Finite Group Databasebeta

← #10695953

Subgroups — #10695953

Order 768.

568 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 97 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#82375C211yesmaximal, characteristic
2#82314C211yesmaximal, characteristic
3#79068C211yesmaximal, characteristic, complemented
4C3×#1486C211yesmaximal, characteristic
5#70793C211yesmaximal, characteristic, complemented
6#70766C211yesmaximal, characteristic, complemented
7#67713C211yesmaximal, characteristic, complemented
8#10001—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#924—21no—
11#5470C2211yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15#5222—21no—
16#5218—21no—
17#5212—21no—
18#5121—21no—
19#5120—21no—
20#5019—21no—
21C4×#214—22no—
22#5008—21no—
23#4983—21no—
24#4971—21no—
25#4944—22no—
26#4936C2211yescharacteristic
27#4925C2211yescharacteristic
28#4851—21no—
29#4850—21no—
30#4829—22no—
31C3×D4⋊C8—22no—
32#4484—22no—
33#4467—22no—
34#3101—31no—
35#3078—31no—
36#2536—31no—
37#1518—31no—
38#1486D311yescharacteristic
39#1479—31no—
40#1286—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C22×Dic6D411yescharacteristic
44C2×C3×M16—21no—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C22⋊C4—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#217—22no—
53C2×#217—22no—
54C2×#214—22no—
55#926—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3D411yescharacteristic
58#924—21no—
59#924—21no—
60#924—42no—
61#924—42no—
62C2×Dic3⋊C4—22no—
63C2×Dic3⋊C4—21no—
64C2×C4×Dic3—21no—
65#870—21no—
66#869D411yescharacteristic
67#869D411yescharacteristic
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C4×Dic6—21no—
72C4×Dic6—21no—
73C4×Dic6—21no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8D412yes—
77C3×C22⋊C8—21no—
78C3×C4×C8—21no—
79#832—22no—
80C3⋊#95—22no—
81C3⋊#95—21no—
82C3⋊#95D412yes—
83#830—21no—
84#818—22no—
85#817—22no—
86#816—22no—
87#816—22no—
88#810—22no—
89#808—22no—
90C4⋊(C3⋊C8)—21no—
91C4×C3⋊C8D412yes—
92#533—31no—
93C2×C22⋊Q8—61no—
94C2×C4×Q8—31no—
95C2×C4×D4C2×D311yescharacteristic
96#483—61no—
97#479—62no—
98#477—61no—
99#476—61no—
100#475—61no—
101#461—61no—
102#450—61no—
103C4×SD16—62no—
104#431—31no—
105#431C2×D311yescharacteristic
106#428—61no—
107#423C2×D311yescharacteristic
108#419—61no—
109#398—61no—
110#394—61no—
111#388—62no—
112#382—31no—
113#380—61no—
114#328—62no—
115Q8⋊C8—62no—
116D4⋊C8—22no—
117C24×C3—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—42no—
121C22×C3×C4—21no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×Dic3—21no—
132C22×Dic3—22no—
133C22×Dic3—21no—
134C2×Dic6—21no—
135C2×Dic6—21no—
136C2×Dic6—21no—
137C2×Dic6—21no—
138C2×Dic6—21no—
139C2×Dic6—21no—
140C2×Dic6—21no—
141C2×Dic6SD1612yescomplemented
142C3×M16—41no—
143C3×M16—41no—
144C2×C3×C8—22no—
145C2×C3×C8—41no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42#4111yescharacteristic
154#217—44no—
155#217—44no—
156#217—41no—
157#214—44no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—41no—
165C4⋊Dic3—41no—
166Dic3⋊C4—44no—
167Dic3⋊C4—41no—
168Dic3⋊C4—41no—
169Dic3⋊C4—41no—
170Dic3⋊C4—41no—
171C4×Dic3—41no—
172C4×Dic3—41no—
173C4×Dic3—41no—
174C2×C3⋊C8—22no—
175C22×Q8—31no—
176C22×D4D1211yescharacteristic
177C23×C4—21no—
178C2×SD16—62no—
179C2×M16—21no—
180C4⋊Q8—31no—
181C4⋊Q8—31no—
182#125—61no—
183C22⋊Q8—61no—
184C22⋊Q8—61no—
185C22⋊Q8—122no—
186C22⋊Q8—122no—
187C4×Q8—61no—
188C4×Q8—61no—
189C4×Q8—61no—
190C4×Q8—61no—
191C4×Q8—61no—
192C4×Q8—61no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—21no—
197#117—61no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—62no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4D1211yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—62no—
205C2×C22⋊C4—62no—
206C2×C42—61no—
207C2×C42C22×D311yescharacteristic
208#107—62no—
209#106—62no—
210#106—62no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—61no—
214C4⋊C8#6712yes—
215#103—62no—
216#103—62no—
217#103—62no—
218#102—62no—
219C22⋊C8—21no—
220C22⋊C8—61no—
221C4×C8—21no—
222C4×C8—32no—
223#95—32no—
224#95—62no—
225#95—61no—
226C23×C3—22no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3#12011yescharacteristic
