Finite Group Databasebeta

← #10691642

Subgroups — #10691642

Order 768.

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

normal subgroups only

This group has 594 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×#4736C211yesmaximal, characteristic
2#73878C211yesmaximal, characteristic, complemented
3#69528C211yesmaximal, characteristic
4#69509C211yesmaximal, characteristic
5#69501C211yesmaximal, characteristic, complemented
6C3×#1289C211yesmaximal, characteristic, complemented
7#67677C211yesmaximal, characteristic, complemented
8#9260—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C22×C4⋊Dic3C2211yescharacteristic
11C2×#877—21no—
12C2×C3×C22⋊C8C2211yescharacteristic
13C3×#380C2211yescharacteristic
14C3×#380C2211yescharacteristic
15C4×#217C2211yescharacteristic
16C2×#830C2211yescharacteristic
17C2×#821C412yescomplemented
18C2×#818C412yescomplemented
19Dic3×C4⋊C4—21no—
20#4972—21no—
21C2×#806—21no—
22C2×#806—21no—
23#4943—21no—
24#4736—21no—
25#4736—21no—
26#4736—22no—
27#4654—22no—
28#4529—22no—
29#4451—22no—
30C2×#480—31no—
31#2344—31no—
32#1964—31no—
33#1823—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—22no—
63C2×C4×Dic3—21no—
64C22×C3⋊C8—21no—
65#877—42no—
66#877—42no—
67C3×C22⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×#95—21no—
70C3×#95D412yes—
71C3⋊#95—22no—
72C3⋊#95—22no—
73C3⋊#95—21no—
74#830—21no—
75#830—21no—
76#821—22no—
77#821—22no—
78#818—22no—
79#818—22no—
80#806—42no—
81#806—42no—
82C2×#121—31no—
83C2×#117—61no—
84C22×C4⋊C4—31no—
85#480—62no—
86#480—61no—
87#480—61no—
88C2×#107—32no—
89C2×#102—32no—
90C2×#99—61no—
91C2×#99—61no—
92C2×C22⋊C8—31no—
93C2×C22⋊C8C2×D311yescharacteristic
94#389—61no—
95#387—62no—
96#381—61no—
97#380C2×D311yescharacteristic
98#380C2×D311yescharacteristic
99C4×C4⋊C4—61no—
100C4×C22⋊C4—31no—
101#340—62no—
102#327—62no—
103C24×C3C2×D411yescharacteristic
104C23×D3—21no—
105C22×C3×C4—22no—
106C22×C3×C4—42no—
107C22×C3×C4—41no—
108C22×C3×C4C22×C411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C2×#67—41no—
115C2×#67—41no—
116C2×#67—41no—
117C2×#67—41no—
118C2×#67—41no—
119C2×#67—41no—
120C22×Dic3—22no—
121C22×Dic3—21no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—22no—
125C22×Dic3—21no—
126C22×Dic3—21no—
127C22×Dic3—41no—
128C22×Dic3—41no—
129C22×Dic3—41no—
130C22×Dic3—41no—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—42no—
134C2×C3×C8—22no—
135C2×C3×C8—42no—
136C3×C4⋊C4—42no—
137C3×C4⋊C4—42no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142#217—21no—
143#217—21no—
144#217—21no—
145#217D812yes—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148C4⋊Dic3—22no—
149C4⋊Dic3—22no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3C22⋊C412yes—
155C4⋊Dic3C22⋊C412yes—
156Dic3⋊C4—44no—
157C4×Dic3—44no—
158C4×Dic3—42no—
159C2×C3⋊C8—42no—
160C2×C3⋊C8—41no—
161C22×D4—31no—
162C22×D4—61no—
163C23×C4—61no—
164C23×C4C22×D311yescharacteristic
165C22×C8—61no—
166C22×C8C4×D312yes—
167#121—61no—
168#121—61no—
169#121—61no—
170#121—61no—
171#117—122no—
172#117—122no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—32no—
176C2×C4⋊C4—32no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—62no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—31no—
187C2×C22⋊C4—31no—
188C2×C22⋊C4—61no—
189C2×C42—62no—
190C2×C42—61no—
191#107—62no—
192#107—62no—
193#102—62no—
194#102—62no—
195#99—122no—
196#99—122no—
197C22⋊C8—21no—
198C22⋊C8—21no—
199C22⋊C8—61no—
200C22⋊C8—61no—
201#95—21no—
202#95—62no—
203#95—62no—
204#95—61no—
205#95D1212yes—
206C23×C3—41no—
207C23×C3—41no—
208C23×C3C4×D411yescharacteristic
209C23×C3#12011yescharacteristic
210C23×C3#12111yescharacteristic
211C23×C3C2×D811yescharacteristic
212C23×C3#13511yescharacteristic
213C23×C3AGL(1,ℤ/8)11yescharacteristic
214C23×C3AGL(1,ℤ/8)11yescharacteristic
215C22×D3—41no—
216C22×D3—41no—
217C22×D3—41no—
218C22×D3—81no—
219C22×D3—81no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—81no—
230C2×C3×C4—81no—
231C2×C3×C4—81no—
232C2×C3×C4C2×C22⋊C411yescharacteristic
