Finite Group Databasebeta

← #10695985

Subgroups — #10695985

Order 768.

572 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C3×#3213C211yesmaximal, characteristic
2#83316C211yesmaximal, characteristic, complemented
3#82976C211yesmaximal, characteristic, complemented
4#71818C211yesmaximal, characteristic, complemented
5#71718C211yesmaximal, characteristic, complemented
6#70802C211yesmaximal, characteristic, complemented
7#67750C211yesmaximal, characteristic, complemented
8#10006—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#496—21no—
13C3×#489—21no—
14C3×#463—22no—
15C3×#448—21no—
16C3×#447—21no—
17C3×#432C2211yescharacteristic
18C2×#837—22no—
19#5229—21no—
20#5221—21no—
21#5091—21no—
22#5075—21no—
23#5040—21no—
24#5033—21no—
25#5018—22no—
26#4953—21no—
27#4947—21no—
28#4945—22no—
29C4×C4⋊Dic3C2211yescharacteristic
30#4926C2211yescharacteristic
31C4×#208C2211yescharacteristic
32#4540—22no—
33#4484—22no—
34#3213D311yescharacteristic
35#3112—31no—
36#2592—31no—
37#1980—31no—
38#1527—31no—
39#1491—31no—
40#1285—31no—
41C2×C3×#41—21no—
42C22×C3×Q8D411yescharacteristic
43C22×C3×D4D411yescharacteristic
44C22×Dic6D411yescharacteristic
45C2×C3×SD16—22no—
46C2×C3×D8—22no—
47C3×C4⋊Q8D411yescharacteristic
48C3×#127D411yescharacteristic
49C3×#124—21no—
50C3×#124—21no—
51C3×#124—21no—
52C3×#124—21no—
53C3×#124—21no—
54C3×#121—21no—
55C3×#121—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C42C2311yescharacteristic
61C2×#217—22no—
62C2×#217—22no—
63C2×#214—22no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3D411yescharacteristic
66C2×Dic3⋊C4—21no—
67C2×C4×Dic3—21no—
68C2×C4×Dic3D412yes—
69C2×#208—21no—
70C4×Dic6—21no—
71C4×Dic6—21no—
72C4×Dic6—21no—
73C4×Dic6—21no—
74C3×C4⋊C8D412yes—
75C3×#103—22no—
76C3×#102—22no—
77C3×C22⋊C8—21no—
78C3×#97—21no—
79#837—44no—
80#832—22no—
81C3⋊#95—22no—
82C3⋊#95—22no—
83C3⋊#95—21no—
84#830—21no—
85#810—22no—
86#807—22no—
87C4⋊(C3⋊C8)—21no—
88#803—21no—
89#803D412yes—
90C4×C3⋊C8—21no—
91#532C2×D311yescharacteristic
92C2×#124C2×D311yescharacteristic
93C2×C4×Q8—31no—
94#496—21no—
95#489—21no—
96#480—61no—
97#477—61no—
98#463—22no—
99#460—61no—
100#448—21no—
101#448—61no—
102#447—21no—
103#440—62no—
104#432—31no—
105#432C2×D311yescharacteristic
106C2×#104—62no—
107C4×M16—31no—
108#397—61no—
109#393—61no—
110#391—61no—
111#390—61no—
112#388—62no—
113C4×C4⋊C4—31no—
114#329—62no—
115Q8⋊C8—62no—
116C24×C3—21no—
117C3×#41—42no—
118C3×#41—42no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—42no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—42no—
129C2×C3×D4—41no—
130C22×C3×C4—21no—
131C22×C3×C4—41no—
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×Dic3—21no—
136C22×Dic3—22no—
137C22×Dic3—21no—
138C2×Dic6—21no—
139C2×Dic6—21no—
140C2×Dic6—42no—
141C3×SD16—82no—
142C3×D8—82no—
143C2×C3×C8—22no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42#4111yescharacteristic
156#217—44no—
157#217—82no—
158#214—44no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—42no—
162Dic3⋊C4—41no—
163Dic3⋊C4—41no—
164C4×Dic3—41no—
165C4×Dic3—41no—
166C4×Dic3—42no—
167#208—22no—
168#208—41no—
169C2×C3⋊C8—22no—
170C2×C3⋊C8—41no—
171C2×#41—21no—
172C22×Q8—31no—
173C22×Q8#6711yescharacteristic
174C22×D4#6711yescharacteristic
175C2×SD16—22no—
176C2×SD16—62no—
177C2×D8—22no—
178C2×M16—61no—
179C4⋊Q8#6711yescharacteristic
180#127#6711yescharacteristic
181#124—21no—
182#124—21no—
183#124—21no—
184#124—21no—
185#124—21no—
186#121—21no—
187#121—21no—
188C4×Q8—61no—
189C4×Q8—61no—
190C4×Q8—61no—
191C4×Q8—61no—
192C4×D4—21no—
193C4×D4—21no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—61no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—62no—
200C2×C22⋊C4—62no—
201C2×C42—32no—
202C2×C42—61no—
203C2×C42C22×D311yescharacteristic
204#107—62no—
205C4⋊C8—61no—
206C4⋊C8#6712yes—
207#104—124no—
208#103—22no—
209#103—62no—
210#102—22no—
211#102—62no—
212C22⋊C8—21no—
213C22⋊C8—61no—
214#97—21no—
215#97—32no—
