Finite Group Databasebeta

← #10695435

Subgroups — #10695435

Order 768.

672 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 115 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#74032C211yesmaximal, characteristic
3C2×#4659C211yesmaximal, characteristic, complemented
4C2×#4469C211yesmaximal, characteristic, complemented
5#69502C211yesmaximal, characteristic, complemented
6C2×#4451C211yesmaximal, characteristic
7C3×#1294C211yesmaximal, characteristic, complemented
8#9921—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C22×C4⋊Dic3C2211yescharacteristic
11C22×Dic3⋊C4C2211yescharacteristic
12C2×#886—21no—
13C2×C3×C22⋊C8C2211yescharacteristic
14C3×#380C2211yescharacteristic
15C2×C3×#95C2211yescharacteristic
16C2×C3⋊#95—21no—
17C2×C3⋊#95—21no—
18C2×#830C2211yescharacteristic
19C2×#821—21no—
20C2×#817—21no—
21#4994—22no—
22#4958—21no—
23#4735—22no—
24#4735—22no—
25#4672—21no—
26#4661—21no—
27#4659—22no—
28#4659—22no—
29#4529—22no—
30#4469—22no—
31#4451—22no—
32C2×#481—31no—
33#2618—31no—
34C2×#400—31no—
35#1824—31no—
36C2×#327—31no—
37C2×#324—31no—
38#1294D311yescharacteristic, complemented
39C23×C3×C4—21no—
40C23×C3×C4C2311yescharacteristic
41C22×#67—21no—
42C23×Dic3—21no—
43C23×Dic3D411yescharacteristic
44C23×Dic3D411yescharacteristic
45C22×D12D411yescharacteristic
46C22×C3×C8—21no—
47C2×C3×C4⋊C4D412yes—
48C2×C3×C22⋊C4—21no—
49#930—21no—
50#930—21no—
51#930—21no—
52#930—21no—
53C2×(C2×D3)⋊C4—22no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—22no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—22no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3D412yes—
63C2×C4⋊Dic3D411yescharacteristic
64C2×Dic3⋊C4—21no—
65C2×Dic3⋊C4—21no—
66C2×Dic3⋊C4—21no—
67C2×Dic3⋊C4—21no—
68C2×Dic3⋊C4—21no—
69C2×Dic3⋊C4—21no—
70C22×C3⋊C8—21no—
71#886—42no—
72#886—42no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8D412yes—
75C3×#95—21no—
76C3×#95—21no—
77C3×#95D412yes—
78C3⋊#95—24no—
79C3⋊#95—24no—
80C3⋊#95—22no—
81#830—21no—
82#830—21no—
83#821—24no—
84#817—24no—
85C2×#123—61no—
86C2×#121—31no—
87C22×C4⋊C4—31no—
88C22×C4⋊C4—31no—
89#481—62no—
90#481—62no—
91C2×#106—61no—
92C2×#102—61no—
93C2×C22⋊C8—31no—
94C2×C22⋊C8C2×D311yescharacteristic
95#400—62no—
96#400—62no—
97#398—61no—
98#396—62no—
99#394—61no—
100#386—61no—
101#380C2×D311yescharacteristic
102C2×#95—61no—
103C2×#95—61no—
104C2×#95C2×D311yescharacteristic
105#340—62no—
106#327—62no—
107#324—62no—
108C24×C3C2×D411yescharacteristic
109C23×D3—21no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—42no—
113C22×C3×C4—42no—
114C22×C3×C4—42no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4#4111yescharacteristic
123C2×#67—41no—
124C2×#67—41no—
125C2×#67—41no—
126C2×#67—41no—
127C2×#67—41no—
128C2×#67—41no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—22no—
137C22×Dic3—21no—
138C22×Dic3—21no—
139C22×Dic3—21no—
140C22×Dic3—42no—
141C22×Dic3—44no—
142C22×Dic3—41no—
143C22×Dic3—41no—
144C22×Dic3SD1612yes—
145C2×D12—21no—
146C2×D12—21no—
147C2×D12—42no—
148C2×C3×C8—22no—