233C2×C3×C4C4×D411yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12311yescharacteristic
236#67—81no—
237#67—81no—
238#67—81no—
239#67—81no—
240C2×Dic3—22no—
241C2×Dic3—22no—
242C2×Dic3—22no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—42no—
252C2×Dic3—42no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259D12—42no—
260D12—81no—
261C3×C8—42no—
262C3⋊C8—81no—
263C24—61no—
264C24D3×D411yescharacteristic
265C2×D4—61no—
266C2×D4—61no—
267C2×D4—122no—
268C2×D4—121no—
269C2×D4—121no—
270C2×D4—121no—
271C2×D4—121no—
272C2×D4—121no—
273C2×D4—121no—
274C22×C4—22no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—62no—
281C22×C4—62no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—121no—
285C22×C4—121no—
286C22×C4—121no—
287C22×C4—121no—
288C22×C4C2×C4×D311yescharacteristic
289C22×C4C2×D1211yescharacteristic
290C22×C4#23511yescharacteristic
291C22×C4D3×D411yescharacteristic
292C22×C4D3×D411yescharacteristic
293C22×C4#23911yescharacteristic
294C2×C8—22no—
295C2×C8—42no—
296C2×C8—122no—
297C2×C8—121no—
298C4⋊C4—32no—
299C4⋊C4—32no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—62no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—62no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—124no—
309C22⋊C4—32no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—61no—
315C22⋊C4—61no—
316C22⋊C4—61no—
317C22⋊C4—121no—
318C22⋊C4—121no—
319C42—122no—
320C42—124no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3#38611yescharacteristic
332C22×C3C4×D811yescharacteristic
333C22×C3#44211yescharacteristic
334C22×C3#44711yescharacteristic
335C22×C3#44911yescharacteristic
336C22×C3#45911yescharacteristic
337C22×C3#46311yescharacteristic
338C2×D3—81no—
339C2×D3—81no—
340C2×D3—81no—
341C2×D3—161no—
342C2×D3—161no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—82no—
347C3×C4—161no—
348C3×C4#41012yes—
349Dic3—42no—
350Dic3—42no—
351Dic3—42no—
352Dic3—81no—
353Dic3—81no—
354C23—41no—
355C23—41no—
356C23—121no—
357C23—121no—
358C23—121no—
359C23—241no—
360C23—241no—
361C23#88211yescharacteristic
362C23#88411yescharacteristic
363C23#89311yescharacteristic
364C23D3×D811yescharacteristic
365C23#91411yescharacteristic
366C23#91411yescharacteristic
367C23#91911yescharacteristic
368D4—122no—
369D4—241no—
370D4—241no—
371D4—241no—
372D4—241no—
373D4—241no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4—62no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—122no—
392C2×C4—122no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—122no—
396C2×C4—121no—
397C2×C4—121no—
398C2×C4—122no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4C4×D1211yescharacteristic
406C2×C4D3×C22⋊C411yescharacteristic
407C2×C4#89411yescharacteristic
408C2×C4#89511yescharacteristic
409C8—42no—
410C8—241no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3—21no—
416C2×C3—21no—
417C2×C3#187911yescharacteristic
418C2×C3#189911yescharacteristic
419C2×C3#190511yescharacteristic
420D3—161no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—241no—
432C22—241no—
433C22—241no—
434C22—481no—
435C22—481no—
436C22#466511yescharacteristic
437C22#477311yescharacteristic
438C22#478611yescharacteristic
439C22#481011yescharacteristic
440C22#481911yescharacteristic
441C22#486911yescharacteristic
442C22#487211yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—82no—
447C4—122no—
448C4—122no—
449C4—122no—
450C4—161no—
451C4—241no—
452C4—241no—
453C4#474012yes—
454C3#926011yescharacteristic, complemented, Sylow
455C2—21no—
456C2—21no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—481no—
462C2#6870711yescharacteristic
463C2#6889711yescharacteristic
464C2#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#9260
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. #73878C2semidirect
5. #69501C2semidirect
6. C3×#1289C2semidirect
7. #67677C2semidirect
17. C2×#821C4semidirect
18. C2×#818C4semidirect
35. #1289D3semidirect
454. C3#9260semidirect

Maximal subgroups

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

subgroupindex
C2×#47362
#738782
#695282
#695092
#695012
C3×#12892
#676772
#92603