216#97—61no—
217C4×C8—61no—
218#95—62no—
219#95—62no—
220#95—61no—
221C23×C3—42no—
222C23×C3—42no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3#12011yescharacteristic
228C3×Q8—42no—
229C3×Q8—42no—
230C3×Q8—81no—
231C3×D4—42no—
232C3×D4—42no—
233C3×D4—42no—
234C3×D4—42no—
235C3×D4—81no—
236C3×D4—81no—
237C3×D4—81no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—22no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—82no—
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12011yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12711yescharacteristic
252C2×Dic3—22no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—41no—
258C2×Dic3—42no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261Dic6—22no—
262Dic6—22no—
263Dic6—41no—
264Dic6—41no—
265C3×C8—42no—
266C3⋊C8—42no—
267C3⋊C8—42no—
268C24—21no—
269#41—42no—
270#41—42no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—42no—
275C2×Q8—61no—
276C2×Q8—61no—
277C2×Q8—122no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—42no—
284C2×D4—41no—
285C22×C4—21no—
286C22×C4—41no—
287C22×C4—61no—
288C22×C4—62no—
289C22×C4—61no—
290C22×C4D3×D411yescharacteristic
291C22×C4D3×D411yescharacteristic
292C22×C4C2×#6711yescharacteristic
293SD16—82no—
294SD16—124no—
295D8—82no—
296M16—62no—
297M16—121no—
298C2×C8—22no—
299C2×C8—62no—
300C2×C8—121no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—122no—
306C4⋊C4—121no—
307C4⋊C4—121no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—124no—
315C22⋊C4—242no—
316C42—122no—
317C42—121no—
318C42—121no—
319C42D3×D411yescharacteristic
320C42#23711yescharacteristic
321C42C2×#6711yescharacteristic
322C42C2×#6711yescharacteristic
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—82no—
326C22×C3—82no—
327C22×C3—81no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3—81no—
331C22×C3—81no—
332C22×C3—81no—
333C22×C3#39211yescharacteristic
334C22×C3#45311yescharacteristic
335C22×C3#45411yescharacteristic
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—81no—
342C3×C4—81no—
343C3×C4#45512yes—
344Dic3—42no—
345Dic3—82no—
346Dic3—81no—
347C23—42no—
348C23—42no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23#93711yescharacteristic
354Q8—42no—
355Q8—42no—
356Q8—62no—
357Q8—62no—
358Q8—81no—
359Q8—121no—
360Q8—121no—
361D4—42no—
362D4—42no—
363D4—42no—
364D4—42no—
365D4—81no—
366D4—81no—
367D4—81no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—62no—
376C2×C4—82no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4#93711yescharacteristic
386C2×C4#93811yescharacteristic
387C2×C4#93911yescharacteristic
388C2×C4#93911yescharacteristic
389C2×C4#94011yescharacteristic
390C2×C4#95311yescharacteristic
391C8—42no—
392C8—122no—
393C8—122no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—161no—
397C2×C3—161no—
398C2×C3#200411yescharacteristic
399C2×C3#200611yescharacteristic
400C2×C3#201011yescharacteristic
401C22—21no—
402C22—21no—
403C22—82no—
404C22—82no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22#519811yescharacteristic
412C22#520111yescharacteristic
413C22#522311yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—81no—
420C4—81no—
421C4—122no—
422C4—242no—
423C4—241no—
424C4#524212yes—
425C3#1000611yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—161no—
429C2—161no—
430C2#7185711yescharacteristic
431C2#7209811yescharacteristic
432C2#7211811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3213C2
socleC22×C3#392
hypercenter#3213D3
hypocenterC3#10006
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3213

Complements

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

NH ≅ G/Ntype
2. #83316C2semidirect
3. #82976C2semidirect
4. #71818C2semidirect
5. #71718C2semidirect
6. #70802C2semidirect
7. #67750C2semidirect
425. C3#10006semidirect

Maximal subgroups

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

subgroupindex
C3×#32132
#833162
#829762
#718182
#717182
#708022
#677502
#100063