Finite Group Databasebeta

← #10705366

Subgroups — #10705366

Order 768.

670 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 107 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#82130C211yesmaximal, characteristic
2C2×C3×#342C211yesmaximal, characteristic, complemented
3C2×#4525C211yesmaximal, characteristic, complemented
4#67491C212yesmaximal, complemented
5#67489C212yesmaximal, complemented
6#11692—31nomaximal, Sylow
7C22×C3×C22⋊C4—21no—
8C22×C3×C22⋊C4C2211yescharacteristic
9C2×#953—21no—
10C2×#930C411yescharacteristic, complemented
11#5213—21no—
12#5212C411yescharacteristic, complemented
13C2×#830—21no—
14#4961—21no—
15#4961—21no—
16#4960C411yescharacteristic, complemented
17#4960C2211yescharacteristic
18#4953C411yescharacteristic, complemented
19#4953C2211yescharacteristic
20C3×#342—21no—
21C3×#342C2212yes—
22#4525—21no—
23#4525C2212yes—
24#2617—31no—
25C2×#342D311yescharacteristic, complemented
26C2×#323—31no—
27#1344—32no—
28#1343—32no—
29C25×C3D411yescharacteristic
30C23×C3×C4—41no—
31C23×C3×C4C2×C411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C22×#67—21no—
34C22×#67—21no—
35C22×#67—21no—
36C22×D12C2×C411yescharacteristic, complemented
37C2×C3×C22⋊C4—24no—
38C2×C3×C22⋊C4—22no—
39C2×C3×C22⋊C4—41no—
40C2×C3×C22⋊C4—41no—
41C2×C3×C22⋊C4—41no—
42C2×C3×C22⋊C4—42no—
43C2×C3×C22⋊C4C2×C411yescharacteristic
44C2×C3×C22⋊C4D411yescharacteristic
45C2×C3×C22⋊C4D411yescharacteristic
46C2×C3×C22⋊C4C2311yescharacteristic
47#953—42no—
48#953—42no—
49C2×#217—21no—
50C2×#217—21no—
51C2×#217—21no—
52#930—41no—
53#930—41no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3C2×C411yescharacteristic, complemented
59C22×C3⋊C8—41no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95C2×C411yescharacteristic, complemented
63C3⋊#95C2×C411yescharacteristic, complemented
64#830—22no—
65#830—41no—
66C2×#121—31no—
67C2×#120—61no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4C2×D311yescharacteristic
70C2×C22⋊C8—61no—
71#397—31no—
72#397—31no—
73#394—61no—
74#394—61no—
75#392—31no—
76#392—31no—
77#386—61no—
78#380—31no—
79#342—21no—
80#342C2×D312yes—
81#323—32no—
82#323—61no—
83C24×C3—21no—
84C24×C3—21no—
85C24×C3—22no—
86C24×C3—81no—
87C24×C3—81no—
88C24×C3C22⋊C411yescharacteristic
89C24×C3C22⋊C411yescharacteristic
90C24×C3D812yes—
91C24×C3SD1612yes—
92C24×C3C2×D411yescharacteristic
93C23×D3—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—44no—
99C22×C3×C4—42no—
100C22×C3×C4—81no—
101C22×C3×C4—81no—
102C22×C3×C4—82no—
103C22×C3×C4—82no—
104C22×C3×C4C22⋊C411yescharacteristic
105C22×C3×C4C22⋊C411yescharacteristic
106C22×C3×C4C22×C411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C2×#67—41no—
109C2×#67—41no—
110C2×#67—42no—
111C2×#67—42no—
112C2×#67—42no—
113C2×#67—41no—
114C2×#67—41no—
115C2×#67—81no—
116C2×#67—81no—
117C2×#67—81no—
118C2×#67—81no—
119C22×Dic3—21no—
120C22×Dic3—21no—
121C22×Dic3—21no—
122C2×D12—41no—
123C2×D12—81no—
124C3×C22⋊C4—24no—
125C3×C22⋊C4—41no—
126C3×C22⋊C4—41no—
127C3×C22⋊C4—41no—
128C3×C22⋊C4—41no—
129C3×C22⋊C4—44no—
130C3×C22⋊C4—42no—
131#217—42no—
132#217—42no—
133#217—41no—
134#217—41no—
135(C2×D3)⋊C4—42no—
136(C2×D3)⋊C4—81no—
137(C2×D3)⋊C4—81no—
