
85C2×C22⋊Q8—31no—
86C2×#121C2×D311yescharacteristic
87#484—61no—
88#480—32no—
89#468—62no—
90#450—61no—
91#448—32no—
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⋊C4C2×D311yescharacteristic
103#342—61no—
104#342—61no—
105#328—61no—
106#328—61no—
107#327—22no—
108C24×C3—21no—
109C24×C3C2×D411yescharacteristic
110C2×C3×D4—22no—
111C2×C3×D4—22no—
112C2×C3×D4—21no—
113C2×C3×D4—81no—
114C2×C3×D4—81no—
115C2×C3×D4—81no—
116C2×C3×D4C22⋊C412yes—
117C22×C3×C4—21no—
118C22×C3×C4—41no—
119C22×C3×C4—81no—
120C22×C3×C4C22×C411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C22×Dic3—21no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—81no—
133C22×Dic3—81no—
134C22×Dic3—81no—
135C2×Dic6—21no—
136C2×Dic6—21no—
137C2×Dic6—81no—
138C3×M16—81no—
139C2×C3×C8—41no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4C22⋊C412yes—
142C3×C22⋊C4—22no—
143C3×C22⋊C4—22no—
144C3×C22⋊C4—22no—
145C3×C22⋊C4—41no—
146C3×C42—81no—
147#217—41no—
148#217—41no—
149#217—41no—
150#217—41no—
151#217—81no—
152#217—81no—
153#217—81no—
154#217—81no—
155#214—82no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—81no—
159C4⋊Dic3—81no—
160Dic3⋊C4—41no—
161Dic3⋊C4—41no—
162C4×Dic3—81no—
163#208—81no—
164C2×C3⋊C8—22no—
165C22×Q8—31no—
166C22×D4—21no—
167C22×D4D1211yescharacteristic
168C23×C4—61no—
169C23×C4C22×D311yescharacteristic
170C2×SD16—62no—
171C2×M16—21no—
172C2×M16—61no—
173C22⋊Q8—61no—
174C22⋊Q8—61no—
175C22⋊Q8—61no—
176C22⋊Q8—61no—
177#121—21no—
178#121—21no—
179#121C4×D312yes—
180#121#6712yes—
181C2×C4⋊C4—31no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4D1211yescharacteristic
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—21no—
196C2×C42—61no—
197#107—62no—
198#103—62no—
199#102—62no—
200C22⋊C8—21no—
201C22⋊C8—21no—
202C22⋊C8—32no—
203C22⋊C8—61no—
204#95—21no—
205#95—31no—
206#95—31no—
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×D4—42no—
226C3×D4—81no—
227C3×D4—82no—
228C3×D4—82no—
229C2×C3×C4—22no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
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—42no—
239C2×C3×C4—81no—
240C2×C3×C4—81no—
241C2×C3×C4C2×C22⋊C411yescharacteristic
242C2×C3×C4C4×D411yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12311yescharacteristic
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—84no—
260C2×Dic3—81no—
261C2×Dic3—81no—
262Dic6—42no—
263Dic6—81no—
264C3×C8—81no—
265C3⋊C8—42no—
266C24—21no—
267C24D3×D411yescharacteristic
268C2×Q8—61no—
269C2×Q8—61no—
270C2×Q8—241no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—21no—
274C2×D4—81no—
275C2×D4—81no—
276C2×D4—81no—
277C2×D4(C2×D3)⋊C412yes—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—81no—
287C22×C4—241no—
288C22×C4—241no—
289C22×C4—241no—
290C22×C4C2×C4×D311yescharacteristic
291C22×C4C2×D1211yescharacteristic
292C22×C4D3×D411yescharacteristic
293C22×C4#23711yescharacteristic
294C22×C4#23711yescharacteristic
295C22×C4C2×#6711yescharacteristic
296SD16—242no—
297M16—81no—
298M16—241no—
299C2×C8—41no—
300C2×C8—62no—
301C4⋊C4—21no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—121no—
305C4⋊C4—121no—
306C4⋊C4—241no—
307C4⋊C4—241no—
308C4⋊C4(C2×D3)⋊C412yes—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—41no—
313C22⋊C4—121no—
314C22⋊C4—121no—
315C22⋊C4—121no—
316C22⋊C4—121no—
317C22⋊C4—241no—
318C22⋊C4—241no—
319C22⋊C4—241no—
320C22⋊C4—241no—
321C42—81no—
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—82no—
349Dic3—81no—
350Dic3—81no—
351Dic3—81no—
352Dic3—81no—
353Dic3—81no—
354Dic3—81no—
355C23—21no—
356C23—21no—
357C23—22no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23#67⋊C411yescharacteristic
367C23#88211yescharacteristic
368C23#93911yescharacteristic
369Q8—122no—
370Q8—241no—
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—42no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
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—244no—
402C2×C4—241no—
403C2×C4—241no—
404C2×C4#67⋊C411yescharacteristic
405C2×C4#88611yescharacteristic
406C2×C4C2×(C2×D3)⋊C411yescharacteristic
407C2×C4#93911yescharacteristic
408C8—81no—
409C8—122no—
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#482511yescharacteristic
436C22#482611yescharacteristic
437C22#495811yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—42no—
443C4—82no—
444C4—241no—
445C4—241no—
446C4—241no—
447C4—241no—
448C4—241no—
449C4—241no—
450C3#934111yescharacteristic, complemented, Sylow
451C2—21no—
452C2—21no—
453C2—41no—
454C2—41no—
455C2—82no—
456C2#7002411yescharacteristic
457C2#7031411yescharacteristic
458C2#7036511yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1507C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#1507D3
hypocenterC3#9341
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1507

Complements

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

NH ≅ G/Ntype
4. #69981C2semidirect
5. #69457C2semidirect
6. #67684C2semidirect
7. #67616C2semidirect
21. #5040C4semidirect
22. #5033C4semidirect
43. C22×Dic6C2×C4semidirect
63. C2×C4⋊Dic3C2×C4semidirect
82. #830C2×C4semidirect
450. C3#9341semidirect

Maximal subgroups

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

subgroupindex
#826392
#826312
C3×#15072
#699812
#694572
#676842
#676162
#93413