Finite Group Databasebeta

← #10695638

Subgroups — #10695638

Order 768.

636 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 101 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#78611C211yesmaximal, characteristic
2C2×#4735C211yesmaximal, characteristic, complemented
3#73519C211yesmaximal, characteristic, complemented
4#69944C211yesmaximal, characteristic, complemented
5#69925C211yesmaximal, characteristic, complemented
6#69456C211yesmaximal, characteristic, complemented
7C3×#1282C211yesmaximal, characteristic, complemented
8#9950—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C22×C4⋊Dic3C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15#5212C2211yescharacteristic
16#5115—22no—
17C2×#821—21no—
18C2×#817—21no—
19#4958—22no—
20C2×#806—22no—
21#4939—21no—
22#4936—21no—
23#4736—21no—
24#4736—21no—
25#4735—22no—
26#4735—22no—
27#4732—21no—
28#4732—21no—
29#4674—22no—
30#4551—22no—
31#4470—22no—
32#3186—31no—
33C2×#481—31no—
34#2504—31no—
35#1964—31no—
36#1824—31no—
37#1519—31no—
38#1282D311yescharacteristic, complemented
39C23×C3×C4C2311yescharacteristic
40C22×#67—21no—
41C22×#67—21no—
42C23×Dic3—21no—
43C22×D12D411yescharacteristic
44C22×D12D411yescharacteristic
45C22×C3×C8—21no—
46C22×C3×C8D412yes—
47C2×C3×C22⋊C4D412yes—
48C2×C3×C42—21no—
49C2×#217—21no—
50#930—21no—
51#930—21no—
52#930—21no—
53#930—21no—
54#930—21no—
55#930—21no—
56#930—21no—
57#930—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—24no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3D412yes—
68C2×C4⋊Dic3D411yescharacteristic
69C2×C4⋊Dic3D411yescharacteristic
70C2×D24—22no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8D412yes—
75C3×#95—21no—
76C3⋊#95—24no—
77C3⋊#95—21no—
78#821—22no—
79#821—22no—
80#821—24no—
81#818—22no—
82#817—24no—
83#806—44no—
84C2×#121—31no—
85C2×#121—31no—
86C22×C4⋊C4—31no—
87#481—62no—
88#481—62no—
89#480—61no—
90#480—61no—
91#466—62no—
92#447—61no—
93#447—61no—
94C2×#106—61no—
95C2×#102—61no—
96C2×#99—62no—
97C2×C22⋊C8C2×D311yescharacteristic
98C2×C22⋊C8C2×D311yescharacteristic
99#397—62no—
100#390—61no—
101#386—62no—
102#382—61no—
103#380—31no—
104C4×C22⋊C4C2×D311yescharacteristic
105#340—62no—
106#327—62no—
107C24×C3C2×D411yescharacteristic
108C23×D3—21no—
109C23×D3—21no—
110C22×C3×C4—22no—
111C22×C3×C4—42no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C2×#67—41no—
119C2×#67—41no—
120C2×#67—41no—
121C2×#67—41no—
122C2×#67—42no—
123C2×#67—42no—
124C2×#67—42no—
125C2×#67—42no—
126C22×Dic3—21no—
127C22×Dic3—22no—
128C22×Dic3—21no—
129C22×Dic3—41no—
130C22×Dic3—41no—
131C22×Dic3—41no—
132C22×Dic3—82no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12—21no—
137C2×D12—42no—
138C2×D12—42no—
139C2×C3×C8—22no—
140C2×C3×C8—22no—
141C2×C3×C8—42no—
142C2×C3×C8—42no—
143C3×C22⋊C4—22no—
144C3×C22⋊C4—22no—
145C3×C42—42no—
146#217—42no—
147(C2×D3)⋊C4—41no—
148(C2×D3)⋊C4—41no—
149(C2×D3)⋊C4—41no—
150(C2×D3)⋊C4—41no—
151(C2×D3)⋊C4—84no—
152C4⋊Dic3—21no—
153C4⋊Dic3—22no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—22no—
157C4⋊Dic3—21no—
158C4⋊Dic3—41no—
159C4⋊Dic3—41no—
160C4⋊Dic3—42no—
161D24—82no—
162C22×D4—31no—
163C22×D4—31no—
164C22×D4—61no—
165C22×D4—61no—
166C23×C4—61no—
167C23×C4C22×D311yescharacteristic
168C2×D8—62no—
169C22×C8—21no—
170C22×C8D1212yes—
171#121—61no—
172#121—61no—
173#121—61no—
174#121—61no—
175#121—61no—
176#121—61no—
177#121—61no—
178#121—61no—
179C2×C4⋊C4—32no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—31no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—61no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—61no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—64no—
192C2×C22⋊C4D1212yes—
193C2×C42—21no—
194#107—62no—
195#106—64no—
196#102—64no—
197#102—62no—
198#102—62no—
199#99—124no—
200C22⋊C8—21no—
201C22⋊C8—21no—
202C22⋊C8—21no—
203C22⋊C8D1212yes—
204#95—21no—
205#95—64no—
206#95—61no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3#12011yescharacteristic
210C23×C3#12111yescharacteristic
211C23×C3#12711yescharacteristic
212C23×C3#13512yes—
213C23×C3AGL(1,ℤ/8)12yes—
214C22×D3—42no—
215C22×D3—42no—
216C22×D3—41no—
217C22×D3—41no—
218C22×D3—42no—
