Finite Group Databasebeta

← #10692209

Subgroups — #10692209

Order 768.

534 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 89 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#78642C211yesmaximal, characteristic
2#74124C211yesmaximal, characteristic
3#69533C211yesmaximal, characteristic, complemented
4#69457C211yesmaximal, characteristic, complemented
5#69444C211yesmaximal, characteristic
6C3×#1284C211yesmaximal, characteristic, complemented
7#67685C211yesmaximal, characteristic, complemented
8#9341—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×#381C2211yescharacteristic
13C3×C4×C22⋊C4C2211yescharacteristic
14#5212C2211yescharacteristic
15#5207C2211yescharacteristic
16#5134—22no—
17#4957—21no—
18#4956—21no—
19#4738C412yescomplemented
20#4736C412yescomplemented
21#4731—21no—
22#4722—21no—
23#4670—21no—
24#4668—21no—
25#4665—21no—
26#4654—21no—
27#4527—21no—
28#4527—21no—
29#4452—21no—
30#4452—21no—
31#4451—22no—
32#3193—31no—
33#2377—31no—
34#1996—31no—
35#1511—31no—
36#1507—31no—
37#1305—31no—
38#1284D311yescharacteristic, complemented
39C23×C3×C4C2311yescharacteristic
40C22×#67—21no—
41C22×D12D411yescharacteristic
42C22×C4×D3—21no—
43C22×Dic6C2×C411yescharacteristic, complemented
44C2×C3×M16—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C42—21no—
48C2×#217—21no—
49C2×#217D411yescharacteristic
50C2×#217D411yescharacteristic
51#930—21no—
52#930—21no—
53#930C2×C412yes—
54#930D412yes—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3C2×C411yescharacteristic, complemented
61C2×C4⋊Dic3D411yescharacteristic
62#924—21no—
63#924—21no—
64#924—21no—
65#924—21no—
66C2×Dic3⋊C4—21no—
67C2×Dic3⋊C4—21no—
68C2×C4×Dic3—21no—
69C2×#208—21no—
70C2×#204—22no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8C2×C412yescomplemented
73C3×#95—21no—
74C3×#95—21no—
75C3×#95D411yescharacteristic
76C3×#95D411yescharacteristic
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80#830—21no—
81#830—21no—
82#821—22no—
83#818—22no—
84#816—22no—
85C2×C22⋊Q8—31no—
86C2×#121—31no—
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#381C2×D311yescharacteristic
101#380—31no—
102C4×C22⋊C4C2×D311yescharacteristic
103#342—61no—
104#342—61no—
105#328—61no—
106#328—61no—
107#327—62no—
108C24×C3C2×D411yescharacteristic
109C23×D3—21no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—41no—
113C22×C3×C4—81no—
114C22×C3×C4C22×C411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C2×#67—22no—
121C2×#67—22no—
122C2×#67—81no—
123C2×#67—81no—
124C22×Dic3—21no—
125C22×Dic3—21no—
126C22×Dic3—21no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C2×D12—21no—
130C2×D12—81no—
131C2×D12C22⋊C412yes—
132C2×C4×D3—81no—
133C2×C4×D3—81no—
134C2×C4×D3—81no—
135C2×Dic6—21no—
136C2×Dic6—21no—
137C2×Dic6—81no—
138C3×M16—81no—
139C2×C3×C8—22no—
140C3×C22⋊C4—22no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—81no—
143C3×C42—81no—
144#217—41no—
145#217—41no—
146#217—41no—
147#217—41no—
148(C2×D3)⋊C4—22no—
149(C2×D3)⋊C4—22no—
150(C2×D3)⋊C4—81no—
151(C2×D3)⋊C4—81no—
152(C2×D3)⋊C4—81no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—81no—
157C4⋊Dic3C22⋊C412yes—
158Dic3⋊C4—41no—
159Dic3⋊C4—41no—
160Dic3⋊C4—81no—
161C4×Dic3—81no—
162#208—81no—
163C2×C3⋊C8—41no—
164#204—82no—
165C22×Q8—31no—
166C22×D4—31no—
167C22×D4—61no—
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—32no—
178#121—32no—
179#121—61no—
180#121—61no—
181C2×C4⋊C4—31no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
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—61no—
202C22⋊C8—61no—
203C22⋊C8C4×D312yescomplemented
204#95—21no—
205#95—21no—
206#95—61no—
207#95—61no—
208#95—61no—
209#95D1211yescharacteristic
210#95D1211yescharacteristic
211C23×C3—21no—
212C23×C3—21no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3C4×D411yescharacteristic
216C23×C3#12011yescharacteristic