149C2×C3×C8—42no—
150C3×C4⋊C4—42no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153(C2×D3)⋊C4—44no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156(C2×D3)⋊C4—82no—
157C4⋊Dic3—21no—
158C4⋊Dic3—22no—
159C4⋊Dic3—22no—
160C4⋊Dic3—21no—
161C4⋊Dic3—44no—
162C4⋊Dic3—41no—
163C4⋊Dic3—41no—
164Dic3⋊C4—41no—
165Dic3⋊C4—41no—
166Dic3⋊C4—41no—
167Dic3⋊C4—41no—
168C2×C3⋊C8—42no—
169C2×C3⋊C8—41no—
170C22×D4—31no—
171C22×D4—61no—
172C23×C4—21no—
173C23×C4—31no—
174C23×C4—31no—
175C23×C4—61no—
176C23×C4C22×D311yescharacteristic
177C22×C8—21no—
178C22×C8—61no—
179#123—122no—
180#123—122no—
181#121—61no—
182#121—61no—
183#121—61no—
184#121—61no—
185C2×C4⋊C4—32no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—62no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4—61no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4D1212yes—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—62no—
202C2×C22⋊C4—61no—
203C2×C22⋊C4—62no—
204#106—64no—
205#102—64no—
206C22⋊C8—21no—
207C22⋊C8—61no—
208C22⋊C8—61no—
209C22⋊C8D1212yes—
210#95—21no—
211#95—21no—
212#95—64no—
213#95—64no—
214#95—62no—
215#95D1212yes—
216C23×C3—22no—
217C23×C3—22no—
218C23×C3#12011yescharacteristic
219C23×C3#12011yescharacteristic
220C23×C3#12111yescharacteristic
221C23×C3C2×SD1611yescharacteristic
222C23×C3#13511yescharacteristic
223C23×C3AGL(1,ℤ/8)11yescharacteristic
224C23×C3#13711yescharacteristic
225C22×D3—41no—
226C22×D3—41no—
227C22×D3—41no—
228C22×D3—42no—
229C22×D3—42no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—82no—
243C2×C3×C4—82no—
244C2×C3×C4#12011yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12711yescharacteristic
248#67—81no—
249#67—81no—
250#67—81no—
251#67—81no—
252C2×Dic3—22no—
253C2×Dic3—22no—
254C2×Dic3—22no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—44no—
260C2×Dic3—42no—
261C2×Dic3—41no—
262C2×Dic3—42no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—42no—
267C2×Dic3—82no—
268C2×Dic3—82no—
269C2×Dic3—81no—
270D12—42no—
271D12—81no—
272C3×C8—42no—
273C3⋊C8—81no—
274C24—61no—
275C24D3×D411yescharacteristic
276C2×D4—61no—
277C2×D4—61no—
278C2×D4—122no—
279C2×D4—121no—
280C2×D4—121no—
281C2×D4—121no—
282C2×D4—121no—
283C2×D4—121no—
284C2×D4—121no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—32no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—61no—
294C22×C4—61no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4—62no—
300C22×C4—61no—
301C22×C4—61no—
302C22×C4—61no—
303C22×C4—61no—
304C22×C4—122no—
305C22×C4—124no—
306C22×C4—121no—
307C22×C4—121no—
308C22×C4C2×D1211yescharacteristic
309C22×C4C2×D1211yescharacteristic
310C22×C4C2×D1211yescharacteristic
311C22×C4D3×D411yescharacteristic
312C22×C4D3×D411yescharacteristic
313C22×C4#23911yescharacteristic
314C2×C8—22no—
315C2×C8—42no—
316C2×C8—122no—
317C2×C8—121no—
318C4⋊C4—42no—
319C4⋊C4—62no—
320C4⋊C4—61no—
321C4⋊C4—62no—
322C4⋊C4—61no—
323C4⋊C4—124no—
324C4⋊C4—121no—
325C4⋊C4—121no—
326C4⋊C4—121no—
327C4⋊C4—121no—
328C4⋊C4—121no—