234C3×D4—22no—
235C3×D4—22no—
236C3×D4—41no—
237C3×D4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—81no—
244C2×C3×C4—81no—
245C2×C3×C4—81no—
246C2×C3×C4—81no—
247C2×C3×C4#12011yescharacteristic
248C2×C3×C4#12011yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12711yescharacteristic
253C2×C3×C4C2×SD1611yescharacteristic
254C2×C3×C4#13511yescharacteristic
255C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
256C2×C3×C4#13711yescharacteristic
257C2×Dic3—22no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—42no—
267Dic6—22no—
268Dic6—22no—
269Dic6—41no—
270Dic6—41no—
271Dic6—42no—
272Dic6—81no—
273C3×C8—42no—
274C3×C8—81no—
275C3⋊C8—42no—
276C24—21no—
277C2×Q8—32no—
278C2×Q8—61no—
279C2×Q8—61no—
280C2×Q8—61no—
281C2×Q8—61no—
282C2×Q8—61no—
283C2×Q8—61no—
284C2×Q8—61no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—42no—
288C22×C4—21no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—61no—
296C22×C4—62no—
297C22×C4—61no—
298C22×C4C2×D1211yescharacteristic
299C22×C4D3×D411yescharacteristic
300C22×C4D3×D411yescharacteristic
301SD16—124no—
302M16—41no—
303M16—41no—
304C2×C8—22no—
305C2×C8—41no—
306C2×C8—62no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—61no—
312C4⋊C4—61no—
313C4⋊C4—61no—
314C4⋊C4—61no—
315C4⋊C4—124no—
316C4⋊C4—121no—
317C4⋊C4—121no—
318C4⋊C4—121no—
319C4⋊C4—121no—
320C4⋊C4—121no—
321C4⋊C4—121no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—124no—
325C22⋊C4—124no—
326C22⋊C4—121no—
327C42—121no—
328C42—121no—
329C42—121no—
330C42#23511yescharacteristic
331C42D3×D411yescharacteristic
332C42D3×D411yescharacteristic
333C42C2×#6711yescharacteristic
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—42no—
337C22×C3—42no—
338C22×C3—42no—
339C22×C3—81no—
340C22×C3—81no—
341C22×C3—81no—
342C22×C3—81no—
343C22×C3#39211yescharacteristic
344C22×C3#45011yescharacteristic
345C22×C3#45211yescharacteristic
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—81no—
351C3×C4#44811yescharacteristic
352C3×C4#44911yescharacteristic
353C3×C4#46511yescharacteristic
354C3×C4#46811yescharacteristic
355Dic3—42no—
356Dic3—82no—
357Dic3—81no—
358Dic3—81no—
359Dic3—81no—
360C23—22no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23#88211yescharacteristic
368Q8—62no—
369Q8—62no—
370Q8—122no—
371Q8—121no—
372Q8—121no—
373Q8—241no—
374D4—22no—
375D4—22no—
376D4—41no—
377D4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—62no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—122no—
391C2×C4—122no—
392C2×C4—122no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—122no—
397C2×C4#88211yescharacteristic
398C2×C4#88411yescharacteristic
399C2×C4#88411yescharacteristic
400C2×C4#91111yescharacteristic
401C2×C4#91211yescharacteristic
402C2×C4D3×SD1611yescharacteristic
403C2×C4#91511yescharacteristic
404C2×C4#93011yescharacteristic
405C2×C4#93711yescharacteristic
406C2×C4#94011yescharacteristic
407C8—42no—
408C8—81no—
409C8—122no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—82no—
413C2×C3#200011yescharacteristic
414C2×C3#200911yescharacteristic
415C2×C3#201011yescharacteristic
416C22—21no—
417C22—21no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22#477611yescharacteristic
426C22#478311yescharacteristic
427C22#496011yescharacteristic
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—81no—
433C4—122no—
434C4—242no—
435C4—241no—
436C4—241no—
437C4—241no—
438C4#477811yescharacteristic
439C4#478011yescharacteristic
440C4#515811yescharacteristic
441C4#517011yescharacteristic
442C3#1000111yescharacteristic, complemented, Sylow
443C2—21no—
444C2—21no—
445C2—82no—
446C2#6999311yescharacteristic
447C2#7020511yescharacteristic
448C2#7041211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1486C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#1486D3
hypocenterC3#10001
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
3. #79068C2semidirect
5. #70793C2semidirect
6. #70766C2semidirect
7. #67713C2semidirect
141. C2×Dic6SD16semidirect
442. C3#10001semidirect

Maximal subgroups

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

subgroupindex
#823752
#823142
#790682
C3×#14862
#707932
#707662
#677132
#100013