138C4⋊Dic3—41no—
139C4⋊Dic3—81no—
140C2×C3⋊C8—42no—
141C2×C3⋊C8—82no—
142C25#6711yescharacteristic
143C22×D4—31no—
144C22×D4—61no—
145C22×D4—61no—
146C22×D4—61no—
147C23×C4—41no—
148C23×C4C4×D311yescharacteristic
149C23×C4D1211yescharacteristic
150C22×C8—121no—
151#121—121no—
152#121—121no—
153#120—122no—
154#120—122no—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—61no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—61no—
164C2×C22⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4—61no—
168C2×C22⋊C4—61no—
169C2×C22⋊C4C4×D311yescharacteristic
170C2×C22⋊C4D1211yescharacteristic
171C2×C22⋊C4#6711yescharacteristic
172C2×C22⋊C4C22×D311yescharacteristic
173C22⋊C8—62no—
174C22⋊C8—121no—
175#95—31no—
176#95—31no—
177#95—61no—
178#95—61no—
179C23×C3—22no—
180C23×C3—22no—
181C23×C3—21no—
182C23×C3—21no—
183C23×C3—22no—
184C23×C3—22no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—41no—
188C23×C3—41no—
189C23×C3—42no—
190C23×C3—42no—
191C23×C3—41no—
192C23×C3—41no—
193C23×C3—41no—
194C23×C3—41no—
195C23×C3—81no—
196C23×C3—81no—
197C23×C3—82no—
198C23×C3—82no—
199C23×C3—84no—
200C23×C3#9911yescharacteristic
201C23×C3#9911yescharacteristic
202C23×C3#10212yes—
203C23×C3#10212yes—
204C23×C3#10412yes—
205C23×C3C2×C22⋊C411yescharacteristic
206C23×C3C2×D811yescharacteristic
207C23×C3C2×SD1611yescharacteristic
208C22×D3—41no—
209C22×D3—41no—
210C22×D3—41no—
211C22×D3—41no—
212C22×D3—41no—
213C22×D3—41no—
214C22×D3—41no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—42no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—44no—
225C2×C3×C4—81no—
226C2×C3×C4—81no—
227C2×C3×C4—81no—
228C2×C3×C4—88no—
229C2×C3×C4—82no—
230C2×C3×C4—82no—
231#67—84no—
232#67—82no—
233#67—82no—
234#67—82no—
235#67—82no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245D12—82no—
246C3⋊C8—82no—
247C24—21no—
248C24—22no—
249C24—21no—
250C24—61no—
251C24—81no—
252C24—81no—
253C24(C2×D3)⋊C411yescharacteristic
254C24(C2×D3)⋊C411yescharacteristic
255C24#21312yes—
256C24#21512yes—
257C24C2×#6711yescharacteristic
258C2×D4—121no—
259C2×D4—122no—
260C2×D4—121no—
261C2×D4—121no—
262C2×D4—122no—
263C2×D4—122no—
264C2×D4—121no—
265C2×D4—121no—
266C2×D4—241no—
267C2×D4—241no—
268C2×D4—241no—
269C2×D4—241no—
270C2×D4—241no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—81no—
281C22×C4—81no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4(C2×D3)⋊C411yescharacteristic
285C22×C4(C2×D3)⋊C411yescharacteristic
286C22×C4C2×C4×D311yescharacteristic
287C22×C4C2×D1211yescharacteristic
288C2×C8—122no—
289C2×C8—242no—
290C4⋊C4—121no—
291C4⋊C4—241no—
292C22⋊C4—24no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—44no—
298C22⋊C4—42no—
299C22⋊C4—122no—
300C22⋊C4—122no—
301C22⋊C4—122no—
302C22⋊C4—121no—
303C22⋊C4—121no—
304C22⋊C4—241no—
305C22⋊C4—241no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—22no—
314C22×C3—21no—
315C22×C3—42no—
316C22×C3—42no—
317C22×C3—42no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—44no—
321C22×C3—44no—
322C22×C3—41no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—81no—
327C22×C3—81no—
328C22×C3—82no—
329C22×C3—82no—
330C22×C3#32712yes—
331C22×C3#32712yes—
