Finite Group Databasebeta

← #10691752

Subgroups — #10691752

Order 768.

522 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 109 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#78617C211yesmaximal, characteristic
2#73473C211yesmaximal, characteristic
3#69946C211yesmaximal, characteristic
4C2×#4467C211yesmaximal, characteristic
5#69457C211yesmaximal, characteristic
6C3×#1282C211yesmaximal, characteristic
7#67675C211yesmaximal, characteristic
8#9276—31nomaximal, Sylow
9C2×#924C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C2×C3×C22⋊C8C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C3×C4×C22⋊C4C2211yescharacteristic
14#5212C2211yescharacteristic
15C2×C3⋊#95C2211yescharacteristic
16#5116—22no—
17#4946—21no—
18Dic3×C22⋊C4—21no—
19#4933—21no—
20C4×Dic3⋊C4—21no—
21#4739—21no—
22#4739C412yescomplemented
23#4722—21no—
24#4722C412yescomplemented
25#4654—21no—
26#4646—21no—
27#4527—21no—
28#4527—21no—
29#4467—22no—
30#4467—21no—
31#4467—21no—
32#3189—31no—
33#2368—31no—
34#1996—31no—
35#1520—31no—
36C2×#328—31no—
37#1294—31no—
38#1282D311yescharacteristic
39C23×C3×C4C2311yescharacteristic
40C23×Dic3—21no—
41C22×Dic6C2×C411yescharacteristic
42C22×Dic6D411yescharacteristic
43C22×C3×C8—21no—
44C22×C3×C8—21no—
45C2×C3×C22⋊C4D411yescharacteristic
46C2×C3×C22⋊C4D411yescharacteristic
47C2×C3×C42—21no—
48C2×#217—21no—
49C2×#217—21no—
50C2×#217—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3C2×C411yescharacteristic
53C2×C4⋊Dic3D411yescharacteristic
54#924—21no—
55#924—21no—
56#924—21no—
57#924—21no—
58#924—21no—
59#924—21no—
60#924C2×C412yes—
61#924D412yes—
62C2×Dic3⋊C4—21no—
63C2×Dic3⋊C4—21no—
64C2×Dic3⋊C4—21no—
65C2×Dic3⋊C4—21no—
66C2×C4×Dic3—21no—
67C2×C4×Dic3—21no—
68C2×Dic12—22no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8C2×C412yes—
73C3×#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95D411yescharacteristic
81C3⋊#95D411yescharacteristic
82#818—22no—
83#816—22no—
84#816—22no—
85C2×C22⋊Q8—31no—
86C2×C22⋊Q8—31no—
87#484—32no—
88#484—61no—
89#467—62no—
90#451—32no—
91#451—61no—
92C2×C22⋊C8C2×D311yescharacteristic
93C2×C22⋊C8C2×D311yescharacteristic
94#391—61no—
95#387—61no—
96#387—61no—
97#385—61no—
98#380—31no—
99C4×C4⋊C4—61no—
100C4×C22⋊C4—61no—
101C4×C22⋊C4C2×D311yescharacteristic
102C2×#95—31no—
103#342—61no—
104#342—61no—
105#328—61no—
106#328—61no—
107#328—62no—
108C24×C3C2×D411yescharacteristic
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—81no—
112C22×C3×C4C22×C411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4#4111yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—21no—
121C22×Dic3—21no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—42no—
125C22×Dic3—41no—
126C22×Dic3—41no—
127C22×Dic3—41no—
128C22×Dic3—81no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C2×Dic6—81no—
133C2×Dic6—81no—
134C2×Dic6C22⋊C412yes—
135C2×C3×C8—22no—
136C2×C3×C8—41no—
137C2×C3×C8—81no—
138C2×C3×C8—81no—
139C3×C22⋊C4—21no—
140C3×C22⋊C4—21no—
141C3×C22⋊C4—21no—
142C3×C22⋊C4Q1612yescomplemented
143C3×C42—81no—
144#217—22no—
145#217—22no—
146#217—41no—
147#217—41no—
148#217—41no—
149#217—41no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—81no—
154C4⋊Dic3C22⋊C412yes—
155Dic3⋊C4—22no—
156Dic3⋊C4—22no—
157Dic3⋊C4—42no—
158Dic3⋊C4—41no—
159Dic3⋊C4—41no—
160Dic3⋊C4—81no—
161C4×Dic3—42no—
162C4×Dic3—81no—
163Dic12—82no—
164C22×Q8—31no—
165C22×Q8—31no—
166C23×C4—61no—
167C23×C4C22×D311yescharacteristic
168C2×Q16—62no—
169C22×C8—21no—
170C22×C8—21no—
171C22⋊Q8—32no—
172C22⋊Q8—32no—
173C22⋊Q8—61no—
174C22⋊Q8—61no—
175C22⋊Q8—61no—
176C22⋊Q8—61no—
177C22⋊Q8—61no—
178C22⋊Q8—61no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4D1211yescharacteristic
190C2×C22⋊C4D1211yescharacteristic
191C2×C42—21no—
192C2×C42—61no—
193C2×C42—61no—
194#107—62no—
195#103—62no—
196#103—62no—
197C22⋊C8—21no—
198C22⋊C8—21no—
199C22⋊C8—21no—
200C22⋊C8C4×D312yes—
201#95—21no—
202#95—31no—
203#95—31no—
204#95—61no—
205#95—61no—
206#95—61no—
207#95—61no—
208#95—61no—
209#95—61no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3C4×D411yescharacteristic
213C23×C3#12011yescharacteristic
