Finite Group Databasebeta

← #10691649

Subgroups — #10691649

Order 768.

566 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#4735C211yesmaximal, characteristic
2#73879C211yesmaximal, characteristic, complemented
3#69528C211yesmaximal, characteristic
4#69509C211yesmaximal, characteristic
5#69502C211yesmaximal, characteristic, complemented
6C3×#1289C211yesmaximal, characteristic, complemented
7#67677C211yesmaximal, characteristic, complemented
8#9261—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C22×C4⋊Dic3C2211yescharacteristic
11C2×C3×C22⋊C8C2211yescharacteristic
12C3×#380C2211yescharacteristic
13C3×#380C2211yescharacteristic
14C4×#217C2211yescharacteristic
15C2×#830C2211yescharacteristic
16C2×#821C412yescomplemented
17C2×#817C412yescomplemented
18Dic3×C4⋊C4—21no—
19#4972—21no—
20C2×#806—21no—
21C2×#806—21no—
22#4942—21no—
23Dic3×C22⋊C4—21no—
24#4735—21no—
25#4735—21no—
26#4735—22no—
27#4653—22no—
28#4529—22no—
29#4451—22no—
30C2×#481—31no—
31#2345—31no—
32#1964—31no—
33#1824—31no—
34#1519—31no—
35#1289D311yescharacteristic, complemented
36#1282—31no—
37C23×C3×C4C2311yescharacteristic
38C22×#67—21no—
39C23×Dic3—21no—
40C22×D12C2×C411yescharacteristic
41C22×C3×C8C2×C412yes—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×#217D411yescharacteristic
47C2×#217D411yescharacteristic
48#930—21no—
49#930—21no—
50#930—21no—
51#930—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3C2×C411yescharacteristic
57C2×C4⋊Dic3C2×C412yes—
58C2×C4⋊Dic3D411yescharacteristic
59C2×C4⋊Dic3D411yescharacteristic
60C2×C4⋊Dic3D412yes—
61C2×Dic3⋊C4—22no—
62C2×C4×Dic3—21no—
63C2×C4×Dic3—22no—
64C22×C3⋊C8—21no—
65C3×C22⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×#95—21no—
68C3×#95D412yes—
69C3⋊#95—22no—
70C3⋊#95—22no—
71C3⋊#95—21no—
72#830—21no—
73#830—21no—
74#821—22no—
75#821—22no—
76#817—22no—
77#817—22no—
78#806—42no—
79#806—42no—
80C2×#121—31no—
81C22×C4⋊C4—31no—
82#481—62no—
83#481—61no—
84#481—61no—
85C2×#106—32no—
86C2×#102—32no—
87C2×#99—61no—
88C2×#99—61no—
89C2×C22⋊C8—31no—
90C2×C22⋊C8C2×D311yescharacteristic
91#389—61no—
92#387—62no—
93#380—61no—
94#380C2×D311yescharacteristic
95#380C2×D311yescharacteristic
96C4×C4⋊C4—61no—
97C4×C22⋊C4—31no—
98C4×C22⋊C4—61no—
99#340—62no—
100#327—62no—
101C24×C3C2×D411yescharacteristic
102C23×D3—21no—
103C22×C3×C4—22no—
104C22×C3×C4—42no—
105C22×C3×C4—41no—
106C22×C3×C4C22×C411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C2×#67—41no—
113C2×#67—41no—
114C2×#67—41no—
115C2×#67—41no—
116C2×#67—41no—
117C2×#67—41no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—21no—
121C22×Dic3—22no—
122C22×Dic3—22no—
123C22×Dic3—21no—
124C22×Dic3—21no—
125C22×Dic3—82no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12—42no—
129C2×C3×C8—22no—
130C2×C3×C8—42no—
131C3×C4⋊C4—42no—
132C3×C4⋊C4—42no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137#217—21no—
138#217—21no—
139#217—21no—
140#217SD1612yes—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—41no—
143C4⋊Dic3—22no—
144C4⋊Dic3—22no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3C22⋊C412yes—
150C4⋊Dic3C22⋊C412yes—
151Dic3⋊C4—82no—
152C4×Dic3—42no—
153C4×Dic3—82no—
154C2×C3⋊C8—42no—
155C2×C3⋊C8—41no—
156C22×D4—31no—
157C22×D4—61no—
158C23×C4—61no—
159C23×C4C22×D311yescharacteristic
160C22×C8—61no—
161C22×C8C4×D312yes—
162#121—61no—
163#121—61no—
164#121—61no—
165#121—61no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—32no—
169C2×C4⋊C4—32no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—31no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—62no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—31no—
180C2×C22⋊C4—31no—
181C2×C22⋊C4—61no—
182C2×C42—61no—
183C2×C42—62no—
184#106—62no—
185#106—62no—
186#102—62no—
187#102—62no—
188#99—122no—
189#99—122no—
190C22⋊C8—21no—
191C22⋊C8—21no—
192C22⋊C8—61no—
193C22⋊C8—61no—
194#95—21no—
195#95—62no—
196#95—62no—
197#95—61no—
198#95D1212yes—
199C23×C3—41no—
200C23×C3—41no—
201C23×C3C4×D411yescharacteristic
202C23×C3#12011yescharacteristic
203C23×C3#12111yescharacteristic
204C23×C3C2×SD1611yescharacteristic
205C23×C3#13511yescharacteristic
206C23×C3AGL(1,ℤ/8)11yescharacteristic
207C23×C3#13711yescharacteristic
208C22×D3—41no—