332C22×C3C2×#9911yescharacteristic
333C22×C3C2×#10211yescharacteristic
334C22×C3C2×#10411yescharacteristic
335C2×D3—81no—
336C2×D3—81no—
337C2×D3—81no—
338C2×D3—81no—
339C2×D3—81no—
340C2×D3—81no—
341C2×D3—81no—
342C3×C4—41no—
343C3×C4—41no—
344C3×C4—41no—
345C3×C4—41no—
346C3×C4—84no—
347Dic3—81no—
348Dic3—81no—
349Dic3—81no—
350C23—21no—
351C23—21no—
352C23—22no—
353C23—22no—
354C23—21no—
355C23—21no—
356C23—22no—
357C23—22no—
358C23—41no—
359C23—41no—
360C23—42no—
361C23—42no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—81no—
367C23—81no—
368C23—82no—
369C23—82no—
370C23—84no—
371C23—121no—
372C23—121no—
373C23—121no—
374C23—121no—
375C23—121no—
376C23—121no—
377C23—121no—
378C23#80512yes—
379C23#80611yescharacteristic
380C23#80611yescharacteristic
381C23#80912yes—
382C23#80912yes—
383C23C2×(C2×D3)⋊C411yescharacteristic
384C23C2×#21311yescharacteristic
385C23C2×#21511yescharacteristic
386D4—242no—
387D4—244no—
388D4—242no—
389D4—242no—
390D4—242no—
391D4—242no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—44no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—88no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—121no—
409C2×C4—121no—
410C2×C4—121no—
411C2×C4—121no—
412C2×C4—121no—
413C2×C4—121no—
414C2×C4—121no—
415C2×C4—121no—
416C2×C4—121no—
417C8—242no—
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—22no—
421C2×C3—22no—
422C2×C3—41no—
423C2×C3—42no—
424C2×C3—41no—
425C2×C3C2×#32711yescharacteristic
426C2×C3#212111yescharacteristic
427C2×C3#212711yescharacteristic
428D3—161no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—22no—
437C22—21no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—41no—
442C22—41no—
443C22—44no—
444C22—44no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—81no—
450C22—81no—
451C22—82no—
452C22—82no—
453C22—241no—
454C22—241no—
455C22—241no—
456C22—241no—
457C22—241no—
458C22—241no—
459C22—241no—
460C22#445112yes—
461C22#445112yes—
462C22C2×#80511yescharacteristic
463C22C2×#80611yescharacteristic
464C22C2×#80911yescharacteristic
465C4—41no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—84no—
470C4—241no—
471C4—241no—
472C4—241no—
473C3#1169211yescharacteristic, complemented, Sylow
474C2—21no—
475C2—22no—
476C2—21no—
477C2—22no—
478C2—41no—
479C2—42no—
480C2—41no—
481C2—481no—
482C2C2×#445111yescharacteristic
483C2#7001511yescharacteristic
484C2#7001911yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC2×C3×#342C2
socleC22×C3C2×#99
prime radical#82130C2
cosocleC2×C3×C22⋊C4C23
hypercenterC2×#342D3
hypocenterC3#11692
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#342

Complements

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

NH ≅ G/Ntype
2. C2×C3×#342C2semidirect
3. C2×#4525C2semidirect
4. #67491C2semidirect
5. #67489C2semidirect
10. C2×#930C4semidirect
12. #5212C4semidirect
16. #4960C4semidirect
18. #4953C4semidirect
25. C2×#342D3semidirect
36. C22×D12C2×C4semidirect
58. C2×C4⋊Dic3C2×C4semidirect
62. C3⋊#95C2×C4semidirect
63. C3⋊#95C2×C4semidirect
473. C3#11692semidirect

Maximal subgroups

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

subgroupindex
#821302
C2×C3×#3422
C2×#45252
#674912
#674892
#116923