214C23×C3#12111yescharacteristic
215C23×C3C2×Q1611yescharacteristic
216C23×C3#13511yescharacteristic
217C23×C3#13711yescharacteristic
218C23×C3#13711yescharacteristic
219C2×C3×C4—22no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4C2×C22⋊C411yescharacteristic
232C2×C3×C4C4×D411yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12311yescharacteristic
235C2×Dic3—22no—
236C2×Dic3—22no—
237C2×Dic3—22no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—82no—
250C2×Dic3—82no—
251C2×Dic3—81no—
252C2×Dic3—81no—
253C2×Dic3—81no—
254C2×Dic3—81no—
255C2×Dic3—81no—
256Dic6—42no—
257Dic6—42no—
258Dic6—81no—
259Dic6—81no—
260C3×C8—42no—
261C3×C8—81no—
262C24C2×D1211yescharacteristic
263C2×Q8—32no—
264C2×Q8—61no—
265C2×Q8—61no—
266C2×Q8—61no—
267C2×Q8—241no—
268C2×Q8—241no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4—81no—
278C22×C4—122no—
279C22×C4—121no—
280C22×C4—121no—
281C22×C4—121no—
282C22×C4—241no—
283C22×C4C2×C4×D311yescharacteristic
284C22×C4#23511yescharacteristic
285C22×C4D3×D411yescharacteristic
286C22×C4D3×D411yescharacteristic
287C22×C4D3×D411yescharacteristic
288C22×C4#23911yescharacteristic
289Q16—242no—
290C2×C8—22no—
291C2×C8—41no—
292C2×C8—81no—
293C2×C8—81no—
294C4⋊C4—32no—
295C4⋊C4—62no—
296C4⋊C4—62no—
297C4⋊C4—61no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C4⋊C4—122no—
301C4⋊C4—121no—
302C4⋊C4—121no—
303C4⋊C4—241no—
304C4⋊C4—241no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—21no—
308C22⋊C4—62no—
309C22⋊C4—62no—
310C22⋊C4—121no—
311C22⋊C4—121no—
312C22⋊C4—121no—
313C22⋊C4—121no—
314C22⋊C4Dic1212yescomplemented
315C42—81no—
316C42—122no—
317C42—241no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3#38611yescharacteristic
329C22×C3C4×Q1611yescharacteristic
330C22×C3#44111yescharacteristic
331C22×C3#45111yescharacteristic
332C22×C3#45211yescharacteristic
333C22×C3#46211yescharacteristic
334C22×C3#46411yescharacteristic
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—42no—
340C3×C4—81no—
341Dic3—42no—
342Dic3—42no—
343Dic3—42no—
344Dic3—81no—
345Dic3—81no—
346Dic3—81no—
347Dic3—161no—
348C23—41no—
349C23—41no—
350C23C4×D1211yescharacteristic
351C23#88211yescharacteristic
352C23#89511yescharacteristic
353C23#90411yescharacteristic
354C23C2×Dic1211yescharacteristic
355C23#90911yescharacteristic
356C23#90911yescharacteristic
357Q8—122no—
358Q8—122no—
359Q8—241no—
360Q8—241no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—62no—
374C2×C4—62no—
375C2×C4—62no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—242no—
388C2×C4—242no—
389C2×C4—241no—
390C2×C4—241no—
391C2×C4—241no—
392C2×C4—241no—
393C2×C4—241no—
394C2×C4D3×C22⋊C411yescharacteristic
395C2×C4#88411yescharacteristic
396C2×C4#89311yescharacteristic
397C2×C4#89411yescharacteristic
398C8—42no—
399C8—81no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3#187811yescharacteristic
407C2×C3#190611yescharacteristic
408C2×C3#191311yescharacteristic
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22#466511yescharacteristic
420C22C4×Dic1211yescharacteristic
421C22#471611yescharacteristic
422C22#473311yescharacteristic
423C22#473911yescharacteristic
424C22#484511yescharacteristic
425C22#484911yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—42no—
431C4—81no—
432C4—122no—
433C4—122no—
434C4—122no—
435C4—241no—
436C4—241no—
437C4—241no—
438C4—481no—
439C3#927611yescharacteristic, complemented, Sylow
440C2—21no—
441C2—21no—
442C2—21no—
443C2—21no—
444C2—21no—
445C2—21no—
446C2#6868511yescharacteristic
447C2#6896411yescharacteristic
448C2#6903911yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1282C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1282D3
hypocenterC3#9276
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1282

Complements

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

NH ≅ G/Ntype
22. #4739C4semidirect
24. #4722C4semidirect
142. C3×C22⋊C4Q16semidirect
314. C22⋊C4Dic12semidirect
439. C3#9276semidirect

Maximal subgroups

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

subgroupindex
#786172
#734732
#699462
C2×#44672
#694572
C3×#12822
#676752
#92763