209C22×D3—41no—
210C22×D3—41no—
211C22×D3—81no—
212C22×D3—81no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—81no—
223C2×C3×C4—81no—
224C2×C3×C4—81no—
225C2×C3×C4C2×C22⋊C411yescharacteristic
226C2×C3×C4C4×D411yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12311yescharacteristic
229#67—81no—
230#67—81no—
231#67—81no—
232#67—81no—
233C2×Dic3—22no—
234C2×Dic3—22no—
235C2×Dic3—22no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—42no—
239C2×Dic3—42no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—82no—
247C2×Dic3—82no—
248D12—42no—
249D12—81no—
250C3×C8—42no—
251C3⋊C8—81no—
252C24—61no—
253C24D3×D411yescharacteristic
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—122no—
257C2×D4—121no—
258C2×D4—121no—
259C2×D4—121no—
260C2×D4—121no—
261C2×D4—121no—
262C2×D4—121no—
263C22×C4—22no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4—62no—
270C22×C4—62no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—242no—
274C22×C4C2×C4×D311yescharacteristic
275C22×C4C2×D1211yescharacteristic
276C22×C4#23511yescharacteristic
277C22×C4D3×D411yescharacteristic
278C22×C4D3×D411yescharacteristic
279C22×C4#23911yescharacteristic
280C2×C8—22no—
281C2×C8—42no—
282C2×C8—122no—
283C2×C8—121no—
284C4⋊C4—32no—
285C4⋊C4—32no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—62no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—62no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—242no—
295C22⋊C4—32no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—61no—
301C22⋊C4—61no—
302C22⋊C4—61no—
303C22⋊C4—121no—
304C22⋊C4—121no—
305C42—122no—
306C42—242no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3#38611yescharacteristic
318C22×C3C4×SD1611yescharacteristic
319C22×C3#44011yescharacteristic
320C22×C3#44811yescharacteristic
321C22×C3#44911yescharacteristic
322C22×C3#46011yescharacteristic
323C22×C3#46411yescharacteristic
324C2×D3—81no—
325C2×D3—81no—
326C2×D3—81no—
327C2×D3—161no—
328C2×D3—161no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—82no—
333C3×C4—161no—
334C3×C4#41012yes—
335Dic3—42no—
336Dic3—42no—
337Dic3—42no—
338Dic3—81no—
339Dic3—81no—
340C23—41no—
341C23—41no—
342C23—121no—
343C23—121no—
344C23—121no—
345C23—241no—
346C23—241no—
347C23#88211yescharacteristic
348C23#88411yescharacteristic
349C23#89311yescharacteristic
350C23#91111yescharacteristic
351C23D3×SD1611yescharacteristic
352C23#91611yescharacteristic
353C23#91811yescharacteristic
354D4—122no—
355D4—241no—
356D4—241no—
357D4—241no—
358D4—241no—
359D4—241no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—62no—
370C2×C4—62no—
371C2×C4—62no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—242no—
386C2×C4—242no—
387C2×C4C4×D1211yescharacteristic
388C2×C4D3×C22⋊C411yescharacteristic
389C2×C4#89411yescharacteristic
390C2×C4#89511yescharacteristic
391C8—42no—
392C8—241no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3#188011yescharacteristic
400C2×C3#189911yescharacteristic
401C2×C3#190411yescharacteristic
402D3—161no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—241no—
414C22—241no—
415C22—241no—
416C22—481no—
417C22—481no—
418C22#466511yescharacteristic
419C22#476211yescharacteristic
420C22#478111yescharacteristic
421C22#479811yescharacteristic
422C22#480611yescharacteristic
423C22#485211yescharacteristic
424C22#488911yescharacteristic
425C4—21no—
426C4—21no—
427C4—21no—
428C4—82no—
429C4—122no—
430C4—122no—
431C4—122no—
432C4—161no—
433C4—241no—
434C4—241no—
435C4#474012yes—
436C3#926111yescharacteristic, complemented, Sylow
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—481no—
444C2#6870411yescharacteristic
445C2#6890911yescharacteristic
446C2#6895411yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1289C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1289D3
hypocenterC3#9261
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
2. #73879C2semidirect
5. #69502C2semidirect
6. C3×#1289C2semidirect
7. #67677C2semidirect
16. C2×#821C4semidirect
17. C2×#817C4semidirect
35. #1289D3semidirect
436. C3#9261semidirect

Maximal subgroups

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

subgroupindex
C2×#47352
#738792
#695282
#695092
#695022
C3×#12892
#676772
#92613