217C23×C3#12111yescharacteristic
218C22×D3—22no—
219C22×D3—41no—
220C22×D3—41no—
221C22×D3—41no—
222C22×D3—41no—
223C22×D3—41no—
224C22×D3—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—81no—
239C2×C3×C4—81no—
240C2×C3×C4C2×C22⋊C411yescharacteristic
241C2×C3×C4C4×D411yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12311yescharacteristic
244#67—82no—
245#67—82no—
246C2×Dic3—22no—
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—81no—
258C2×Dic3—81no—
259D12—42no—
260D12—81no—
261C4×D3—84no—
262Dic6—42no—
263Dic6—81no—
264C3×C8—42no—
265C3⋊C8—81no—
266C24—61no—
267C24D3×D411yescharacteristic
268C2×Q8—61no—
269C2×Q8—61no—
270C2×Q8—241no—
271C2×D4—32no—
272C2×D4—62no—
273C2×D4—62no—
274C2×D4—61no—
275C2×D4—241no—
276C2×D4—241no—
277C2×D4—241no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—41no—
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×C4#23511yescharacteristic
293C22×C4D3×D411yescharacteristic
294C22×C4D3×D411yescharacteristic
295C22×C4#23911yescharacteristic
296SD16—242no—
297M16—81no—
298M16—241no—
299C2×C8—22no—
300C2×C8—121no—
301C4⋊C4—32no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—121no—
306C4⋊C4—121no—
307C4⋊C4—241no—
308C4⋊C4—241no—
309C22⋊C4—22no—
310C22⋊C4—41no—
311C22⋊C4—62no—
312C22⋊C4—62no—
313C22⋊C4—81no—
314C22⋊C4—121no—
315C22⋊C4—121no—
316C22⋊C4—121no—
317C22⋊C4—121no—
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—81no—
332C22×C3—81no—
333C22×C3#38611yescharacteristic
334C22×C3#45511yescharacteristic
335C22×C3#45611yescharacteristic
336C2×D3—42no—
337C2×D3—42no—
338C2×D3—42no—
339C2×D3—81no—
340C2×D3—81no—
341C2×D3—81no—
342C2×D3—81no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—81no—
348C3×C4—81no—
349C3×C4—82no—
350Dic3—42no—
351Dic3—81no—
352Dic3—81no—
353Dic3—81no—
354Dic3—81no—
355C23—21no—
356C23—21no—
357C23—41no—
358C23—41no—
359C23—62no—
360C23—121no—
361C23—121no—
362C23—121no—
363C23—121no—
364C23—121no—
365C23—121no—
366C23#88211yescharacteristic
367C23#88411yescharacteristic
368C23#89311yescharacteristic
369Q8—122no—
370Q8—241no—
371D4—122no—
372D4—241no—
373D4—242no—
374D4—242no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—62no—
389C2×C4—81no—
390C2×C4—81no—
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×C4C4×D1211yescharacteristic
405C2×C4D3×C22⋊C411yescharacteristic
406C2×C4#89411yescharacteristic
407C2×C4#89511yescharacteristic
408C8—42no—
409C8—241no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—41no—
413C2×C3—41no—
414C2×C3#188211yescharacteristic
415C2×C3#190411yescharacteristic
416C2×C3#191311yescharacteristic
417D3—82no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—81no—
427C22—81no—
428C22—122no—
429C22—122no—
430C22—122no—
431C22—241no—
432C22—241no—
433C22—241no—
434C22—241no—
435C22#466511yescharacteristic
436C22#482511yescharacteristic
437C22#482611yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—81no—
443C4—81no—
444C4—82no—
445C4—122no—
446C4—241no—
447C4—241no—
448C4—241no—
449C4—241no—
450C3#934111yescharacteristic, complemented, Sylow
451C2—21no—
452C2—21no—
453C2—41no—
454C2—41no—
455C2—242no—
456C2#6870411yescharacteristic
457C2#6877511yescharacteristic
458C2#6896411yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1284C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1284D3
hypocenterC3#9341
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
3. #69533C2semidirect
4. #69457C2semidirect
6. C3×#1284C2semidirect
7. #67685C2semidirect
19. #4738C4semidirect
20. #4736C4semidirect
38. #1284D3semidirect
43. C22×Dic6C2×C4semidirect
60. C2×C4⋊Dic3C2×C4semidirect
72. C3×C22⋊C8C2×C4semidirect
203. C22⋊C8C4×D3semidirect
450. C3#9341semidirect

Maximal subgroups

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

subgroupindex
#786422
#741242
#695332
#694572
#694442
C3×#12842
#676852
#93413