329C4⋊C4—121no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—124no—
333C22⋊C4—121no—
334C22⋊C4—121no—
335C22⋊C4—242no—
336C22×C3—22no—
337C22×C3—22no—
338C22×C3—22no—
339C22×C3—22no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3#39211yescharacteristic
345C22×C3#44811yescharacteristic
346C22×C3#44911yescharacteristic
347C22×C3#45011yescharacteristic
348C22×C3#45211yescharacteristic
349C22×C3#45412yes—
350C22×C3#45812yes—
351C22×C3#46511yescharacteristic
352C22×C3#46811yescharacteristic
353C2×D3—82no—
354C2×D3—82no—
355C2×D3—81no—
356C2×D3—81no—
357C2×D3—81no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—82no—
363C3×C4—82no—
364Dic3—42no—
365Dic3—81no—
366Dic3—82no—
367Dic3—81no—
368C23—22no—
369C23—22no—
370C23—121no—
371C23—122no—
372C23—121no—
373C23—121no—
374C23—122no—
375C23#88211yescharacteristic
376C23#88211yescharacteristic
377C23#89511yescharacteristic
378C23#91111yescharacteristic
379C23D3×SD1611yescharacteristic
380C23#91611yescharacteristic
381C23#91811yescharacteristic
382D4—122no—
383D4—241no—
384D4—241no—
385D4—241no—
386D4—241no—
387D4—241no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—62no—
401C2×C4—62no—
402C2×C4—62no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—121no—
406C2×C4—121no—
407C2×C4—121no—
408C2×C4—121no—
409C2×C4—122no—
410C2×C4—124no—
411C2×C4—121no—
412C2×C4—122no—
413C2×C4—121no—
414C2×C4—122no—
415C2×C4—121no—
416C2×C4—121no—
417C2×C4—242no—
418C2×C4—242no—
419C2×C4—241no—
420C2×C4#87411yescharacteristic
421C2×C4#88211yescharacteristic
422C2×C4#89511yescharacteristic
423C2×C4#89511yescharacteristic
424C8—42no—
425C8—241no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3#161512yes—
431C2×C3#161612yes—
432C2×C3#200011yescharacteristic
433C2×C3#200311yescharacteristic
434C2×C3#202211yescharacteristic
435D3—161no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—241no—
445C22—241no—
446C22—241no—
447C22—242no—
448C22—242no—
449C22#467111yescharacteristic
450C22#474212yes—
451C22#477611yescharacteristic
452C22#478111yescharacteristic
453C22#480611yescharacteristic
454C22#480811yescharacteristic
455C22#482712yes—
456C22#486411yescharacteristic
457C22#488611yescharacteristic
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—82no—
463C4—82no—
464C4—122no—
465C4—241no—
466C4—241no—
467C4—242no—
468C3#992111yescharacteristic, complemented, Sylow
469C2—21no—
470C2—21no—
471C2—21no—
472C2—21no—
473C2—481no—
474C2#6829912yes—
475C2#6833012yes—
476C2#6876711yescharacteristic
477C2#6888711yescharacteristic
478C2#6896611yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC23×C3#120
prime radicalC2×#930C22
hypercenter#1294D3
hypocenterC3#9921
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
3. C2×#4659C2semidirect
4. C2×#4469C2semidirect
5. #69502C2semidirect
7. C3×#1294C2semidirect
38. #1294D3semidirect
468. C3#9921semidirect

Maximal subgroups

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

subgroupindex
C2×#47352
#740322
C2×#46592
C2×#44692
#695022
C2×#44512
C3×#12942
#99213