219C22×D3—42no—
220C22×D3—81no—
221C22×D3—81no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12111yescharacteristic
235#67—82no—
236#67—82no—
237#67—81no—
238#67—81no—
239#67—81no—
240#67—81no—
241C2×Dic3—44no—
242C2×Dic3—42no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—82no—
248C2×Dic3—82no—
249C2×Dic3—81no—
250C2×Dic3—81no—
251C2×Dic3—81no—
252D12—42no—
253D12—42no—
254D12—42no—
255D12—81no—
256C3×C8—42no—
257C3×C8—42no—
258C24—61no—
259C24—61no—
260C24C2×D1211yescharacteristic
261C2×D4—61no—
262C2×D4—61no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—122no—
266C2×D4—122no—
267C2×D4—121no—
268C2×D4—121no—
269C2×D4—121no—
270C2×D4—121no—
271C2×D4—122no—
272C2×D4—122no—
273C2×D4—122no—
274C2×D4—122no—
275C22×C4—22no—
276C22×C4—42no—
277C22×C4—61no—
278C22×C4—62no—
279C22×C4—61no—
280C22×C4—121no—
281C22×C4—121no—
282C22×C4—121no—
283C22×C4—242no—
284C22×C4C2×D1211yescharacteristic
285C22×C4C2×D1211yescharacteristic
286C22×C4D3×D411yescharacteristic
287C22×C4D3×D411yescharacteristic
288C22×C4D3×D411yescharacteristic
289C22×C4#23911yescharacteristic
290D8—242no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—42no—
294C2×C8—42no—
295C4⋊C4—62no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C4⋊C4—61no—
299C4⋊C4—62no—
300C4⋊C4—61no—
301C4⋊C4—122no—
302C4⋊C4—121no—
303C4⋊C4—121no—
304C22⋊C4—22no—
305C22⋊C4—22no—
306C22⋊C4—121no—
307C22⋊C4—121no—
308C22⋊C4—121no—
309C22⋊C4—121no—
310C22⋊C4—122no—
311C22⋊C4—244no—
312C42—42no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—42no—
320C22×C3—42no—
321C22×C3#39211yescharacteristic
322C22×C3#44912yes—
323C22×C3#46312yes—
324C22×C3#49511yescharacteristic
325C22×C3#49611yescharacteristic
326C2×D3—84no—
327C2×D3—81no—
328C2×D3—81no—
329C2×D3—81no—
330C2×D3—81no—
331C2×D3—81no—
332C2×D3—81no—
333C2×D3—162no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—82no—
338C3×C4#45812yes—
339Dic3—82no—
340Dic3—81no—
341Dic3—81no—
342Dic3—161no—
343C23—41no—
344C23—41no—
345C23—122no—
346C23—122no—
347C23—121no—
348C23—121no—
349C23—122no—
350C23—122no—
351C23—241no—
352C23—241no—
353C23#87411yescharacteristic
354C23#88211yescharacteristic
355C23#89511yescharacteristic
356C23#90412yes—
357C23#90812yes—
358D4—122no—
359D4—122no—
360D4—122no—
361D4—241no—
362D4—241no—
363D4—241no—
364D4—241no—
365D4—241no—
366D4—242no—
367D4—242no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—124no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—242no—
384C2×C4—242no—
385C2×C4—241no—
386C2×C4—241no—
387C2×C4—241no—
388C2×C4#88211yescharacteristic
389C2×C4#88211yescharacteristic
390C2×C4#89511yescharacteristic
391C2×C4#89511yescharacteristic
392C8—42no—
393C8—42no—
394C2×C3—21no—
395C2×C3—22no—
396C2×C3—22no—
397C2×C3—21no—
398C2×C3#200811yescharacteristic
399C2×C3#201211yescharacteristic
400C2×C3#202211yescharacteristic
401D3—161no—
402D3—161no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—42no—
410C22—42no—
411C22—244no—
412C22—241no—
413C22—241no—
414C22—241no—
415C22—241no—
416C22—241no—
417C22—241no—
418C22—482no—
419C22#467111yescharacteristic
420C22#469611yescharacteristic
421C22#471211yescharacteristic
422C22#473412yes—
423C22#484412yes—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—82no—
428C4—241no—
429C4—241no—
430C4—242no—
431C4—481no—
432C4#474212yes—
433C3#995011yescharacteristic, complemented, Sylow
434C2—22no—
435C2—21no—
436C2—22no—
437C2—21no—
438C2—481no—
439C2—481no—
440C2#6867511yescharacteristic
441C2#6896611yescharacteristic
442C2#6902811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1282C2
socleC22×C3#392
prime radical#78611C2
hypercenter#1282D3
hypocenterC3#9950
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1282

Complements

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

NH ≅ G/Ntype
2. C2×#4735C2semidirect
3. #73519C2semidirect
4. #69944C2semidirect
5. #69925C2semidirect
6. #69456C2semidirect
7. C3×#1282C2semidirect
38. #1282D3semidirect
433. C3#9950semidirect

Maximal subgroups

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

subgroupindex
#786112
C2×#47352
#735192
#699442
#699252
#694562
C3×#12822
#99503